Historic, Archive Document

Do not assume content reflects current
scientific knowledge, policies, or practices.

IK), n

I The Use of Multivariate Statistics
in Studies of Wildlife Habitat

ABSTRACT

This report contains edited and reviewed versions of
papers presented at a workshop held at the University of
Vermont in April 1980. Topics include sampling avian
habitats, multivariate methods, applications, examples, and
new approaches to analysis and interpretation.

Limited numbers of reprints are available from the authors
of each paper contained in this report.

USDA Forest Service

General Technical Report RM-87

August 1981

The Use of Multivariate Statistics
in Studies of Wildlife Habitat'

Edited by

David E. Capen
University of Vermont

Tlie worl(shop at whiich the papers inciuded in this report
were presented was jointly sponsored by

School of Natural Resources, University of Vermont

U.S. Fish and Wildlife Service
USDA Forest Service

^This report was published by the Rocky Mountain Forest and Range Experiment Station.
Supervision was provided by Thomas W. Hoel<stra, project leader for the National Resource
Analysis Techniques Project of the Resource Evaluation Techniques Program. Station head-
quarters is in Fort Collins, in cooperation with Colorado State University.

ACKNOWLEDGMENTS

The editor acknowledges the following
contributions to this report: Hugo H. John,
Director, School of Natural Resources, University
of Vermont, was an early supporter of the workshop
on which this report is based, providing financial
and staff support. Stanley Anderson of the U.S.
Fish and Wildlife Service and Tom Hoekstra and
Hans Schreuder of the USDA Forest Service served
on the planning committee.

The following session chairmen evaluated
submitted abstracts and reviewed manuscripts:
James Karr, University of Illinois; Barry Noon,
U.S. Fish and Wildlife Service; Robert Whitmore,
West Virginia University; Jake Rice, Arizona State
University and Memorial University of Newfound-
land; Mark Boyoe, University of Wyoming, and Ray
Dueser, University of Virginia.

Graduate assistants and research technicians,
Allen Boynton, Douglas Inkley, Stephen Parren,
Bill Roberts, Douglas Runde, Mark Scott, Diane
Tessaglia, and Frank Thompson contributed greatly
to the organization of the workshop.

Trish McLaren, Lisa Klein, and Richard Lent
provided editorial assistance. Richard Lindeborg
rendered valuable service as the editorial liaison
with the Forest Service.

Diane Tessaglia designed and contributed the
cover artwork.

Maureen Douglas put all the manuscripts and
revisions on the word processor and produced the
camera copy.

PREFACE

The commonly recognized multidimensional
nature of wildlife habitat has led to a rapidly
increasing use of multivariate statistical
techniques in studies of wildlife ecology.
Techniques such as discriminant function analysis,
principal component analysis, factor analysis, and
canonical correlation have been applied in studies
of habitat selection and resource partitioning,
ordination of habitats and simulation of habitat
change, and in development of habitat inventory
systems. Although mul;tivariate methods are not
understood easily, they may be employed with
little difficulty if one has access to a computer.
Such convenience tempts researchers to employ
sophisticated analytical techniques but overlook
important statistical assumptions, experimental
design, and biological interpretation. With these
temptations in mind, a meeting was organized to
bring research biologists and statisticians
together to discuss multivariate methods and their
applications to studies of wildlife habitat.

The meeting (held at the University of
Vermont, Burlington, April 23-25, 1980) was called
a workshop, although it was not unlike a research

symposium. It was a working conference and
participants took an active role in discussing and
critiquing topics of concern. Biologists learned
from statisticians and vice-versa. Interactive
sessions and productive interchanges of ideas
satisfied the workshop's purpose of encouraging
the use of improved statistical methods in studies
of wildlife habitat and fostering better
interpretation of research results. It is hoped
that these published proceedings will encourage
other investigators to improve design of their
research, analyses of their data, and
interpretation of their results.

The papers presented at the workshop were
either invited or submitted by abstract. Those
papers are the basis of this report. Following
the workshop, authors revised manuscripts and
incorporated discussion from the meeting.
Additional comments and critiques were recorded,
edited, and added to papers where appropriate.
Manuscripts were subsequently reviewed and revised
so that these proceedings could be drawn together
to form a cohesive volume, rather than a mere
collection of papers.

This report was printed from camera-ready pages supplied by
the University of Vermont, which is responsible for the
accuracy and style of the contents. Statements of con-
tributors may not necessarily reflect the policies of the
U.S. Department of Agriculture.

CONTENTS

Pref ace

Acknowledgments

Summary of the Workshop 1

Stanley H. Anderson and David E. Capen
An Overview of Multivariate Methods and Their Application to

Studies of Wildlife Habitat 4

H.H. Shugart, Jr.

The Use and Misuse of Statistics in Wildlife Habitat Studies 11

Douglas H. Johnson
Random Numbers and Principal Components:

Further Searches for the Unicorn? 20

James R. Karr and Thomas E. Martin

Special Session: Sampling Avian Habitats

Rationale and Techniques for Sampling Avian Habitats: Introduction 26

James R. Karr

Why Measure Bird Habitat? , 29

John T. Rotenberry

Theoretical Aspects of Habitat Use by Birds 33

Richard T. Holmes

Applied Aspects of Choosing Variables in Studies of Bird Habitats 38

Robert C. Whitmore

Techniques for Sampling Avian Habitats ... 42

Barry R. Noon

How to Measure Habitat — A Statistical Perspective 53

Douglas H. Johnson

Multivariate Methods

Discriminant Analysis in Wildlife Research: Theory and Applications 59

Byron Kenneth Williams

Theory and Methods of Factor Analysis and Principal Compenents 72

Helen Bhattaaharyya

Canonical Correlation Analysis and Its Use in V/ildlife Habitat Studies 80

Kimherly G. Smith

Data-Based Transformations in Multivariate Analysis 93

James E. Dunn

Applications: Ecological Theory, Habitat Management, Inventory

Multivariate Analysis of Niche, Habitat and Ecotope 104

Andrew B. Carey

FORHAB: A Forest Simulation Model to Predict Habitat Structure

for Nongame Bird Species 114

T.M. Smith, H.H. Shugart, and D.C. West
A Windshield and Multivariate Approach to the Classification,

Inventory and Evaluation of Wildlife Habitat: An Exploratory Study 124

C.E. Grue, R.R. Reid, and N.J. Silvy

Examples: Multivariate Analyses of Wildlife Habitats

Interspecific Differences in Nesting Habitat of Sympatric

Woodpeckers and Nuthatches 142

Martin G. Raphael

Robust Canonical Correlation of Sage Grouse Habitat 152

Mark S. Boyae

Ecological Relationships of Grassland Birds to Habitat and

Food Supply in East Africa 160

L. Joseph False, Jr.
Habitat Associations of Birds Breeding in Clearcut Deciduous

Forests in West Virginia 167

Brian A. Maurer, Lawrence B. MaArthur, and Robert C. Whitmore
Principal Components Analysis of Deer Harvest-Land Use

Relationships in Massachusetts 173

Phillip J. Sazerzenie

An Application of Factor Analysis in an Aquatic Habitat Study 180

T.J. Harshbarger and H. Bhattaaharyya

New Approaches to Analysis and Interpretation

Bird Community Use of Riparian Habitats: The Importance of

Temporal Scale in Interpreting Discriminant Analysis 186

Jake Riae, Robert D. Ohmart, and Bertin W. Anderson
A Synthetic Approach to Principal Component Analysis of

Bird/Habitat Relationships 197

John T. Rotenberry and John A. Wiens
Robust Principal Component and Discriminant Analysis of

Two Grassland Bird Species Habitat 209

E. James Earner and Robert C. Whitmore
Use of Discriminant Analysis and Other Statistical Methods in

Analyzing Microhabitat Utilization of Dusky-footed Woodrats 222

Janet I. Cavallaro, John W. Menke, and William A. Williams

A Descriptive Model of Snowshoe Hare Habitat 232

Kathryn A. Converse and Bernard J. Morzuah

A Discussion of Robust Procedures in Multivariate Analysis 242

Lyman L. McDonald

Workshop Participants 245

SUMMARY OF THE WORKSHOP'
Stanley H. Anderson^ and David E. Capen*

BACKGROUND

Wildlife habitat evaluation studies have
progressed from purely descriptive work involving
a discussion of community types and plant species
present through concepts of vegetation function,
physical structure, and vegetation structure.
Many forms of special habitat descriptors, such as
life forms or physiognomy, have been used.

Recently, resource agencies have been looking
at means of evaluating wildlife habitat with ideas
of describing changes that occur in wildlife
populations as a result of habitat alteration.
Five federal agencies (Forest Service, Fish and
Wildlife Service, Soil Conservation Service,
Geological Survey, and Bureau of Land Management)
are working out a joint agreement for gathering
and classifying wildlife habitat on a regional
basis. Some federal agencies, many states, and
some private organizations are developing data
systems that include large amounts of wildlife
habitat data. Objectives of these systems are to
quickly classify wildlife habitat and indicate
what changes might occur in wildlife as a result
of widespread changes in the environment.
Obviously, the utility of these systems depends on
the type of data they contain.

Most papers presented at the workshop and
included in these proceedings relate to species
and groups of species. The application of
wildlife habitat evaluation techniques has,
therefore, jumped ahead of standardization of
methods and development of analysis techniques.
Development was the primary point of discussion in
the workshop.

The computer age has revolutionized the
analysis of research data. Statistical techniques

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-35, 1980, Burlington, Vt.

^Leader, Wyoming Cooperative Fish and
Wildlife Research Unit, University of Wyoming,
Laramie, WY 82071.

'Assistant Professor, Wildlife Biology
Program, University of Vermont, Burlington, VT
05401.

are easily available to assist the biologist in
examining relationships between wildlife and
wildlife habitat. Techniques only theories two
decades ago, are now employed routinely through
the use of "canned" computer programs. Users of
multivariate statistical methods requiring matrix
algebra must rely on computers to analyze very
large sets of data. Unfortunately, computer
programs do not have control over either the
quality of data analyzed or interpretation of
results. Also, the researcher may have little
knowledge of the proper type of program or method
to be used for different forms of data. Yet
multivariate analyses are becoming standard
procedure in ecological studies.

CONTENT

Overview and Cautions

H.H. Shugart keynoted this meeting by
pointing out that multivariate statistical
techniques as methods of choice in analyzing
habitat data among animals have three distinct
advantages over alternative methodologies: 1)
multivariate procedures intrinsically fit
ecological problems (and data) of this sort; 2)
many multivariate methods seem to be robust in the
face of mild deviations from underlying
assumptions; and 3) there already exists a
hypergeometr ic interpretation of the relationship
among animals (niche theory) that is essentially
based on a multivariate sample space. He also
discussed the need for more information on
density, which has not been emphasized in most
multivariate studies. His opening remarks
initiated a recurring theme that biologists have
not always planned statistical analysis prior to
collecting data. Rather, they have sought
statistical advice after massive amounts of data
were gathered.

Douglas Johnson, an experienced biometrician ,
initially was invited to respond in a general
sense to papers presented at the meeting. His
response evolved into a comprehensive review of
essential considerations which should preceed
multivariate analyses; considerations such as non-
linear response functions of wildife species to
their habitats. This paper should be read before

1

wildlife habitat studies are designed, and before
multivariate methods are even selected to explore
relationships or confirm hypotheses.

A more dramatic cautionary presentation was
made by James Karr and Thomas Martin. These
authors describe a study of bird/habitat
relationships where habitat variables were reduced
to "meaningful" axes by principal components
analysis (PCA). Surprisingly, their habitat data
were collected from a random numbers table, but
findings compared favorably with those from
published reports of real — we assume — bird/habitat
studies. This paper emphasizes the importance of
objective interpretation of PCA.

Habitat Measurements

A satellite session presented the opportuntiy
to back away from data analysis and focus on
habitats per se . This session, although
restricted to avian habitats, provided both a
theoretical and practical framework for designing
studies of wildlife habitat relationships. The
six speakers presented first a historical review
of why we measure habitat and a theoretical
perspective of how birds use multidimensional
resources. Then there were discussions of how to
select the proper habitat variables; how to
measure them; and how to design statistical
treatment for the data. Statisticians with little
knowledge of biological processes will appreciate
this chapter of the proceedings.

Theory and Methods

Biologists must have an understanding of
multivariate statistical theory and methodology
before employing these techniques in research
endeavors. Papers given by Williams,

Bhattacharr ya , and Smith addressed the more
commonly used methods employed in ecological
studies. Ken Williams discussed problems in using
discriminant function analysis (DA) in terms of
habitat variables approching normality, problems
of covariance equality in canonical analysis, and
possible statistical violations found in the
literature that have resulted in misinterpretation
of data. His paper is of particular consequence
in satisfying objectives of the workshop and the
proceedings. Following Williams' presentation of
the structure of canonical variates, Kim Smith
gave an extensive review of the uses of canonical
correlation in ecological work and formulated some
important recommendations for improved use of this
technique in our discipline. Research biologists
are directed, in particular, to Smith's recom-
mendations on sample size. Helen Bhattacharrya ' s
presentation was an easily understood explanation
of principal component analysis and its many
variations. James Dunn prepared a particularly
comprehensive treatment of transformations for
univariate and multivariate data; this paper
emphasizes the need for a thorough understanding
of statistics before taking advantage of th$ more
advanced options available in this area of
quantitative science.

Appl ication

Fourteen speakers gave papers which
illustrated how multivariate techniques have been
applied to field studies of a variety of different
organisms. Andrew Carey uses PCA of a montane
ecosystem to illustrate a suggested terminology
for such ecological work. On a more applied
basis, Tom Smith and co-authors Shugart and West
show how important elements of a forest habitat
can be identified by DA and incorporated into
simulation models which predict habitat
availability for selected species of birds. Chris
Grue and co-investigators Reid and Silvy used
stepwise multiple regression and DA to condense a
large number of habitat variables into workable
models for a large-scale classification and
inventory system. Workshop participant Paul
Geissler critiqued their study design by
emphasizing the potential for prediction bias when
the number of variables exceeds sample size. Like
Karr and Martin, Geissler used a random number
universe to illustrate his point.

Six of these 14 examples of multivariate
applications may be studied as a collection of
contrasting approaches to a variety of ecological
problems. Martin Raphael used DA and cluster
analysis to study nesting habitat of sympatric
cavity-nesting birds. Mark Boyce and Joe Folse
used canonical correlation in their studies of
bird habitats. Boyce describes a robust analysis
of sage grouse habitat, while Folse employs
canonical correlation as an ordination technique,
an unus'ual application. Brian Maurer and
co-authors McArthur and Whitmore used PCA in what
has become a common format in bird/habitat
studies, but introduced a procedure of obtaining
weighted mean values for habitat variables. Phil
Sczerzenie applied PCA and PC-regression to deer
harvest-land use relationships and illustrated the
use of these techniques on an expanded spatial
scale. Tom Harshbarger and Helen Bhattacharrya
contributed the only paper where multivariate
techniques were used in a study of an aquatic
species, trout. Their conclusion was that
regression models based on derived factors were no
better than models composed of original variables.

New Approaches

Five research papers were particularly
relevant to the purposes of the workshop; these
presentations introduced both new statistical
applications to familiar problems and new field
approaches to familiar statistics. Jake Rice,
R.D. Ohmart and B.W. Anderson illustrated the
importance of seasonal and annual variation when
using DA to classify avian habitats. The
application of their findings cannot be over-
looked. John Rotenberry and John Wiens also used
a familiar approach, PCA, but reported an
innovative technique of combining species
abundance and habitat relationships in the same
environmental space.

The remaining papers dealt with the important

2

topic of robust procedures. Jim Harner and R.C.
Whitmore described new techniques which are robust
to outlying data; their computer programs will be
sought by many investigators. The problem of
multicollinearity is addressed by Janet Cavallaro,
J.W. Menke, and W.A. Willimas; their use of ridge
regression to deal with this problem is applaud-
able. Multicollinearity is also highlighted in
Kathryn Converse and B.J. Morzuch's paper.

Lyman McDonald, like Doug Johnson, is a
statistician who has worked extensively with
wildlife research problems. He too was asked to
respond in general to papers presented at the
workshop. Dr. McDonald reviewed papers and chose
to concentrate on robust procedures, indicating
the importance of this topic to wildlife habitat
studies. His comments are found after those
papers which address robust techniques.

CONCLUSION

Although insights often occur when techniques
do not work, we must adhere to basic scientific
premises. Multivariate analyses are useful tools
for describing wildlife habitat only when properly
applied; when misused or abused they become not
only ineffective, but also disastrously
misleading. It is important to remember that
multivariate statistics alone do not solve
problems; they only assist us in using our
knowledge to interpret large amounts of data. We
should not try to make sense out of nonsense;
however, exploratory studies are valid when they
follow the stated purpose.

To competently continue our work in examining
wildlife habitat, it is most important for us to
clearly define the problem at the outset of each
study. This means setting objectives,

establishing hypotheses that can be tested, and
determining the forms of data to be collected and
how they should be analyzed statistically. These
steps will help determine data collection
procedures. At this point it is extremely
important to know the assumptions of tests to be
used and make sure that these assumptions are met
in the data collection procedure. Finally, we
must recognize that results of the tests are not
an end in themselves, they only provide direction.
We must return to the field and validate the
results, frequently through manipulative
experiments. We as investigators, teachers, and
scientific reviewers, must encourage proper
interpretation through our selection of tests.

Site-specific studies must be used as one
example of regional and national application.
Researchers must be keenly aware of their need to
design experiments and translate results into
meaningful information for teachers, managers, and
lawmakers. A cautionary question: are we
publishing our results too soon? Maybe studies
should go beyond the typical one or two years and
proper verification should follow.

We hope this symposium of biologists and
mathematicians is of help to all involved. The
demands for the information we gather are great
but it is important that we pass on only the best
information. The type of interaction brought
about by this workshop needs to continue.

3

AN OVERVIEW OF MULTIVARIATE METHODS AND THEIR
APPLICATION TO STUDIES OF WILDLIFE HABITAT'

Abstract. — Multivariate statistical techniques as
methods of choice in analyzing habitat relations among
animals have three distinct advantages over competitive
methodologies: 1) Multivariate procedures intrinsically fit
ecological problems (and data) dealing with habitat
selection. 2) Many of the multivariate methods seem to be
robust in the face of mild deviations from the underlying
assumptions. 3) There already exists a hypergeoraetr ic
interpretation of relations among animals (niche theory) that
is essentially based on a multivariate sample space. These
considerations, joined with a reduction in the cost of
computer time, the increased availability of multivariate
statistical "packages," and an increased willingness on the
part of ecologists to use mathematics and statistics as
tools, have created an exponentially increasing interest in
multivariate statistical methods over the past decade. The
earliest multivariate statistical analyses in ecology did
more than introduce a set of appropriate and needed
methodologies to ecology. These studies emphasized different
spatial and organizational scales from those typically
emphasized in habitat studies. The traditional wildlife
habitat study was based on measuring the density of a
population in a homogeneous plant community. New studies,
using multivariate methods, emphasized individual organisms'
responses in a heterogeneous environment. This philosophical
(and to some degree, methodological) emphasis on
heterogeneity has led to a potential to predict the
consequences of disturbances and management on wildlife
habitat. One recent development in this regard has been the
coupling of forest succession simulators with multivariate
analysis of habitat to predict habitat availability under
different timber management procedures.

Key words: Habitat selection; multivariate statistics,
quantitative ecology; succession models; wildlife-vegetation
interactions .

H.H. Shugart, Jr.^^

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Senior Research Staff Member, Environmental
Sciences Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37830.

4

INTRODUCTION:

THE EVOLUTION OF MULTIVARIATE HABITAT ANALYSIS

Investigators in research centers all over
the world, and particularly in the United States,
are discovering applications of multivariate
statistical techniques in studying the habitat
relations of a diverse range of animals and
plants. Given the suitability of multivariate
statistics for habitat analysis coupled with the
need for a wider understanding of animal/
environment relations mandated by man's increasd
use of the earth's resources, this is a logical
situation. Indeed, the logic of using
multivariate analysis to manage animal habitat
would seem to make such applications inevitable.
Yet only a decade ago, no studies that are
methodologically and philosophically equivalent to
those of today were in evidence. This paper takes
as its central tenet that multivariate statistical
analysis of habitat requirements of animals is the
product of a synthesis, occurring in the early
1970's at several different research centers, that
united different lines of scientific research.

Figure 1 illustrates the main elements of
this synthesis. Two developments that were
independent of ecological studies, the increased
availability of computer time on high-speed
digital computers and the development of
multivariate statistical techniques, were combined
with three ecological developments: 1) the
hyperspace theory of the niche, 2) the realization
that small spatial-scale studies could reveal much

on animal interrelations, and 3) an emphasis on
individual organism response as being important in
determining species distributions. A number of
papers developed- various aspects of what was to
become multivariate analysis published before the
1970's, but most of these papers did not combine
all elements of the synthesis (table 1).
MacArthur's (1958) classic work emphasized a
relation between individual microhabitat
utilization and the hyperspace niche theory model
(Hutchinson 19^4, 1957, 1965). Klopfer (1965)
developed an impressive line of research in
associating individual organism responses to
microhabitat, as did Wiens (1969). Cody (1968)
and Fujii (1969) both developed the idea of using
multivariate statistics in studies of animal
niches. In 1971 and 1972 a number of papers, by
individuals working independently, appeared that
combined these elements and attempted to quantify
niches of various organisms (Green 1971,
Hespenheide 1971, James 1971, Martinka 1972,
Shugart and Patten 1972). The management
implications of these works became obvious to
various individuals and within a few years several
papers had appeared that discussed the use of
multivariate habitat analysis as a management
tool. Because of its synthetic origins,
multivariate habitat analysis can be considered in
terms of the elements that formed this synthesis
and in terms of the problems intrinsic to these
synthesis elements.

ELEMENTS OF SYNTHESIS
Quantitative Elements: Statistics and Computers

ORNL DWG 80 -7522 ESD

MULTIVARIATE
STATISTICAL
TECHNIQUES

IMPORTANCE
OF

MICROHABITAT

CONSIDERATION
OF

INDIVIDUAL
RESPONSE

STATISTICAL
NICHE
QUANTIFICATION

I

MULTIVARIATE
HABITAT
ANALYSIS

Figure 1. Schematic diagram of the scientific
research elements that combined in a synthesis
to produce multivariate habitat analysis.

Two of the most important elements that led
to the development of multivariate habitat
analysis were the state of development of
multivariate statistical procedures and the
increased availability of high-speed digital
computers. It is interesting to note that current
user-oriented statistical analysis procedures
(e.g., Cooley and Lohnes 1971, Dixon 197^*, Barr et
al . 1975) are further accelerating the
"computerization" of ecological studies in
general. Many of the statistical procedures used
in multivariate habitat selection studies require
rather involved numerical analysis programs that:
1) use computers and 2) would be difficult for
ecologists to develop de nova .

The availability of multivariate statistical
analysis programs and computer time created a
situation in which the initial work in developing
habitat analysis techniques in the 1970's was
often done in conjunction with a consulting
statistician (e.g.. Dr. J.E. Dunn, a contributor
to these proceedings). This has had a positive
effect in that the rigor in testing to meet
statistical assumptions, in using the "right"
methodologies, and in interpreting results
correctly is much higher in multivariate habitat
analysis than in botanical ordination and
classification procedures. Ecologists interested
in plant ordination developed a number of
analytical techniques with little rigor relative

5

Table 1. Papers involved with animal/habitat relationships before and during the synthesis period for
multivariate habitat analysis. A "yes" to Niche theory indictes the paper is strongly oriented toward
niche theory; "yes" to Microhabitat indicates a use of a small spatial scale sample size; "yes" to
Individuals indicates an emphasis on individual organisms; "yes" to Management indicates management
potential is discussed in paper.

Multivariate

Niche

paper

Tax a

Methods

theory

Microhabitat

Ind ividusls

MacArthur (1958)

Birds

Yes

Yes

Yes

Klopfer (1965)

Birds

_

Yes

Yes

Wiens (1969)

Birds

_

_

Yes

Yes

_

Cody (1968)

Birds

Discriminant
function (DF)

Yes

-

Yes

-

Fujii (1969)

Insects

Principal
components
analysis (PCA)

Yes

Yes

_

Green (1971)

Molluscs

DF

Yes

Yes

Hespenheide (1971)

Birds

DF

Yes

Yes

_

_

James (1971)

Birds

DF, PCA

Yes

Yes

Yes

Martinka (1972)

Birds

DF

Yes

Yes

Shugart and Patten

Birds

DF, other

Yes

Yes

Yes

(1972)

techniques

Anderson and Shugart

Birds

DF, PCA

Yes

Yes

Yes

Yes

(1974)

Green (1974)

Benthic
Animals

DF

Yes

Yes

Yes

Shugart et al .

Birds

DF, PCA

Yes

Yes

Yes

Yes

(1974)

Shugart et al .

Birds

DF

Yes

Yes

Yes

Yes

(1975)

Mammals

Conner and Adkisson

Birds

DF

Yes

Yes

Yes

to underlying statistical assumptions. The
difference in rigor between plant ordination and
animal habitat analysis is quite pronounced to
anyone who has worked in both areas. While this
would appear to be a positive attribute relative
to habitat studies, this is not necessarily the
case as I will discuss below.

the pattern of the means in a sample or
discriminant space can be interpreted relative to
theories of community structure (Shugart and
Patten 1972, Dueser and Shugart 1979).

Consideration of Individual Responses
and the Importance of Microhabitat

Niche Theory

Hutchinson (1944, 1957, 1965) formulated a
hypergeometr ic model of the niche of an organism
that captured the imagination of theoretical
ecologists for several decades (e.g., Horn 1966,
Maguire 1967, McNaughton and Wolf 1970, Colwell
and Futuyma 1971, Pielou 1972, May 1974, 1975),
although this geometric interpretation has tended
to be lost from more recent interpretations. The
hypergeometric concept of the niche and the
n-dimensional sample space are analogous in many
respects, and this analogy was noticed by several
early investigators (Table 1). The existence of a
set of theories (niche theory) has led to an
extended level of interpretation of multivariate
habitat studies. Thus, one can interpret lack of
overlap in terms of theories of "limiting
similarity" or "competitive exclusion" (Harner and
Whitmore 1977, Dueser and Shugart 1978). Further,

Population-level censuses of bird populations
have for years been coupled with plant community
surveys to obtain a measure of species response to
vegetation. While these surveys will continue to
be important in the future as they have in the
past, they do not necessarily provide much insight
into the detailed aspects of why one location
seems more suitable for a species than some other
location. Interest in this microscale problem as
well as the theoretical underpinnings of habitat
selection had been published quite early (e.g.,
von Uexkull 1909, Kohler 1947, Tinbergen 1951,
Harris 1952). In the mid-1960's, there was an
intensified interest in habitat selection as a
behavioral phenomena (e.g., Wecker 1963, Klopfer
1965, MacArthur and Pianka 1966). This emphasis
on individual organisms tended to produce
observational data sets with very high degrees of
freedom and with more than one variable recorded
for each observation. Such data created a need to

6

explore multivariate statistics as an analytical
tool .

ORNL DWG 80-7521 ESD

PROBLEMS INTRINSIC TO THE SYNTHESIS ELEMENTS

The synthesis of multivariate habitat
selection methodologies proceeded from the
combining of several elements of research from
different fields in the 1970 's. As a product of a
rapid synthesis, today's procedures suffer from
certain problems intrinsic to the component
elements.

Statistical Assumptions Versus Niche Theory

Multivariate habitat selection studies have a
strong emphasis on using the "correct" statistical
procedures. Most letters and phone calls that I
receive regarding habitat selection involve
deciding what method to use, not on how to
interpret analytical results. Further, as
contrasted to plant ordination studies in which
authors typcially reference their methods to other
ordination studies, multivariate habitat studies
refer their methods to reputable multivariate
statistical textbooks. This healthy respect for
statistics is probably quite good, but it also
makes papers in the field proceed with ponderous
inevitability. I am of the opinion that rigorous
adherence to methodological correctness probably
has prevented incorporation of the heuristically
rich geometric theories of the hyperspace niche
concept from 6eing fully developed in wildlife
habitat studies.

STATISTICAL
NICHE
QUANTIFICATION

HABITAT
SIMULATION
MODELS

PREDICTION OF
ANIMAL DISTRIBUTIONS
IN SPACE AND
TIME

MAPPING
TECHNIQUES

Figure 2. Schematic diagram of scientific
research elements that may combine to produce a
second synthesis extending from multivariate
habitat analysis.

Inclusion of Population-Level Aspects

Most multivariate habitat analyses are so
focused on the responses of individual organisms
to microhabitats that concepts of animal density
are almost lost. Some species appear to be quite
uncommon even in the face of an apparent abundance
of suitable microhabitat . The Swainson's warbler
(Limnothlypis swainsonii) is a possible example of
such a species. It is difficult to include
population dynamics in a detailed microhabitat
study, yet such dynamics are essential to
management of habitat for selected species. I
believe that this will be one of the challenges to
workers in this field over the coming decade.

Inclusion of Macrohabitat Considerations

The consideration of the pattern of micro-
habitats at larger spatial scales (macrohabitat)
is difficult to include in the present multi-
variate habitat analysis methodologies. In fact,
macrohabitat considerations are often viewed as an
unfortunate sampling situation in which the sample
space has a pattern of internal clusters. There
may be considerable importance in arrangement of
elements of the landscape that are suitable for
maintaining viable populations of a species.
Rosenzweig's (1 973, 1 97'4), Rosenzweig and
Winakur's (1969), and Schroder and Rosenzweig's

(1975) pioneering work on the theoretical and
experimental responses of populations to altered
patterns of microhabitats is an important
benchmark set of studies that needs to be repeated
in other communities.

THE ELEMENTS OF A NEW SYNTHESIS

In the previous section, I have tried to
identify briefly some areas of new fruitful
exploration for studies of habitat selection.
There is a need to use the powerful methodology of
multivariate habitat analysis to develop new
theory, as well as to contine its use in difficult
applications. In the former area, I believe that
work I have been involved with in trying to
develop a concept of niche patterns for
communities (Shugart and Patten 1972) and
particularly the development of this idea by my
colleague Dr. R. D. Dueser (Dueser and Shugart
1979) are examples of attempts to extend from
simple data analysis to theory. These proceedings
hold great promise for stimulating other
theoretical work and should definitely provide
some classic examples of applications.

There is, in my opinion, a need for a second
synthesis in the field and I would like to briefly

7

identify what may be the important elements of
this synthesis (fig. 2). First, there is a need
to meld the exciting developments made in
population biology over the past decade with the
equally exciting developments in multivariate
habitat analysis. The potential for cross-seeding
these lines of research is great, and the only
limitation in uniting the field is in the
formidable mathematical development that must be
unified between the two. Two important
quantitative developments (the ability to simulate
changes in microhabitats through time and computer
mapping of habitats) are already beginning to be
included in habitat studies, and two examples are
provided below.

Habitat Simulation Models

One interesting development in the field of
ecosystem modeling over the past few years has
been the development of forest succession
simulation models capable of providing extremely
detailed predictions on the future states of
forests (reviewed by Shugart and West 1980). The
level of detail and spatial scale of the output of
some of these models is similar to that used in
multivariate habitat selection studies. This
convenient parallel development opens the
possibility of projecting the temporal pattern of
habitat availability following either man-made or
natural disturbances. Figure 3 is an example of
such an application using the Appalachian
deciduous forest succession (FORET) model (Shugart
and West 1977) to project habitat conditions for

OVENBIRD

100

YEAR

Figure 3. Ovenbird habitat availability over 500
years on Walker Branch Watershed under two
treatments. Results are from the FORHAB model
(Smith et al . 1981 ) .

the ovenbird (Seiurus aurocapillus) on Walker
Branch watershed, a site of intensive ecological
investigation located on the Department of Energy
(DOE) reservation in Oak Ridge, Tennessee. The
FORET model simulates the birth, death, and growth
of each tree on a 0.085-ha circular plot. By
collecting habitat data on the presence of the
ovenbird that has corresponding habitat
information to that predicted by the model, one
can use the model as a habitat simulator. In this
particular example, we used discriminant function
analysis to classify habitat versus non-habitat.
As can be seen from an example, one can also use
the detailed succession simulator to perform model
experiments such as projecting the habitat
dynamics of a 22.9 cm (9-in) diameter-limit cut of
commercially valuable species (fig. 3). Details
of this particular methodology will be treated
later in this volume (Smith et al . 1981). I
mention this example here to identify a need for
adding dynamics to our currently largely static
methodologies .

Mapping Techniques and Data Sets

There are presently a large number of data
sets (e.g., the USDA Forest Service (ksntinuou?
Forest Inventory [C.F.I.]) that could be used in
conjunction with multivariate habitat data to make
state- or continental-scale maps of distributions
of wildlife habitat. The problems here involve
keying habitat variables associated with a given
species in a multivariate habitat study to the
variables that can be obtained from inventory data
sets. A fine example of this appraoch is Lennartz
and McClure's (1979) application of C.F.I, data to
map the potential extent of the red-cockaded
woodpecker (Dendrocopus borealis) in the
southeastern U.S. The determination of the level
of variation over the continent of various
species' habitat selection would be a valuable set
of information for developing these maps.

THE SECOND SYNTHESIS

In this paper I have taken a broad view of
the ontogeny of a still-developing understanding
of habitat selection of several animals. The
excitement of being involved in a young field is
contagious, and I hope that this meeting will, by
virtue of the increased interaction of scientists
involved in multivariate habitat studies, help
create a period of reviewed synthesis. The
primary elements of such a synthesis might include
the elements that I have mentioned (fig. 2), or
they may take some entirely different direction —
only time will tell. Whatever the case, the
present symposium should provide much fuel for the
fire.

The healthiest aspects of multivariate
habitat analysis have been of a philosophical
rather than of a methodological nature. Studies
have attempted to be rigorous in the sense of
statistics while, at the same time, they have
attempted to be both theoretical and explanatory.

8

This balance should not be lost. The studies have
traditionally emphasized an understanding of
ecological mechanisms at a fine scale of
resolution. If such detail can be related to
regional maps, the contribution to biogeography
could be considerable.

In my opinion, the current research direction
and velocity could, within the decade, provide
such research products as dynamic maps of regional
habitat availability for a great number of
species, with the potential to determine the
changes in these maps due to altered land-use
policies .

ACKNOWLEDGMENTS

Research supported by the National Science
Foundation under Interagency Agreement 40-700-78
with the U.S. Department of Energy under contract
W-7405-eng-26 with Union Carbide Corporation.
Publication No. 1615. Environmental Sciences
Division, Oak Ridge National Laboratory.

LITERATURE CITED

Anderson, S.H., and H.H. Shugart. 1974 Habitat
selection of breeding birds in an East
Tennessee deciduous forest. Ecology
55:828-837.

Barr, A.J., J.H. Goodnight, J. P. Sail, and J.T.

Helwig. 1976. A User's Guide to SAS-76 .

329 p. SAS Institute, Inc., Raleigh, N.C.
Conner, R.N., and C.S. Adkisson. 1976.

Discriminant function analysis: a possible

aid in determining the impact of forest

management on woodpecker nesting habitat.

Forest Science 22:122-127.
Cody, M.J. 1968. On methods of resource division

in grassland bird communities. American

Naturalist 102:107-147.
Colwell, R.K., and D.J. Futuyma. 1971. On the

measurement of niche breadth and overlap.

Ecology 52:567-576.
Cooley, W.W., and P.R. Lohnes. 1971. 364 p.

Multivariate Data Analysis. John Wiley and

Sons, New York, N.Y.
Dixon, W.J., editor. 1974. BMD Biomedical

computer programs. 773 p. University of

California Press, Berkeley, Calif.
Dueser, R.D., and H.H. Shugart. 1 978.

Microhabitats in forest floor small mammal

fauna. Ecology 59:89-98.
Dueser, R.D., and H.H. Shugart. 1979. Niche

pattern in a forest floor small mammal fauna.

Ecology 60: 108-118.
Fujii, K. 1969. Numerical taxonomy of ecological
characteristics and the niche concept.

Systematic Zoology 18:151-153.
Green, R.H. 1971. A multivariate statistical

approach to the Hutchinsonian niche: bi-
valve mollusks of central Canada, Ecology

52:543-556.

Green, R.H. 1974. Multivariate niche analysis
with temporally varying environmental
factors. Ecology 55:73-83.

Harner, E.J., and R.C, Whitmore. 1977.
Multivariate measures of niche overlap using
discriminant analysis. Theoretical
Population Biology 12:21-36.

Harris, V.T. 1952. An experimental study of
habitat selection by prairie and forest
races of the deermouse, Peromyscos
mani culatus . 103 p. Contribution of
Laboratory of Vertebrate Biology 56.
Univiversity of Michigan Press, Ann Arbor,
Mich.

Hespenheide, H.A. 1971. Flycatcher habitat
selection in the eastern deciduous forest.
Auk 88:61-74.

Horn, H.S. 1966. Measurement of overlap in
comparative ecological studies. American
Naturalist 100:419-424.

Hutchinson, G.E. 1944, Limnological studies in
Connecticut. VII, A critical examination of
the supposed relationship between phyto-
plankton periodicity and chemical changes in
lake waters. Ecology 25:3-26. (see footnote
5, p. 20),

Hutchinson, G.E. 1957. Concluding remarks. Cold
Spring Harbor Symposium of Quantitative
Biology 22:415-427.

Hutchinson, G.E. 1965. The ecological theater
and the evolutionary play. 134 p. Yale
University Press, New Haven, Conn.

James, F.C. 1971. Ordinations of habitat
relationships among breeding birds. Wilson
Bulletin 83:215-236.

Klopfer, P. 1965. Behavioral aspects of habitat
selection. A preliminary report on stereo-
typy in foliage preferences in birds. Wilson
Bulletin 75:15-22.

Kohler, W. 1947. Gestalt psychology. 482 p.
Liverright Publishing Corp., New York, N.Y.

Lennartz, M.R., and J. P. McClure. 1979.
Estimating the extent of Red-cockaded
woodpecker habitat in the Southeast,
p. 48-62. In Frayer, W.E., editor. Forest
resource inventories: Proceedings of a
workshop. [Fort Collins, Co., July 22-27,
1979] Department of Forest and Wood Science,
Colorado State University. Unnumbered
publication.

MacArthur, R.M, 1958. Population ecology of some

warblers of northeastern coniferous forests.

Ecology 39:599-619.
MacArthur, R.M., and E. Piauka. 1966. On optimal

use of a patch environment. American

Naturalist 100:603-609.
Maguire, B., Jr, 1972, A partial analysis of the

niche. American Naturalist 101:515-523.
Martinka, R.R. 1972. Structural characteristics

of Blue Grouse territories in southwestern

Montana. Journal of Wildlife Management

36:498-510.

May, R.M. 1975. Some notes on estimating the
competition matrix. Ecology 56:737-741.

McNaughton, S.J., and L.L. Wolf. 1972. Dominance
and the niche in ecological systems. Science
167: 131-139,

Pielou, E.L. 1972. Niche width and overlap: a
method for measuring them. Ecology
53:687-692.

9

Rosenzweig, M.L. 1973. Habitat selection
experiments with a pair of coexisting
heteromyid rodent species. Ecology
54: 1 1 1-1 17.

Rosenzweig, M.L. 1974. On the evolution of
habitat selection. p. 401-404. Iji
Proceedings of First International Congress
on ecological structure, function and
management of ecosystems. Centre Agriculture
Publication and Documents, Waginingen, The
Netherlands.

Rosenzweig. M.L,, and J. Winakur. 1969.
Population ecology of desert rodent
communities: habitats and environmental
complexity. Ecology 50:558-572.

Schroder, G.D., and M.L. Rosenzweig. 1975.
Perturbation analysis of competition and
overlap in habitat utilization between
Dip'odomys or d i i and Dipodomys merriami .
Oecologia 19:9-28.

Shugart, H.H., S.H. Anderson, and R.H. Strand.
1975. Dominant patterns of bird populations
of the eastern deciduous forest biome. p.
90-95. In Smith, D.R., technical
coordinator. Management of forest and range
habitats for nongame birds: Proceedings of a
symposium [Tucson, Ariz., May 6-9, 1975]
USDA Forest Service General Technical Report
WO-1 , 343 p. Washington, D.C.

Shugart, H.H., R.D. Dueser, S.H. Anderson. 1974.
Influence of habitat alterations on bird and
small mammal populations. p. 92-96. I_n
Slusher, J. P., and T.M. Hinckley, editors.
Timber-wildlife management: Proceedings of a
symposium [Columbia, Mo., Jan. 22-24, 1974]
Missouri Academy of Science Occasional Paper
3, 131 p. Columbia, Mo.

Shugart, H.H., and B.C. Patten. 1972. Niche
quantification and the concept of niche
pattern. p. 283-327. In Patten, B.C.,
editor. Systems analysis and simulation in
ecology. Volume 2. Academic Press, New
York, N.Y.

Shugart, H.H., and D.C. West. 1977. Development
of an Appalachian deciduous forest succession
model and its application to assessment of
the impact of the chestnut blight. Journal
of Environmental Management 5:161-179.

Shugart, H.H., and D.C. West. 1980. Forest
succession models. Bioscience 30:308-313.

Smith, T.M., H.H. Shugart, and D.C. West. 1981.
FORHAB: A forest simulation model to predict
habitat structure for non-game bird studies.
In Capen , D.E., editor. The use of
multivariate statistics in studies of
wildlife habitat: Proceedings of a workshop
[Burlington, Vt., April 23-25, 1980]. USDA
Forest Service General Technical Report.
Rocky Mountain Forest and Range Experiment
Station, Fort Collins, Colo. (In press).

Tinbergen, N. 1951. The study of instinct. 228 p.

Oxford University Press, London.
Von Uexkull, J. 1909. Umwelt und Innenwelt der

Tiere. Springer-Verlag, Berlin.
Wecker, S.C. 1963. The role of early experience

in habitat selection by the prairie

deermouse, Peromyscus maniculatus bai rdi .

Ecological Monographs 33:307-325.
Wiens, J. A. 1969. An approach to the study of

ecological relationships among grassland

birds. 93 p. Ornithological Monographs 8.

DISCUSSION

MICHAEL C.S. KINGSLEY: Multivariate analysis
tends to assume multivariate normality. Are there
considerations in niche theory, particularly with
respect to niche packing, which imply that niches
should be a) similar in shape and orientation, b)
similarly oriented multivariate normal density
spaces?

H.H. SHUGART: Yes. R.M. May (1973. Stability
and complexity in model ecosystems. 265 p.
Princeton University Press, Princeton, N.J.) used
the standard deviation and separation of means of
species distributions along a continuum as an
index of packing and overlap. He then determined
ratios of these two statistics from derivations
based on a general n-species non-linear
competition model. He (Chapter 6) also provided a
fair number of citations on niche shapes. In a
somewhat less abstract vein, the late R.H.
Whittaker generally pictured the distributions of
species in response to gradients as unimodel and
of shapes that are easily approximated by normal
distributions. This is true both in his data
papers, as well as in his more theoretical works.
Whittaker used fairly abundant literature
citations and his work serves as a useful point of
reference. Maguire (1972) studied niche shapes in
protozoa using non-parametric methods and found
niches to have tendencies to vary along similar
environmental axes and to have similar shapes.
R.H. MacArthur and E.P. Wilson (1967. The theory
of island biogeography . 203 P. Princeton
University Press, Princeton, N.J.) produced a
"compression hypothesis" regarding the expected
similarity in shape and/or orientation of niches
in different communities. W.E. Westman (1980.
Gaussian analysis: identifying environmental
factors influencing bell-shaped species
distributions. Ecology 61:733-739.) provides a
discussion on shapes of species' responses to
gradients and also provides several references to
individuals who have noted normal distributions in
nature. A discussion with several citations of
theories that would lead to such shapes is found
in Westman' s introduction.

10

THE USE AND MISUSE OF STATISTICS IN WILDLIFE
HABITAT STUDIES'
Douglas H. Johnson^

Abstract. — This paper briefly surveys the application of
various multivariate statistical techniques in studies of
wildlife and their habitats. Several methods are widely
employed, but with little regard for the requisite
assumptions and often without full appreciation of what the
methods do and whether they are appropriate for the problem
or not .

The well known fact that species typically respond to an
environmental gradient in a nonlinear fashion is poorly
accounted for in many statistical treatments. Moreover, most
analyses are not well validated. The few valiant attempts at
validation have suggested that the models produced, often for
predictive or management purposes, are less successful than
might have been anticipated.

The use of multivariate methods in developing
recommendations for wildlife management calls for special
caution, for it is a major step from describing the
relationships observed between a species and some habitat
features to predicting the response of that species as the
habitat changes.

Key words: Canonical correlation analysis; discriminant
function analysis; multiple regression; nonlinear response
function; principal components analysis; transformations;
val idation .

INTRODUCTION

The application of multivariate analysis to
studies of wildlife habitat offers an exciting
opportunity to statisticians. They have a major
role to play as wildlife studies become
increasingly complex and as greater numbers of
environmental variables are investigated. A basic
tenet of ecology is that "everything is connected
to everything else." Although that generalization
is a bit extreme, it does express the ecologist's

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop. April 23-25, 1980, Burlington, Vt .

^Statistician, U.S. Fish and Wildlife
Service, Jamestown, ND 58^*01.

conviction that the variables affecting a species
or a system are numerous.

Faced with this acknowledged need for
statistical service and counsel, statisticians
must respond appropriately, which means carefully
and thoughtfully, with a full understanding of the
biological problem, and with an appreciation of
the consequences of the statistical analysis and
interpretation. Merely adding variables to an
analysis will not do. Multivariate analysis must
not be a masquerade for ignorance.

In this paper I survey some of the potential
applications of multivariate analysis to
wildlife-habitat studies. I try to offer
biologists some guidance through the maze of
multivariate statistics. My viewpoint is that of

11

1

a statistician, with one eye on the mathematical
requirements of the methodology, and one eye on
the needs of the ultimate user — the wildlife
manager. While looking in these two directions,
we also must not forget the animal; its biology is
of fundamental importance.

IS MULTIVARIATE ANALYSIS APPROPRIATE?

Green (1971) identified the need for a
multivariate approach when he mentioned three
operational problems in defining the niche of a
species: 1) Not all potentially important
environmental parameters can be measured. 2) Many
of the parameters measured are likely to be
correlated, relatively invariant, or irrelevant to
the problem. 3) The many potentially relevant
variables result in a mass of multidimensional
data that is difficult to interpret.

These considerations have led ecologists to
the troughs of principal component analysis and
discriminant function analysis, where they have
drunk freely. These analyses have produced "new"
variables that are linear combinations of the old
ones, and which are fewer in number, to eliminate
the third problem, a large mass of data. Further,
these new variables are uncorrelated , which
overcomes the second problem. It is left to the
ecologist to handle the first problem by carefully
including variables that are potentially important
to the species; statistics will be of minimal help
here.

These multivariate analysis overcome two
obstacles, but not without their own disadvant-
ages. One drawback is that, by involving linear
combinations of all the measured variables, they
do not really reduce the parameter space; all the
original variables must be measured in order to
calculate the new ones. In that sense, the
analyses only provide guidance for future
research. Another problem is that the linear
combinations themselves may be extremely difficult
to interpret. Many of the published ones lend
themselves to meaningful interpretation, although
some do not ; and I suspect that a lot of
unintelligible linear combinations have been lost
somewhere between analysis and publication.

I wonder if we can take a slightly different
approach. Can we objectively define meaningful
variables a priori , instead of doing so
statistically, a posteriori? These new variables
should be few in number, more or less uncorrelated
with one another, and of potential importance to
the animal. These variables would account for
what is known (or believed) about the animal, and
any additional variables that the analysis
identified as significant would represent new
findings worthy of additional investigation. For
example, a bird might find James' (1971) outline
drawings of niche-gestalts to be meaningful, and
would be willing to select its habitat based on
those drawings. But the bird would be
hard-pressed to plug the values of 15 or 20
variables into a number of linear combinations.

compare the calculated values to one another, and
select a habitat with value closest to its liking.

Multivariate analysis, while evidently
appropriate for wildlife-habitat studies, is
difficult to apply and understand; Gnanadesikan
(1977:2) suggested that univariate difficulties
are raised to the pth power. He also identified
some of the added problems and noted that much of
the theoretical work in multivariate analysis,
oriented toward formal procedures such as
hypothesis tests on means, is of limited value for
actual data analysis in the multivariate case. I
next discuss some of the usual multivariate
methods and their role in wildlife-habitat work,
as identified by speakers in this workshop and
earlier publications.

Discriminant Function Analysis

One of the most widely used multivariate
techniques is discriminant function analysis
(DFA), which is used to separate observations into
groups on the basis of a set of measurements. In
wildlife-habitat studies the observations are
usually sites, and the groups denote whether a
species was present or absent from the site, or
whether species A was present as opposed to
species B present. The variables are habitat
measurements at the site. Lachenbruch (1975) is a
valuable general reference and Tatsuoka (1970)
presented a nontechnical exposition on DFA.
Williams (1981) noted that DFA is applicable when
the groups are well-defined and the set of
measurements is ecologically meaningful. Groups
of habitat sites defined by the presence or
absence of a species are not always well-defined.
At least three reasons for a species' absence can
be identified: 1) the habitat at the site is
unsuitable; 2) the habitat is suitable but the
species is absent for other reasons, such as
numbers in the population inadequate to occupy all
suitable habitat or interspecific competition; or
3) the habitat is suitable and the site occupied,
but the sampling procedure failed to detect it.
DFA based on presence-absence data involves the
implicit assumption that absence is due to the
first reason above, but even in that situation the
group corresponding to "species absent" may
include sites where the habitat is unsuitable for
different reasons (see subsequent section on
Nonlinear Response Functions).

The requirement of DFA for ecologically
meaningful measurements has already been
addressed. My experience with biologists suggests
that on the basis of their prior knowledge of a
species and its habitats, they often can develop a
very limited number of ecological variables that
characterize occupied habitat. These variables
tend to be combinations of two or more habitat
measurements; for example, the breeding habitat of
American woodcock ( Phi lohela minor ) might be
characterized by a single variable expressing the
presence of fertile soil supporting earthworms,
vegetation dominated by shrubs or young trees, and
the nearby presence of forest openings.

12

E6ologists should use their best information when
defining habitat variables.

Williams (1981) emphasized the importance of
equality of covariance matrices in DFA. More
widespread is the notion, expressed by Klecka
(1975:435) that DFA "is very robust and these
assumptions (multivariate normality and equal
covariance matrices) need not be strongly adhered
to." Lachenbruch (1975) reviewed several studies
and concluded that linear discriminant functions
are satisfactory if the covariance matrices are
not too different. If they differ considerably,
then the appropriate technique is quadratic
discrimination, which employs the unequal
covariance matrices in the discriminant functions.
Unfortunately, quadratic DFA requires large
samples and is itself not robust to nonnormality .
Most of the work involving DFA and its sensitivity
to underlying assumptions has focused on the
misclassif ication rates of the discriminant
functions. In ecological studies the

interpretation of coefficients is also important,
but little is known of their robustness.

Principal Components Analysis

Principal components analysis (PCA) and
closely related factor analysis are multivariate
procedures designed to reduce the dimensionality
of a data set. The purpose of the reduction might
be to "stabilize scales of measurements"
(Gnanadesikan 1977:6) by compounding several
measurements of a similar nature into a fewer
number that may be more stable, but I am aware of
no ecologist who stated this as an intention.
More often the purpose is for exploratory
analysis, to screen out of many variables a few
that are important. Bhattacharrya (1981) provided
a general introduction to these methods;
Gnanadesikan (1977) compared several methods
including nonmetric and nonlinear ones.

Three problems specific to the application of
principal components analysis in ecological work
can be identified. First, I see little
justification for selecting a linear combination
of variables simply because it maximizes the
variance within the total set, that is, it is a
"best" summary. As Holmes et al. (1979) noted,
one can increase the percentage of variance
explained by the first principal component merely
by adding redundant variables to the data set. As
more and more of these are included, the principal
components analysis appears better and better, but
in fact only noise is being added to the system,
and interpretation becomes increasingly awkward.
A large percent of variance explained may reflect
only ignorance in the selection of variables.

Second, PCA is not always useful in
discovering the underlying structure of the
variables. Armstrong (1967) presented an example
involving 1 1 measures on rectangular boxes of
various metals. All variables were functions of
five underlying (and independent) factors:
length, width, thickness, density and cost per

pound. The PCA identified three factors that
summarized 90.7 percent of the information in the
original 11 variables. Despite an orthogonal
rotation, the factors were not easy to interpret
and the PCA was unable to reveal the rather simple
basic structure of the data. It would seem
unreasonably sanguine to expect PCA to be more
enlightening in a complex ecological system. Karr
and Martin (1981) presented another example, this
one pertinent to biological data, that illustrated
some hazards associated with the use and
interpretation of PCA.

Third, there is no reason to assume that a
principal component necessarily relates to the
animal or its needs. The animal could be
responding to one of the variables that is a minor
component of all those that were measured. It
seems more appropriate in most circumstances to
use regression analysis with some measure of
population density or "fitness" as the dependent
variable. Smith (1981) suggested that canonical
correlation analysis may also be useful in this
regard .

In general, PCA offers a convenient way of
summarizing a data set containing many variables.
If the ecologist 's purpose in measuring those many
variables is to relate them to the presence/
absence or density of a wildlife species, it is
not obvious that the major principal components
are the important ones. If most variables were
selected because of prior knowledge about their
relationship to the species, then these variables
are likely to appear in the major components. It
is conceivable, however, that a minor component
may be important to the animal, and for this
reason it is suggested that the species' response
be examined in relation to each of the components.
Factor analysis, which essentially deletes minor
components, would not permit this examination and
should thus be avoided in initial analyses.

Canonical Correlation Analysis

Canonical correlation analysis involves the
linear relationship of one set of variables to
another; it has had limited application to
ecological problems. Smith (1981) reviewed these
applications and discussed the technique and its
shortcomings. He also provided a useful
introduction to the literature. My own belief is
that canonical correlation will continue to play
only a small role in habitat studies. In addition
to the reasons given by Smith, viz., difficulties
in interpretation, large sample size requirements,
lack of robustness to nonnormality, and assumption
of linearity, the rationale for the technique
rests upon the ecologist showing concern for a
linear combination of "dependent" variables.
Moreover, the coefficients in that combination are
not based upon the biologist's intuition or
interests, but are generated by the technique.
How does one justify an interest in 2.07 robins +
0.16 brown thrashers - 1.92 chickadees? More
likely the ecologist has a single dependent
variable, or set of them, in mind, and univariate

13

or multivariate regression is the more appropriate
method .

Other Techniques

Regression has not been specifically
addressed in this workshop, although it remains
one of the most popular and powerful statistical
tools for examining relationships among variables.
The usual formulation of the regression model,

Y = Bq + B ^ + ^2 ••• * error,

where the X's are fixed input (or "independent")
explanatory variables, is not strictly a
multivariate technique; there is only one random
variable, the error term. In actual practice,
however, the X's are not fixed quantities (Johnson
1981) and the model can be viewed profitably in a
multivariate light. If several species are to be
examined in relation to a set of habitat
variables, multivariate regression appears to be
the appropriate method. Gnanadesikan (1977)
discussed the method and noted (p. 81) that a
multivariate viewpoint may be preferable to
considering each dependent variable separately,
because of intercorrelations among the dependent
variables. Unfortunately, most multivariate
regression work has focused on hypothesis testing
(the general linear model) rather than model
building .

In contrast to their botanical counterparts,
wildlife ecologists have made little use of
clustering techniques. Cluster analysis differs
from DFA (a classification technique) by the lack
of groups defined a priori ; the groups are
essentially defined by the clustering algorithm
and the user is responsible for providing a
reasonable interpretation of the resulting groups.
Cluster analysis differs from ordination in the
implicit assumption that units fit into neat and
discrete groups, rather than being arranged in a
continuum along one or more principal component or
discriminant function axes.

Although multivariate techniques can be and
have been rewarding in habitat studies, ecologists
should not ignore the simple and powerful methods
of univariate statistics. Careful analyses begin
with graphical displays of the data for several
purposes; detecting outlying observations that
may be erroneous or would be inordinately
influential on the analysis; selecting variables
that appear to be important; checking the
assumptions underlying particular analyses, such
as normality; and suggesting appropriate
transformations of the variables. Valuable
references on graphical techniques include Daniel
and Wood (1971), who emphasized the use of plots
for detecting relationships between variables and
evaluating fitted models; Mosteller and Tukey
(1977), who employed graphs both for general
display and for fitting of models; and Green
(1979), who discussed and illustrated many graphs
useful in ecological studies. Plotting is more
awkward in the multivariate situation than in the

univariate case, but methods are available (e.g.,
Gnanadesikan 1977). Among these are 1) two- and
three-dimensional scatterplots of subsets of the
data for studying separation within the sample,
outliers, general shape and interrelationships; 2)
probability plots of the observations on each
response, useful for suggesting transformations;
3) scatterplots or probability plots onto
principal components, and others. The longest
journey begins with a single step. To insure that
the direction of the trip is appropriate, that
first step should be graphing the data.

Transformations

It is often necessary to transform variables
in order to meet more closely the assumptions of
various statistical methods, specifically those of
normality, constant error variance, and uncor-
related errors. Other purposes are independent of
the statistical properties, such as simplifying
the relationship between variables or quantifying
qualitative, count, or percentage data. Trans-
formations may also maximize the separation
between groups of observations.

Univariate transformations have received much
more attention than multivariate ones; good
expositions are given by Green ( 1979:13-5')),
Kruskal (1978:1044-1055). and references they
cited. Kruskal (1978) offered some "clues" that
suggest a transformation might be in order.
Variables that approach an intrinsic boundary may
be candidates for transformation; examples are
percentages that closely approach 0 or 100 percent
or correlation coefficients that come close to
A nonnormal distribution can be detected by
plotting observations on normal probability paper
or by graphical procedures available in SAS (SAS
Institute Inc. 1979) and other statistical
packages. Nonconstancy of variance can be
statistically tested for, but graphical procedures
are more informative. Plots of residuals from
fitted models may be made to suggest the presence
of correlated error terms. Plots not only
indicate the need for transformations, they also
point out possible outlying observations. These
may be data errors or at least data points more
influential than the others.

The choice of an appropriate transformation,
once one is recognized as necessary, is not quite
as confusing as it might seem from the variety of
those available. In many situations the choice
can be made from theoretical considerations, such
as the arcsine transformation for binomial data.
In other situations a suitable selection can be
made by examining the data; for example, the well
known Box and Cox (1964) power transformation
is a family of transformations indexed by a
parameter that is itself estimated from the data.

Less guidance is available for
transformations of multivariate data sets. This
is unfortunate, because multivariate situations
may require transformations more often in order to
use familiar and simple analytic methods (Machado

14

1976). The choice of models in multivariate
situations is more limited due to the general
dependence on strict normality. The simplest
approach to multivariate transformation is to
treat each variable separately; this is likely to
be a good first step at least. As Dunn (1981)
noted for normalizing transformations, marginal
normality does not insure multivariate normality,
but the exercise is harmless at worst and is often
quite adequate. Andrews et al . (1971) and Machado
(1976) generalized the Box and Cox transformation
to the multivariate case, and Dunn (1981) extended
it to cover situations with more than one sample.

Transformations are often useful for
obtaining stricter compliance with certain
assumptions of statistical analysis. My own
experience has demonstrated little advantage from
analysis of transformed, as opposed to
untransformed, data. Careful design to insure
randomness overcomes many problems with correlated
errors, and reasonably large and well-balanced
samples mitigate the effects of nonnormality and
nonhomogeneous variance (Green 1979:165).
Transformations are advised, however, if they will
clarify relationships or if the statistical
assumptions are obviously violated. It certainly
will not hurt to analyze the data both
untransformed and transformed, and judge which
analysis is superior.

NONLINEAR RESPONSE FUNCTIONS

It seems generally agreed that a species
responds to an environmental variable, or to a
gradient, in a nonlinear fashion. The form of the
function could be normal, or of another symmetric
shape, or it could be asymmetrical but unimodal,
or it could even be bimodal. But it is nonlinear.
Despite this wide acknowledgment, relatively
little has been said about the effect of
nonlinearity on the results of multivariate
analysis (Westman 1980).

I wonder if we are like the several blind
men, each touching a different part of an
elephant, and reaching widely discrepant
conclusions about the total shape of the animal.
Suppose, for example, that a species responds to
an environmental variable X as shown by the
symmetric curve in figure 1. An investigator
studying values of the variable only in region A
would conclude that the species responds favorably
to the variable; it is an "increaser." A study in
region C would lead to just the opposite
conclusion, while a study in region B would
probably find that the species did not correlate
at all with the variable. All conclusions are
correct, and yet all are wrong. Even a study
involving the full range of the variable, regions
A plus B plus C, would detect no linear
association .

A possible example is that of Converse and
Morzuch (1981), who found that hare activity was
correlated positively with the number of hardwood
trees on one of their study areas, while the

111
CO

z
o

CO
LU

X

Figure 1. The nonlinear response of a species to
an environmental gradient X.

correlation with hardwood trees was negative in
the other area.

In regard to discriminant analysis, Green
(1971:514) suggested that a nonlinear response
function might not be troublesome, because
"species are groups separated in ecological space
by linear additive functions of ecological
parameters, rather than dependent variables
supposedly related to the ecological parameters in
a linear additive manner." I am not convinced
that there is no difficulty, at least in some
applications. Suppose, referring back to the
response curve (fig. 1), that we form two groups:
species Y present, and species Y absent. We want
to distinguish these groups on the basis of their
value of X. Most X values in regions A and C
would fall into the species absent group, vjhile
region B would donate mostly species present. But
it is evident that average values of X in the two
groups might be very similar, even identical. For
example, if the species is a mesic one and X
represents a moisture gradient, then some sites
might be too dry and others too wet. There are
thus two groups in which the species is absent,
and they differ more from each other than they do
from the species present group. In higher-
dimension space the problem only gets worse.
Quadratic discriminant functions are theoretically
applicable to such situations, but separating
groups on the basis of variance when means are
similar is tenuous at best.

In principal components analysis, or factor
analysis, two problems related to nonlinearity
occur. First, the methods produce linear
combinations of the variables; if a nonlinear
function is a better summary of the data, we will
not detect it by the usual methods, although some
nonlinear techniques have been developed
(Gnanadesikan 1977). Second, if a species
responds nonlinearly to a principal component, we
had better be alert if we want to detect it, e.g.,
by plotting species responses against each
component .

Nonlinear response functions can be detected
by graphical tools and can be treated
appropriately by nonlinear regression models (cf.

15

Boyce [1981], who eliminated from analysis a
variable with a recognized nonlinear effect).

We must be particularly careful about
nonlinearity when presenting our results to
resource managers, who look assiduously for the
"bottom line": What can a manager do? He might
be told that opening up a closed forest yields
more woodpeckers. He also needs to be told that
this practice works only up to a point: complete
elimination of trees will certainly have the
opposite effect. We must remind him, and
ourselves, that a management activity is like
aspirin: just because two pills are good for us,
it does not follow that four are twice as good,
and a whole bottle is ideal.

STATISTICAL ASSESSMENT

I now turn to the question of how we might
assess the statistical properties of habitat
analyses. As a statistician, I am frequently
appalled by the small sample sizes reported in
many studies, particularly those involving large
numbers of variables. Small samples of a complex
system cannot support sound conclusions.

We are taught early in our statistical
training to look, not only at the mean of a
distribution, but also at its variability. In
many published studies, however, we see species
means plotted on axes of principal components or
discriminant functions. Relatively few

investigators mention the variation about those
means; the few who have done so were very
informative.

Conner and Adkisson (1977) plotted means of
principal components for five species of
woodpeckers; niche separation was shown among the
species. When values for individual birds were
plotted, however, they exhibited more overlap than
might have been expected. Raphael (1981) also
found considerable overlap among species on
discriminant axes. Similarly, Smith's (1977)
analysis of summer birds along a moisture gradient
showed clear separation of ^% confidence regions
for the means of eight species on two principal
component axes. When plotted on the discriminant
function representing a moisture gradient,
however, values for individual birds of five
species were scattered throughout, and relatively
minor separation was evident. Researchers should
meticulously examine the variability among
animals .

Too little attention is given to annual
variation in bird populations, fluctuations that
take place seemingly without regard to habitat
conditions, and certainly without regard to the
unfortunate ecologist who is trying to determine
relationships between habitats and populations.
Ornithologists familiar with long-term studies can
document the magnitude of this variation (e.g..
Lack 1966, Wiens 1975), and I strongly suspect
that it impacts habitat analysis. Rice et al.

(1981) illustrated this problem nicely;
discriminant functions developed from one year's
data had few successes at predictions during the
next year.

The use of stepwise procedures is viewed by
many statisticians as a "fishing expedition." The
standard significance tests are invalid (Draper et
al. 1971, Pope and Webster 1972, Rencher and
Larson 1980) and results are questionable,
particularly in the usual situation of numerous
variables. Automated computer procedures are no
substitute for careful biological reasoning.

I believe that the application of robust
statistical methods will offer considerable help
in habitat analysis. McDonald (1981) gave details
and references to the literature. Harner and
Whitmore (1981) exemplified these methods and some
internal validation tools, such as the jackknife
and leaving-one-out procedures. In multivariate
situations, applications of robust methods thus
far have been primarily the treatment of one
variable at a time (Gnanadesikan 1977:136), but
this is likely to change in the near future.
Particularly appealing is robustness of result
with respect to method. If the same general
conclusions are reached through the use of several
different methods, each resting upon a somewhat
different set of assumptions, we should feel more
comfortable about the validity of those
conclusions (Green 1977: I'J).

Validation is an important final product of
an analysis. Whitmore (1977:263) stated that "the
validity of ordination work can be tested by
subsequent field observation." I agree, but such
testing is rarely done. Noon and Able (1978)
applied their discriminant functions for thrushes,
developed for five species on Mount Mansfield in
Vermont, to the two species occurring in the Great
Smoky Mountains in Tennessee and North Carolina.
They found little predictive ability.

The validation of models is particularly
important if we are to present them to resource
managers for their use. The need for caution on
our part is obvious. In statistical parlance,
management is really control, which is farther up
the methodological ladder than description,
inference, and prediction. As ecologists, our
abilities on the lower rungs are as yet unproven.

A question that is rarely raised: What is
the universe to which our results are to pertain?
If it is a single study area, in a single year,
with the measurements we have observed, there is
no problem. If we want to generalize, let us be
careful. Our study area must be representative of
the area we want to extrapolate to, similarly the
year, and the habitat. Noon and Able ( 1978) and
Converse and Morzuch (1981) described how naive
application of results from one area to another
could be erroneous.

I see a clear need for experimentation. As
observers of (more-or-less) natural populations of
wild animals that do as they please, our options

16

here' are unforunately limited. But they are not
curtailed. Is there a role for removal studies
such as Stewart and Aldrich (1951) performed? An
area which was repopulated soon after birds were
removed would exhibit more of whatever the birds
need than would an area that remained vacant.
Habitat manipulation is another possibility. Some
imaginative thinking might produce very valuable
experiments to test the results we obtain from
multivariate analysis of passive and uncontrolled
observations.

MANAGEMENT APPLICATIONS

Multivariate analysis has been suggested as
an important and useful tool in the management of
habitats (e.g., Lennartz and Bjugstad 1975, Conner
and Adkisson 1976, Evans 1978, Noon and Able 1978,
Niemi and Pfannmuller 1979). It promises enhanced
ability to examine a multitude of variables at
once, just as a resource manager manipulates a
multitude of habitat variables with the swing of
an ax or the drop of a match. Clear-cutting a
forest does not merely change the standing biomass
of trees, it also affects stem counts, ground
cover, litter, and a host of other measurable
features: clearly a multivariate situation.

Again, I must sound a cautionary note. Are
we looking at the right variables, and are we
confusing association with causation? Consider
the following paradigm. An action A, which may be
either a natural phenomenon or a specific
management activity, causes certain effects,
denoted X^, X^, X^, upon the environment.

One of these environmental effects, say X^, in

turn triggers a population response by a
particular species, Z. Suppose we are observing
the phenomenon, and record many of the
environmental variables, but not X^, and we record

the response of species Z. Our analysis would
show a relationship between the X's and Z; a
multivariate analysis might reduce the set of X's
to a smaller set of principal components, or
discriminant functions, that also relate to Z. In
truth, however, these associations are all
spurious; the real association involves the
"lurking variable" X^, which correlates with the

other X's and causes the response by species Z.

Even if we are clever enough to measure X^

along with the other X's, the true relationship
between X^ and Z is almost certain to be clouded

by the plethora of other variables. This paradigm
is certainly not new; we have all been exposed to
it in one form or another. But I believe we need
to remind ourselves of it regularly; I am deeply
concerned about the soundness of management
recommendations based upon associations that are
interpreted as causations.

The analysis may not always be misleading.
In the paradigm just presented we might conclude
that a particular configuration of X values is
conducive to good numbers of species Z. The way
to reach that X-conf iguration is by applying
action A. This advice will work. Action A
affects variable X^, which in turn results in an

increase of species Z. ■ ■

But suppose there are other ways to obtain
the desired configuration of X values without the
appropriate value of X^ . Then that management

action, even though it succeeds in producing the
"correct-appearing" habitat, will fail to increase
species Z.

As an example, fire produces certain effects
in North Dakota grassland habitats, including
removal of most standing vegetation and reduction
of litter. The effects of the fire in turn
produce a response by shorebirds; killdeer
(Charadrius vociferus) , marbled godwits (Limosa
f edoa ) and upland sandpipers (Bartramia
longicauda ) come to the burned areas to forage.
Some range ecologists claim that appropriate
regimens of cattle grazing will produce habitat
changes similar to those caused by fire. Grazing
could in fact produce similar measurements on a
variety of vegetative parameters. But the
shorebirds do not seem to respond in the same way
to grazing as they do to fire. They must be keyed
into one or more of the "lurking variables."

Other examples might contrast the effects of
forest fires to those of clear-cutting, natural
food supplies to artificial feeding stations, or
natural pest control versus chemical control.

Rice et al . (1981) also pointed out how
management practices directed toward one season
can have possibly unintended and undesirable
ramifications during other parts of the year.

Where we cannot design experiments freely,
select optimal levels of variables at will, and
replicate as often as we desire, we must be
reserved about our findings. If management
practices are adopted as a result of a
multivariate habitat analysis, a thorough
evaluation should be designed and conducted to
determine if predictions from the proposed model
are accurate and the model thereby remains
tenable.

CONCLUSIONS

Multivariate analysis has been claimed to be
useful in studies of wildlife and their habitats,
despite the fact, clearly pointed out in this
workshop, that the assumptions of the various
methods are largely unmet. I see two possible
explanations for this seeming inconsistency. The
first is that biologists might be reporting the
results of sophisticated multivariate techniques
only when they are in accord with their biological

17

intuition, previous knowledge, or results of
simpler univariate analyses. A technique that
produces conclusions in conflict with other
knowledge may be rejected as inappropriate. If
this is the case, then the multivariate methods
are not truly useful; they only lend an appearance
of further credibility to conclusions already
reached from other directions.

A second possibility is that the multivariate
methods might actually be more robust than is
recognized. They may yield generally correct
answers despite rather flagrant violations of
their assumptions. Statisticians could be looking
at finer detail than biologists do when they
evaluate multivariate methods. A technique can be
biased, of low power, inefficient, and defective
in other statistical ways, but still provide
valuable insight into the biological problem. One
might define a technique to be useful if it
produces more nearly correct answers than would be
available without it.

It is for statisticians and biologists to
work together in determining how useful various
techniques are, relaxing assumptions that may not
be critical, and designing studies to meet the
assumptions that are truly needed. The papers and
discussions have given a real stimulus to further
cooperative efforts, and they will furnish
guidance to researchers for many years to come.

ACKNOWLEDGMENTS

I appreciate comments on an earlier draft of
this report supplied by D. E. Capen, J. C. Lewis,
S. G. Machado and G. J. Niemi.

LITERATURE CITED

Andrews, D.F., R. Gnanadesikan , and J.L. Warner.

1971. Transformations of multivariate data.

Biometrics 27:825-840.
Armstrong, J.S. 1967. Derivation of theory by

means of factor analysis or Tom Swift and his

electric factor analysis machine. American

Statistician 21:17-21.
Bhattacharrya , H. 1981. Theory and methods of

factor analysis and principal components.

Proceedings of this workshop.
Box, G.E.P., and D.R. Cox. 1964. An analysis of

transformations. Journal of the Royal

Statistical Society B 26:211-252.
Boyce , M.S. 1980. Robust canonical correlation

of sage grouse habitat. Proceedings of this

workshop.

Conner, R.N., and C.S. Adkisson. 1976.
Discriminant function analysis: a possible
aid in determining the impact of forest
management on woodpecker nesting habitat.
Forest Science 22:122-127.

Conner, R.N., and C.S. Adkisson. 1977. Principal
component analysis of woodpecker nesting
habitat. Wilson Bulletin 89:122-129.

Converse, K.A. , and B.J. Morzuch. 1981. A
descriptive model of snowshoe hare habitat.
Proceedings of this workshop.

Daniel, C, and F.S. Wood. 1971. Fitting
equations to data. 342 p. John Wiley and
Sons, New York, N.Y.

Draper, N.R., I. Guttman, and H. Kanemasu. 1971.
The distribution of certain regression
statistics. Biometrika 58:295-298.

Dunn, J.E. 1981. Data-based transformations in
multivariate analysis. Proceedings of this
workshop.

Evans, K.E. 1978. Forest management

opportunities for songbirds. Transactions
North American Wildlife and Natural Resources
Conference 43:69-77.

Gnanadesikan, R. 1977. Methods for statistical
data analysis of multivariate observations.
311 p. John Wiley and Sons, New York, N.Y.

Green, R.H. 1971. A multivariate statistical
approach to the Hutchinsonian niche: bivalve
molluscs of central Canada. Ecology
52:543-556.

Green, R.H. 1979. Sampling design and
statistical methods for environmental
biologists. 257 p. John Wiley and Sons, New
York, N.Y.

Harner, E.J., and R.C. Whitmore. 1981. Robust

principal component and discriminant analyses

of two grassland bird species' habitat.

Proceedings of this workshop.
Holmes, R.T., R.E. Bonney, Jr., and S.W. Pacala.

1979. Guild structure of the Hubbard Brook

bird community: a multivariate approach.

Ecology 60:512-520.
James, F.C. 1971. Ordinations of habitat

relationships among breeding birds. Wilson

Bulletin 83:215-236.
Johnson, D.H. 1981. How to measure habitat — a

statistical perspective. Proceedings of this

workshop.

Karr, J.R., and T.E. Martin. 1981. Random
numbers and principal components: further
searches for the unicorn. Proceeding of this
workshop.

Klecka, W.R. 1975. Discriminant analysis. p.
434-467. In Nie, N.H., C.H. Hull, J.G.
Jenkins, K. Steinbrenner , and D.H. Bent,
editors. SPSS: statistical package for the
social sciences. Second edition.

McGraw-Hill Book Co., New York, N.Y.

Kruskal, J.B. 1978. Transformations of data,
p. 1044-1056. In Kruskal, W.H., and J.M.
Tanur, editors. International encyclopedia
of statistics. Volume 2. The Free Press,
New York, N.Y.

Lachenbruch, P. A. 1975. Discriminant analysis.
128 p. Hafner Press, New York, N.Y.

Lack, D. 1966. Population studies of birds.
341 p. Clarendon Press, Oxford.

Lennartz, M.R., and A.J. Bjugstad. 1975.
Information needs to manage forest and range
habitats for nongame birds. p. 328-333. Jjl
Smith, D.R., technical coordinator.
Management of forest and range habitats for
nongame birds: Proceedings of a symposium
[Tucson, Ariz., May 6-9, 1975]. USDA Forest
Service General Technical Report WO-1, 343 p.
Washington, D.C.

Machado, S.B.G. 1976. Transformations of
multivariate data and tests for multivariate
normality. Ph.D. Dissertation. 279 p.
University of Chicago, Chicago.

18

McDonald, L. 1981. A discussion of robust
procedures in multivariate analysis.
Proceedings of this workshop.

Hosteller, F., and J.W. Tukey. 1977. Data
analysis and regression. 588 p.

Addison-Wesley Publishing Co., Reading, Mass.

Niemi, G.J., and L. Pfannmuller. 1979. Avian
communities: approaches to describing their
habitat associations. p. 15^-178. I_n
DeGraaf, R.M., and K.E. Evans, compilers.
Management of north central and northeastern
forests for nongame birds: Proceedings of a
workshop [Minneapolis, Minn., January 23-25,
1979]. USDA Forest Service General Technical
Report NC-51, 268 p. North Central Forest
Experiment Station, St. Paul, Minn.

Noon, B.R., and K.P. Able. 1978. A comparison of
avian community structure in the northern and
southern Appalachian Mountains. p. 98-117.
In DeGraaf, R.M., technical coordinator.
Management of southern forests for nongame
birds: Proceedings of a workshop [Atlanta,
Ga., January 24-26, 1978]. USDA Forest
Service General Technical Report SE-14, 176
p. Southeastern Forest Experiment Station,
Asheville, N.C.

Pope, P.T., and J.T. Webster. 1972. The use of
an F-statistic in stepwise regression
procedures. Technometrics 14:327-3^0.

Raphael, M.G. 1981. Interspecific differences in
nesting habitat of sympatric woodpeckers and
nuthatches. Proceedings of this workshop.

Rencher, A.C., and S.F. Larson. 1980. Bias in
Wilk's A in stepwise discriminant analysis.
Technometrics 22:349-356.

Rice, J., R.D. Ohmart, and B.W. Anderson. 1981.
Bird community use of riparian habitats: the
importance of temporal scale in interpreting
discriminant analysis. Proceedings of this
workshop.

SAS Institute, Inc. 1979. SAS user's guide.

1979 edition. 494 p. Raleigh, N.C.
Smith, K.G. 1977. Distribution of summer birds

along a forest moisture gradient in an Ozark

watershed. Ecology 58:810-819.
Smith, K.G. 1981. Canonical correlation analysis

and its use in wildlife habitat studies.

Proceedings of this workshop.
Stewart, R.E., and J.W. Aldrich. 1951. Removal

and repopulation of breeding birds in a

spruce-fir forest community. Auk 68:471-482.
Tatsuoka, M.M. 1970. Discriminant analysis: the

study of group differences. 57 p. Institute

for Personality and Ability Testing,

Champaign, 111.
Westman, W.E. 1980. Gaussian analysis:

identifying environmental factors influencing

bell-shaped species distributions. Ecology

61:733-739.

Whitmore, R.C. 1977. Habitat partitioning in a
community of passerine birds. Wilson
Bulletin 89:253-265.

Wiens, J. A. 1975. Avian communities, energetics,
and functions in coniferous forest habitats,
p. 226-265. In Smith, D.R., technical
cooordinator . Management of forest and range
habitats for nongame birds: Proceedings of a
symposium [Tucson, Ariz., May 6-9, 19751.
USDA Forest Service General Technical Report
WO-1, 343 p. Washington, D.C.

Williams, B.K. 1981. Discriminant analysis in
wildlife research: theory and applications.
Proceedings of this workshop.

DISCUSSION

E. JAMES HARNER: I disagree that multivariate
analyses are almost always just exploratory. The
jackknife and bootstrap techniques offer rather
robust ways to make statistical inferences.

DOUGLAS JOHNSON: Multivariate techniques

certainly can be used for inferential or, as I
termed it, confirmatory research. As they are
usually employed in wildlife-habitat studies,
however, their proper role is exploratory.

JAKE RICE: From your figure it seemed that you
pointed out that nonlinearity was a major problem
in PCA, and you were talking largely about
nonlinear responses of species (say abundances) to
the environmental gradient represented by the
principal component. Does not that sort of
nonlinearily show up by simply graphing abundance
against PCA scores? In that sense nonlinear
responses are no more problem than in any
univariate study where a species may also have a
nonlinear response to any measure of an
environmental attribute, and nonlinear regression
methods are readily available. Is not the
nonlinearity problem in PCA (and other multi-
variate techniques) one that the variables used in
the PCA (say several environmental or habitat
measures) may be interrelated in nonlinear ways?

DOUGLAS JOHNSON: You are correct on all counts.
Nonlinearity of response is a potential problem
whether the technique be univariate or multi-
variate. Nonlinearity can be examined by simple
graphing, as you suggest and as I heartily
recommend. A further difficulty with multivariate
techniques, particularly principal components
analysis, is nonlinear relationships among the
explanatory (in our case, habitat) variables. If
the relationships are present and strongly
nonlinear, then the best linear combination of
habitat variables may not be very good at all.

CHARLES SMITH: What is meant by "robustness"?

DOUGLAS JOHNSON: Robustness generally means
insensiti vity to violation of the assumptions of a
technique. It has been further refined (Mallows,
C.L. 1979. Robust methods — some examples of
their use. American Statistician 33:179-184) to
incorporate three concepts: 1) resistance--
insensitivity to a moderate number of bad values
and to inadequacies in the model; 2) smoothness —
a characteristic of a technique in that it
responds only gradually to the introduction of a
few errors; and 3) breadth — the extent to which a
technique can be applied in a wide variety of
circumstances. A technique is called robust if it
yields at least approximately correct answers
despite having its assumptions not fully met.

19

RANDOM NUMBERS AND PRINCIPAL COMPONENTS:
FURTHER SEARCHES FOR THE UNICORN?"
James R. Karr^ and Thomas E. Martin^

Abstract. — Analysis of biological data with an array of
multivariate procedures has increased in recent years. While
these powerful tools have considerable potential for
producing more rigorous biological conclusions, they are
subject to misuse. We have analyzed real biological data and
random number matrices using principal components analysis
(PCA) and have shown that: 1) percent of variation accounted
for may be similar for both, especially for the second and
higher principal axes; 2) loadings of the original variables
on principal axes are often as high as those for real data;
and 3) matrix size is an important determinant of amount of
variation extracted by PCA. The relevance of these and other
points is discussed in light of the need for more objective
grounds for the interpretation of PCA.

Key words: Bird/habitat relationships; principal
components analysis; random numbers; sphericity test.

INTRODUCTION

This volume on multivariate statistics and
wildlife habitat is the most recent in a series of
publications demonstrating the power and value of
multivariate procedures in analysis of ecological
data. But, like so many other tools and/or dogmas
in the ecological sciences, we feel it is
necessary to reflect a bit on the potential for
misuse of these "new" procedures. We urge caution
in the use of one of the hottest items in the
arsenal of multivariate procedures — principal
components analysis (PCA).

Many researchers have used this procedure in

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Professor, Department of Ecology, Ethology,
and Evolution, University of Illinois, 606 E.
Healey, Champaign, IL 61820.

'Ph.D. candidate. Department of Ecology,
Ethology, and Evolution, University of Illinois,
606 E. Healey, Champaign, IL 61820.

the past decade for its major intended purpose — to
reduce the complexity of multivariate data to a
more manageable set of compound variables. This
is not without some danger as no testing
procedures are commonly used to allow measurement
or evaluation of the significance of results
generated by use of principal components analysis.
Interpretation of pattern has commonly been based
on relatively subjective grounds.

TWO DATA SETS

Our objective was to compare results of PCA
on a matrix of real biological data With results
from analysis of a random number table of the same
dimensions. The size of data matrices used for
analysis of bird/habitat relationships varies
significantly among studies. For illustrative
purposes we selected a matrix size of 10 x 2^
because of an early use of PCA in bird/habitat
literature (10 vegetation variables and 24 bird
species). (Throughout this paper we refrain from
citing specific studies. The questions we raise
here are directed generally rather than at any
particular study.)

20

Table 1. Percent of variation accounted for by
the first three principal components for two 10
X 2^ matrices.

Percent of variation by component

Matrix

II

III

Cumulative

10 vegetation
variables, 24
bird species

10 X 24 matrix
of random
numbers

57

25

16

14

12

14

85

53

RANKED VARIABLES

1 23456789 10
"T 1 1 1 1 1 \ 1 1 1

For the real data matrix, 85% of the total
variance was accounted for by the first three
principal axes (table 1). However, use of PCA on
an equivalent-sized matrix of random numbers
extracted 53% of the variation in the first three
components. Amount of variation accounted for in
the real data was clearly very high, but it is
instructive to view the situation from the
opposite perspective. The percent of variation
accounted for in a random number table was well
above that which might be expected by the naive
reader. A greater percentage of variation was
accounted for by the first component for real data
than for random numbers, but the second and third
components accounted for similar amounts of
variation in both real and random number matrices
(table 1).

These results might tempt the reader to argue
that loadings of original variables on the
principal components must be much lower in the
random numbers than the real data. This, it
turned out, was not the case (fig. 1). Factor
loadings had approximately the same maxima in the
two matrices. We believe that most biologists
would be able to generate a plausible post facto
biological explanation of the high loadings in the
random numbers for each of the three principal
axes .

Although the maximum loadings of the original
variables were about the same across the three
principal axes for random and real data, the array
of loadings over the ten variables was different
for the first axis (fig. 1). Loadings for the
random number and real data matrices were more
similar in subsequent principal components. These
results were due presumably to correlations among
variables. The correlations among variables for
real data include a greater number of high
correlations than for random numbers (fig. 2).
These high correlations allow the first component
to account for a greater percentage of variation

10

5-

JM

REAL

nrm

15

Z)

O

Li_

RANDOM

ML

O- 1- 2- 3- 4- 5- 6- 7- 8- 9-
CORRELATION COEFFICIENTS

Figure 1. Distributions of loadings of original
variables for principle axes I, II, and III for
real biological data and random numbers (10 x 24
matrix).

Figure 2. Distribution of correlations among
original variables for real biological data and
random numbers (10 x 24 matrix).

21

Table 2. Percent of variation accounted for by the first three principal
components for real and random number (in parentheses) matrices of the
same dimensionality. The real data examples are from published studies
using principal components analysis.

Percent of variation by component

Matrix size I II III Cumulative

6

X

56

33(23)

22(20)

18(18)

73(61)

7

X

5

45(44)

23(28)

11(18)

79(91)

8

X

21

46(27)

17(20)

10(17)

73(64)

10

X

24

57(25)

16(14)

12(14)

85(53)

10

X

46

65(18)

12(16)

12(16)

85(50)

15

X

15

— »(22)

— (16)

11(13)

58(51)

* ( — ) indicates not available in original publication.

in real data. Thus, real data are more
ellipsoidal in multivariate space than are random
data which tend to be more spherical.

How, then, can we develop some confidence in
the typically post facto explanations generated
when real data are analyzed? As a first step, we
suggest that authors and readers carefully
interpret the percent of variation accounted for
in a PCA by comparing the results with those
obtained from a random number matrix of equal
dimensions. One possibility might be a table of
expected amount of variation accounted for by
random number matrices that could be used as a
test relative to the amount of variation accounted
for in real data. Another possibility is the
sphericity test described by Pimentel (1979).
This procedure tests the equality of components.
When three or more axes are of similar length,
they define a sphere in which axes are arbitrarily
placed. Thus, components may be uninterpretable
when they are equal. Finally, there are other
tests available when analysis is based on the
covariance matrix. For instance, the equality of
a vector based on biological data and a
theoretically derived vector can be tested
(Pimentel 1979). However, since most biological
PCA's have been based on correlation matrices, we
confine discussion to these analyses.

The sphericity test simply tests the
homogeneity of the last p - k factors, where p is
the total number of factors and k is the first k
factors not being compared. The homogeneity of
the second and third factors can be tested if the
first three factors are transformed by subtracting
the mean eigenvalue of the last p - 3 factors.
Analysis of results from the 10 x 24 matrix (table
1) with the sphericity test shows that the second

and third axes are equal for real data (x^ =
0.604, P > 0.05). Thus, real data appear to
exhibit a primary ellipsoidal pattern (axis 1) but
not in subsequent axes, so interpretation of axes
2 and 3 may be invalid.

OTHER MATRICES OF BIOLOGICAL DATA

We selected several published studies to
illustrate the similarity of results from real and
random number data. Much variation is explained
by random number matrices relative to real data of
the same dimensions, especially in situations with
few variables (table 2). These results lead us to
urge caution in using PCA for small matrices, a
proviso on sample size similar to that for normal
univariate statistical procedures.

As in the earlier example, the trend for a
greater difference in the first axis but similar
results for the second and third axes of random
and real numbers seems to apply across a variety
of matrix sizes (table 2). However, note that
although the percent of variation accounted for in
the real and random data is similar for the second
and third axes, this is partly due to the first
component of real data accounting for a greater
percent of variation than in random numbers. This
leaves less variation to be accounted for by the
second and third components in real data. Real
data account for a greater proportion of the
remaining variation than in random data for large
matrix sizes. For example, in the 10 x 46 matrix,
the proportion of remaining variation accounted
for by the second (real - 34%; random - 20%) and
third (real - 52%; random - 24%) components is
higher for real data. However, for the 7x5
matrix, the proportion of remaining variation

22

Matrix
Size
4x12

I II III

PRINCIPAL COMPONENT

Matrix
Size

12x12

12x36
12x60

I II III

PRINCIPAL COMPONENT

dimensions. If this is true, interpretations of
components subsequent to the first may be invalid;
at the least, they should be tested by the
sphericity test before they are interpreted in
great detail.

THE EFFECTS OF MATRIX SIZE

Obviously, the amount of variation explained
from random numbers varies with number of derived
axes since 100% of the variation must be extracted
from a number of axes equal to or less than the
number of variables in the analysis. For example,
a matrix of 10 vegetation variables with 24 bird
species must yield 100% variation in 10 or fewer
axes. Typically, all variation is extracted with
fewer components than variables. When there are
few variables, the problem becomes especially
critical (e.g., 89% on the first three principal
axes in a 4 X 12 matrix; fig. 3a).

Similarly, when the number of variables is
held constant but the number of cases varies, the
percent of variance accounted for declines with
increasing matrix size (fig. 3b). Note also that
variability among random number matrices of the
same size is rather small (fig. 3b).

DISCUSSION

We must emphasize that our purpose is not to
convince the reader to avoid using principal
components analysis. Rather, we hope to inspire
more critical evaluation of the uses and misuses
of PCA. When the amount of variation accounted
for from an axis or set of axes is similar to the
amount from random numbers, we question the
biological validity of interpreting those axes.

Figure 3. Cumulative percent of variation
accounted for in principle component analysis of
random number tables of various sizes.

accounted for by the second (real - 42%; random -
50%) and third (real - 34%; random - 64%)
components is greater for the random data. While
the pattern is of interest, we know of no
statistical procedures that can be used to test
its significance. At the least, the pattern
emphasizes the need for caution when using PCA on
small matrices.

Even if the second and third components
explain a greater proportion of the remaining
variation in real data than in random numbers, it
may be difficult to apply biological interpreta-
tions to these axes if they are equal. The
sphericity test indicates that the second and
third components are not significantly (P > 0.05)
different in any of the studies in table 2. One
is tempted to conclude that, in bird/habitat data,
the first axis reflects the presence of a primary
ellipsoid while subsequent axes are nearly as
spherical as a random numbers matrix of the same

Canonical correlation (CanCorr) analysis is
similar to PCA in its analytical procedures except
that CanCorr is developed to analyze the
relationship between two data sets. In addition,
CanCorr includes a test of significance of the
canonical variates. We have applied the test to
random number matrices of a variety of sizes; no
significant (P > 0.05) canonical variates were
found. Most biologists would probably not try to
apply biological interpretations to such results.
Use of a similar decision-making process is needed
in the use of PCA.

Finally, we note that similar results from
principal components analysis of random and real
data do not necessarily mean that biologically
important relationships are not present. However,
it is important to use sound biological judgment
in the interpretation of the results. The final
test of the procedure in any circumstance is the
biological insight developed. That is, the
results of PCA should not be used simply for post
facto interpretations, but rather to aid
researchers in generating reliable predictions and
viable hypotheses which are testable with
additional research.

23

ACKNOWLEDGMENTS

J. Blake and I. Schlosser made helpful
comments on an early draft of the manuscript. M.
Tatsuoka helped to clarify some of the problems
associated with the use of the sphericity test.

LITERATURE CITED

Pimentel, R.A. 1979. Morphometries. The
multivariate analysis of biological data.
276 p. Kendall/Hunt Publishing Co., Dubuque,
la.

DISCUSSION

JAKE RICE: In defense of the use of these
methods, in this case PCA, I can at least say that
the field of statistical ecology has progressed to
the point where one usually cannot get a paper
published Just by doing a PCA on a data set and
writing up the result. If the factor loadings
make biological sense, like Smith's forest
moisture gradient (Smith, K.G. 1977. Distri-
bution of summer birds along a forest moisture
gradient in an Ozark Watershed. Ecology 58:
810-819), then one has some results to work with.
If the loadings and the scores do not make sense,
one goes back, or should go back, and look hard at
the system; obviously one's understanding (and
data base) of the system are incomplete.

If you also randomly assigned habitat names
to "variables" and site names to "samples," in
your random matrix; factored it; and could put a
plausible biological explanation on the results,
then I would be surprised.

JIM KARR: Many users of PCA and other multi-
variate procedures have progressed beyond the
"fishing expedition" approach but others have not.
The questions posed in this paper are directed
toward both groups — primarily as a caution about

interpretation and use of results. They should
not be construed as suggestions to avoid the
procedures and the new insights they can produce
if used wisely. A problem inherent in PCA is the
lack of objective statistical tests to determine
the significance of the patterns which "make
sense." Note that both statistical and biological
significance are of concern; they may not be the
same .

We clearly disagree with your final point. I
am confident that most biologists could generate
plausible "post-facto" explanations for high
loadings after randomly assigning habitat names to
the "variables" in random number tables. The
small sample of examples that I have seen clearly
demonstrate the ingenuity of biologists in that
regard.

CHARLES SMITH: We typically assume that there is
a pattern of some kind; and when none is detected,
we question the validity of the techniques. What
if the most easily detected pattern is one in
time, not space, and the spatial case is not
detectably different from a random pattern, in the
absence of a time-series sample?

JIM KARR: This is a real problem. Indeed, I have
fallen into that trap myself. After one year of
study in forests in Panama, I concluded that
individuals and species moved extensively among
microhabitats in lowland forest. Long-term data
show that to be true but the subtlety of habitat
selection is exceptional. The use of specific
microhabitats varies between seasons and years in
ways that are quite predictable from knowledge of
environmental conditions (wet vs. dry dry season)
at the time. Lumping of data for many kinds of
statistical analysis can result in the error you
mention (i.e., assuming a lack of pattern when a
pattern, in fact, exists); that simply reinforces
the point that statistical analyses should always
be guided by biological knowledge (insight).

24

Special Session: Sampling Avian Habitats

25

RATIONALE AND TECHNIQUES FOR SAMPLING
AVIAN HABITATS: INTRODUCTION'
James R. Karr^

Three more or less cfistinct (but overlapping)
stages might be recognized in the development of
studies of avian habitats. The first "Catalog
Stage" began with efforts to identify birds and
determine their phylogenetic and biogeographic
affinities. Habitat descriptions were generally
cursory and nonquantitative . For example, the
monumental Manual of Neotropical Birds (Blake
1977) describes the habitat of the red-tailed hawk
(Buteo jamaicensis ) as "Mainly woodland and
semiopen country."

During the second "Natural History Stage"
scientists were interested in general biology or
natural history of species: nest type and
location, food habits, clutch size, incubation
period and general habitat were of primary
interest. Few efforts were made to provide
quantitative information on avian habitats; the
main focus was the bird itself. The classic works
of Nice ( 1937) on the song sparrow and Skutch
(e.g. 1969) on neotropical birds are examples
which focus on the natural history of birds,
including nonquantitative studies of their
habitats .

In the third "Ecology of Habitat" stage
emphasis shifted to interest in both the birds
(often as communities) and their habitat
(vegetation type). The importance of foliage
structure in determining avian use of habitat has
long been recognized (Merriam 1890, Gfinnel 1917,
Lack 1937, Pitelka 1941, Kendeigh 1915, Svardson
1919). The first quantitative and graphical
demonstration that vegetation of increasing height
and complexity typically supports increasingly
diverse avifaunas was provided by MacArthur and
MacArthur (1961). Many refinements of this
approach have been developed in the past two
decades. Some are appropriate for consideration

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Professor, Department of Ecology, Ethology,
and Evolution, University of Illinois, 606 E.
Healey, Champaign, IL 61820.

of single species, while others are more
appropriate for community level studies.

Increased emphasis on quantitative approaches
to the study of avian habitats has precipitated
some innovative uses of quantitative methods. Use
of information theory was pioneered by Robert
MacArthur and has continued with the work of
others (Recher 1969, Karr and Roth 1971, Blondel
et al . 1973). Other efforts to demonstrate a
relationship between foliage height diversity and
bird species diversity have been less successful.
Some failed because of inappropriate measures of
habitat structure. Others failed in habitats
where the rules of plant geometry and distribution
are different. For example, foliage volume is
important in some situations (Sturman 1968,
Laudenslayer and Balda 1976), while life form
diversity of plants is more important in others
(Tomoff 1974). Further, precision of

relationships between habitat structure and avian
diversity is often less precise when narrow ranges
of habitat structure are examined (Lovejoy 1974,
Willson 1974).

An array of multivariate statistical
procedures also are being used in studies of
bird-habitat relations. In general, these
techniques (principal components, discriminant
function analysis, reciprocal averaging, canonical
correlation) have the attribute of reducing many
variables to a small set of complex variables.
Other attractive features include easy development
of graphical presentations and studies of how
variation in one variable affects birds when other
variables are held constant.

Multivariate methods are not without
problems, however. First, like most univariate
methods, they are merely descriptive procedures.
They are essentially correlation techniques and
cannot be used to determine casual (ultimate or
even proximate) relationships between birds and
their habitats. Although complex data

transformations result from use of multivariate
procedures, it is not always clear how to extract
biological (vs. statistical) meaning from these
correlations. Further, the caution that ecology
is the art of describing the obvious is also

26

relevant. There is a tendency to use multivariate
methods to demonstrate phenomena which are already
obvious from univariate studies. The true test of
a procedure is its ability to generate new
predictions about biological phenomena, to go
beyond the obvious. Application of multivariate
methods cannot substitute for in-depth
consideration of the biology of the organisms
under investigation. Careless use of these
procedures has stimulated one biologist to comment
that so much mathematical formality combined with
so much ecological casualness is puzzling (Seals
1972) .

THE DEFINITION OF HABITAT

Use of the concept of habitat by a single
investigator (and among investigators) is often
inconsistent (Karr 1980). At least three major
meanings of habitat commonly can be discerned:

1. Habitat = vegetation type (grassland,
forest, etc.).

2. Habitat = the living and non-living
surroundings of an organism.

3. Habitat = specific horizontal (vegetation
configuration) or vertical (twig angle,
leaf density, etc.) components of
vegetation structure. These are often
referred to as the microhabitat of the
species.

Obviously, it is not possible to address all
of these and other difficulties involved in the
analysis of avian habitats in a few short papers.
However, the papers that follow outline many of
the problems and propose some innovative solutions
as guides for the future. Presentations are
organized around three major questions:

1. Why do we measure habitat?

2. lifhat habitat variables should be
measured?

3. How do we measure those variables?

The "why" question must generally be answered
first. The answer will place constraints on the
"what" question (and so on to the "how" question).

THE "WHY" QUESTION

In addressing this question Rotenberry
provides the simple answer: It works. He argues
that it works because habitat forms the background
upon which all adaptive patterns are expressed.
Further, it works for a wide range of studies.
Successes using the habitat approach include both
basic and applied objectives; single species,
several species, or entire communities. The need
for quantitative habitat assessment is especially
critical in efforts to preserve endangered
species. Since the type of "why" question helps
define approach, sampling protocols, and nature of
data analysis in any study, dealing with the "why"
question is an important first step.

THE "WHAT" QUESTION

Since not all relevant parameters can be
measured in any study, researchers must restrict
the "what" to strike a balance between time and
money constraints and the need of a sound research
design with high probability of developing useful
conclusions. Exact variables to be measured, as
well as how precisely they will be measured, vary
with study objectives. The emphases of the two
contributions here are toward theoretical and
applied considerations. In the applied situation,
a reliable correlation between a habitat variable
and avian patterns of interest may often be
sufficient for management decisions. Whitmore's
presentation makes the point that sound research
design directed toward a distinct management
objective can yield results of major significance
for management to enhance wildlife populations.

In a theoretical context, it may be more
important to understand the causal links relating
habitat and avian species or communities. Thus, a
theoretician may need to sample insect (or other
food) abundances in space and time to understand
complex relations between physical environment,
vegetation structure, food supply, and avian
habitat use. This more complex picture may be
necessary for a rewarding exploration of
ecological and evolutionary causes of patterns in
avian species and communities. In addressing this
question, Holmes outlines examples to show the
need for more rigorous procedures in determining
which habitat variables should be measured. He
argues strongly for the need to consider natural
history of the study organism(s) in determining
variables to be measured. Based on this
principle, he outlines several difficulties with
current approaches and cautions that more detailed
and precise information is needed before truly
scientific management can be accomplished.

THE "HOW" QUESTION

The answers in a specific case to the "why"
and "what" questions narrow the options in the
"how" question. In some cases, superficial
(perhaps even nonquantitative) measures of very
general factors are all that is required; in other
cases, detailed and comprehensive measurements may
be necessary. Both biological and statistical
constraints must be kept in mind at this stage.
Noon's extensive experience with sampling avian
habitats leads him to propose a variety of
specific protocols for use in several circum-
stances. His proposals are a valuable mix of
techniques which have proved successful for a
number of researchers. He calls for efforts to
standardize some procedures to facilitate
comparisons among studies. While there is some
merit to this point, I urge caution in too firmly
establishing a sampling protocol. It is premature
for protocols to be "chipped-in-stone" in such a
rapidly developing area. Some adoption of
sampling conventions should be attempted, but the
way for improvement of those procedures should be
kept open as knowledge of the relations between

27

birds and their habitats is improved. In the
final paper of the series, Johnson urges
biologists to make study design decisions with a
clear view of the objectives and goals of each
specific study. He contrasts exploratory and
confirmatory research, while urging that
conclusions be based on sound scientific design
(hypothesis testing, etc.) and not fishing
expeditions for significant correlations.

Finally, I would like to reiterate the point
that all insights will not depend on use of
complex multivariate procedures. Although that is
the general subject of these proceedings, it is
important that we reflect on the reality that the
final measure of value for a study or technique is
whether or not it yields insight into the dynamics
of the ecological systems that we are studying.

LITERATURE CITED

Beals, E.W. 1972, Ordination: mathematical
elegance and ecological naivete. Journal of
Ecology 60:23-35.

Blake, E.R. 1977. Manual of neotropical birds,
volume 1. 67'* p. University of Chicago
Press, Chicago, 111.

Blondel, J., C. Ferry, and B. Frochot. 1973.
Avifaune et vegetation. Essai d 'analyse de
la diversite. Aluda 41:63-84.

Grinnell, J. 1917. Field tests of theories
concerning distributional control. American
Naturalist 51:115-128.

Karr , J.R. 1980. Strip-mine reclamation and bird
habitats. p. 88-96. In DeGraaf R.M., and
N.G. Tilghman, compilers. Management of
western forests and grasslands for nongame
birds: Proceedings of a workshop [Salt Lake
City, Utah, Feb. 11-14,1980]. USDA Forest
Service General Technical Report. INT-86,
535 p. Intermountain Forest and Range
Experiment Station, Ogden, Ut .

Karr. J.R., and R.R. Roth. 1971. Vegetation
structure and avian diversity in several New
World areas. American Naturalist 105:
423-435.

Kendeigh, S.C. 1945. Community selection by

birds in the Helderberg Plateau of New York.

Auk 62:418-436.
Lack, D. 1937. The psychological factor in bird

distribution. British Birds 31:130-136.
Laudenslayer , W.P., and R.P. Balda. 1976.

Breeding bird use of a pinyon- juniper-

ponderosa pine ecotone. Auk 93:571-586.
Lovejoy, T.E. 1974. Bird diversity and abundance

in Amazon forest communities. Living Bird

13: 127-191.

MacArthur, R.H., and J.W. MacArthur. 1961. On

bird species diversity. Ecology 42:594-598.
Merriam, C.H. 1890. Results of a biological

survey of the San Francisco Mountain region

and desert of the Little Colorado in Arizona.

U.S. Department of Agriculture, North

American Fauna 3:1-136.
Nice, M.M. 1937. Studies in the life history of

the song sparrow. I. Transactions of the

Linnean Society of New York 4:1-247.
Pitelka, F.A. 1941. Distribution of birds in

relation to major biotic communities.

American Midland Naturalist 25:113-137.
Recher, H.F. 1969. Bird species diversity and

habitat diversity in Australia and North

America. American Naturalist 103:75-80.
Skutch, A.F. 1969. Life histories of Central

American birds. III. Pacific Coast Avifauna

35: 1-580.

Sturman, W.A. 1968. Description and analysis of
breeding habits of the chickadees, Parus
atricapillus and P_. r uf escens . Ecology
49:1118-431.

Svardson, G. 1949. Competition and habitat
selection in birds. Oikos 1:157-174,

Tomoff, C.S. 1974. Avian species diversity in
desert scrub. Ecology 55:396-403.

Willson, M.F. 1974. Avian community organization
and habitat structure. Ecology 55: 1017-1029.

28

WHY MEASURE BIRD HABITAT?'
John T. Rotenberry^

Abstract. — The rationale behind studies that attempt to
assess bird/habitat relationships quantitatively is
structured around the premise that a bird species' habitat
selection is of adaptive significance. This rationale was
originally posited by Grinnell and led to the development of
the concept of "niche". Although the definition and use of
the term "niche" have undergone substantial expansion, it
seems clear that habitat variables (especially those
associated with vegetation structure) do represent an
integral part of bird species' niches. By comprising either
proximate or ultimate factors to which species must respond,
habitat forms the background upon which all adaptive patterns
are expressed. Thus a full understanding of the evolution of
species' ecological attributes can come only in association
with precise quantitative descriptions of environmental
conditions .

Key words: Adaptation; birds; Grinnell; habitat
measurement; habitat selection; niche.

INTRODUCTION

With maturation of natural history as a
science, it is apparent that information of a
general descriptive or quantitative nature about
habitat is absolutely essential for a full
understanding of patterns of life history,
adaptation, and evolution of any species. This
seems especially apparent for birds. As I
perceive the role of this paper, it is not to
state that the reason you should measure bird
habitat is because it "works," then proceed to
provide you with an exhaustive list of examples;
indeed, many of the contributors to these
proceedings are responsible for those examples.

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Research Associate, Shrubsteppe Habitat
Investigation Team, Department of Biology,
University of New Mexico, Albuquerque, NM 87131.
Current address: Department of Biological
Sciences, Bowling Green State University, Bowling
Green, OH 43^03.

and their reiteration would prove superfluous.
Instead, I shall attempt to summarize briefly some
of the rationale behind why we think it "works,"
structured around the notion that an organism's
habitat reflects aspects of its adaptations.

NICHE CONCEPTS

With birds, perhaps more than for any other
taxa, there has been a historical as well as a
theoretical component to why we gather habitat
data. The seminal paper in bird/habitat
relationships, and indeed for much of ecology
today, is Joseph Grinnell 's classic study of the
California thrasher (Toxostoma redi vivum)
(Grinnell 1917). Grinnell introduced two
important concepts that have formed the foci for
the way we think about birds and their habitat
relationships. The first of these, of course, was
the creation of the term "niche" and its
conceptualization as a close association between
distributional patterns of a species and the
underlying environmental conditions. The second
was the notion that niche relationships were
important not only in telling us about adaptation

29

and natural history of whatever organism we are
describing, but also in revealing aspects of its
relationships to other organisms and, ultimately,
of the structure of the community in which it
resides. The latter idea has received more
emphasis in a theoretical context, largely under
the construct of the n-dimensional hypervolume
model of Hutchinson (1958). Both of these
concepts, the niche as habitat and as something
that reflects individual adaptation and community
organization, form the basis for the sorts of
questions addressed in these proceedings.

Grinnell's original concern was to explain
the rather restricted ecological distribution of
the California thrasher. He thought that reasons
could be found in the various physiological and
behavioral adaptations of the bird to a narrow
range of environmental conditions. It seemed
evident that the nature of these critical
conditions was to 6e learned through an
examination of the bird's habitat. From this
Grinnell went on to relate various aspects of
thrasher biology and distribution to certain
physiognomic, floristic, faunistic, and climatic
features of the environment, and described these
as the "niche-relationships" of the species.

Since Grinnell, niche has been redefined
several times, with most of these reformulations
stressing some aspect of the functional role of
the organism within the community (e.g. Elton
1927). Some have even argued for a strict
separation of a species' habitat and function,
with the former representing environmental
relations and the latter the "true" niche
(Whittaker et al. 1973). It seems more probable,
however, that these ideas simply represent ends of
a continuum, with substantial intergradation
between the two. Insofar as birds can be shown
either to partition or to select different subsets
of habitat within a community (e.g. Cody and
Walter 1976), those differences in habitat among
them will provide indications of the differences
in their functional roles. It seems clear, then,
that habitat variables do represent an integral
part of a bird species' niche regardless of the
presumed rigor of one's definition of the niche in
general; indeed, numerous aspects of niche theory
have been elaborated in just this context (Shugart
and Patten 1972).

HABITAT SELECTION

Of course, the ornithologist's approach to
the concept of habitat as niche has evolved
through the years as well, but it still embodies
the basic assumption that a predictable
relationship exists between the occurrence of a
species and certain characteristic habitat
requirements. The current approach is often to
focus on the floristic or vegetational structural
aspects of avian habitat, what James (1971) has
termed the "niche-gestalt . " The concept of
niche-gestalt is predicated on the theory that
there is some basic configuration or pattern in
the environment that an individual animal will

seek out and settle in. This habitat selection
process may be based on a specific search image,
early learned experience, the particular genetic
make-up of the individual, or any combination of
these factors (Klopfer 1970). Presumably, then,
habitat selection is an evolutionarily derived
mechanism that insures that individuals seek out
and remain in the particular environment to which
they are adapted. This implies, in turn, that the
result of habitat selection, that is, the associ-
ation of a particular species with a particular
subset of an otherwise vast environmental land-
scape, will reflect major aspects of a species'
ecology and behavior.

The recognition stimuli that induce a bird or
any other animal to select a particular habitat
may often appear to be unrelated to the actual
survival and successful reproduction of that
organism (Hilden 1965). Such proximate factors
achieve importance less through direct influences
on the animal's biology than through correlation
or association with the ultimate factors that are
influential, in an adaptive sense. It is the
latter that are essential to survival and
constitute the underlying selective pressures to
which the species is adapted. In many respects,
then, the niche-gestalt is that set of proximate
factors that elicits a habitat selection response
(Smith 1977). These factors are presumed, in
turn, to provide the birds with predictive
evidence of ultimate factors. For birds, the
physical structure of the habitat has long been
considered to be an important proximate niche
dimension, either directly, as it provides
shelter, nesting substrate, or protection from
predators, or indirectly, as it provides cues to
the potential availability and diversity of prey
(Wiens 1969).

In dealing with habitat physiognomy,
ornithologists generally consider the measured
parameters to be representative of proximate
aspects of the niche. From the readily observable
proximate relationships, however, we then
inductively create hypotheses or models that often
deal with functional elements of the niche, or the
species' role within the community. Again the
"habitat" and "functional role" aspects of the
niche concept seem intertwined rather than
exclusive alternatives. If there is a relatively
good correspondence between the proximate and
ultimate factors, then from these hypotheses and
models we can deductively create testable
predictions about other relationships between
birds and their environment, or even between sets
of bird species.

Although the very concept of "niche" has been
challenged by some ecologists as being trivial
(e.g., Ricklefs 1973: 522), it is clearly on the
merit of its usefulness as a model or predictive
tool that it should stand or fall. It seems quite
clear for birds, at least, that the notion of
habitat as niche has been extremely useful.
Perhaps the most direct manifestation of this is
the relative ease with which numbers of
individuals or relative densities of many bird

30

species may be predicted from measurement of
habitat variables (e.g., Anderson and Shugart
1974, Robbins 1978). Likewise, the simple
presence or absence of a species may often be
strongly correlated with a suite of environmental
measures (e.g., Smith 1977). Such relationships
have been empirically and statistically verified
many times, and some bird/habitat relationships
are so strong that field identification guides nay
use habitat occupancy as a key character (Robbins
et al . 1966). If in fact habitat selection does
represent a major evolutionary component of avian
natural history, then these correlations may allow
us to make predictions about the adaptations of
particular species. We may also identify species
that depend on rather specific habitat conditions
(what we call narrow-niched species, or
specialists) versus those that are associated with
a wide variety of conditions (broad-niched, or
generalist species). Through this relationship we
may again infer adaptive relationships.

Such correlations and associations between
single species of birds and habitat variables are
also of substantial practical value. They allow
us to predict with varying degrees of accuracy the
response of a species to natural or artificial
habitat alterations. They permit us to identify
species that, because they are relatively specific
in their habitat requirements, are most likely to
undergo major population declines if certain
portions of that habitat are altered. And
finally, they allow us to identify environmental
management practices that may increase the amount
of appropriate habitat for species that have
become endangered or rare as a result of previous
habitat loss.

COMMUNITY COMPOSITION

On a large scale, niche theory and the
distribution of species along environmental
gradients allow us to make predictions about the
coexistence or co-occurrence of species in
communities. To the extent that species are
distributed along a habitat gradient more or less
independently of one another, the community
composition of a site is predicted merely by its
location along that gradient (Rotenberry and Wiens
1980). Niche theory suggests, however, that under
certain environmental conditions species may not
vary independently from one another, but instead
that inter-populational interactions, such as
competition, will contribute to the nonrandom
distribution of species along habitat gradients
(Terborgh 1971). This, of course, has proven to
be a fertile area for theoretical speculation and
prediction, and the role of competition in
organizing bird communities, expressed through
either habitat displacement or, conversely,
coexistance mechanisms, has been investigated
extensively (see especially Cody 1974).
Regardless of whether one chooses to look at
community composition as a result of the
independent distribution of species on
environmental continua or the product of intense
competitive interactions leading to tightly

ordered distributional patterns, it seems clear
that the nature of the underlying habitat will
have a profound effect on community composition.
This observation is perhaps best exemplified by
what has become virtually a truism in avian
ecology: bird species diversity may be regularly
predicted from vegetational complexity, as
expressed through either plant species diversity
(Lovejoy 197**), life form diversity (Tomoff 1974),
or structural diversity (MacArthur and MacArthur
1961). Of course, it will be only through careful
measurement of appropriate habitat variables
coupled with knowledge of species' biologies that
we will be able to distinguish the independent
versus competitive alternatives.

' CONCLUSION

Quite beyond all these other reasons that I
have outlined as providing a rationale for
measuring bird habitat is the simple observation,
as I indicated at the outset, that the measurement
of habitat does seem to "work" for birds. By
"work," I mean that there appear to be regular,
repeatable patterns of associations or
correlations between birds and habitat variables,
regardless of any theoretical expectations or
interpretations. Empirically this is verified by
the literally hundreds of papers that have
appeared since Grinnell that demonstrate nonrandom
occupancy of habitat. These documentations may
range from things as simple as the correlation
between sagebrush coverage and sage sparrow
(Amphispiza belli ) density in the Great Basin
(Rotenberry and Wiens 1978), on up through a
detailed elaboration of the relationship between
foliage height diversity and bird species
diversity in a tropical forest (Karr and Roth
1971 ).

The message I wish to convey is simple: if
we are to discuss any bird species' ecology in an
adaptive context, information about its habitat is
essential. This is because habitat forms the
background on which all adaptive patterns are
expressed. Virtually all attributes of a species,
from its internal physiology on up through its
interaction with other members of its community,
have evolved for certain environmental conditions.
Without knowledge of those conditions, which is
expressed through our quantitative or qualitative
description of habitat, the adaptive nature of
these attributes is unknown. It seems apparent,
therefore, that the necessity of defining these
environmental conditions will result in the
continued intertwining of bird populations and
habitat measurements throughout all phases of
avian ecology.

ACKNOWLEDGMENTS

John Wiens, Linda Heald, and Dennis Heinemann
provided constructive criticism of an earlier
version of this paper. Recent support has been
provided by NSF 75-11898 to John Wiens.

31

LITERATURE CITED

Anderson, S.H., and H.H. Shugart. ^9T^. Habitat
selection of breeding birds in an east
Tennessee deciduous forest. Ecology
55:828-837.

Cody, M.L. 197^. Competition and the structure
of bird communities. 318 p. Princeton
University Press, Princeton, H.J.

Cody, M.L., and H. Walter. 1976. Habitat
selection and interspecific interactions
among Mediterranean sylviid warblers. Oikos
27:210-238.

Elton, C. 1927. Animal ecology. 209 p.

Sidgwick and Jackson, London.
Grinnell, J. 1917. The niche-relationships of

the California thrasher. Auk 3^*: '427-433.
Hilden, 0. 1965. Habitat selection in birds.

Annales Zoologici Fennici 2:53-75.
Hutchinson, G.E. 1958. Concluding remarks. Cold

Spring Harbor Symposium on Quantitative

Biology 22:415-427.
James, F.C. 1971. Ordination of habitat

relationships among breeding birds. Wilson

Bulletin 83:215-236.
Karr, J.R., and R.R. Roth. 1971. Vegetation

structure and avian diversity in several New

World areas. American Naturalist 105:

423-435.

Klopfer, P. 1970, Behavioral ecology. 173 p.

Duke University Press, Durham, N.C.
Lovejoy, T.E. 1974. Bird diversity and abundance

in Amazon forest communities. Living Bird

13: 127-191.

MacArthur, R.H., and J.W. MacArthur. 1961. On
bird species diversity. Ecology 42:594-598.

Ricklefs, R.E. 1973. Ecology. 861 p. Chiron
Press, Newton, Mass.

Robbins, C.S. 1978. Determining habitat
requirements of nongame species. Transactions
North American Wildlife and Natural Resources
Conference 43:57-68.

Robbins, C.S., B. Bruun, H.S. Zim, and A. Singer.
1966. Birds of North America. 340 p.
Golden Press, New York, N.Y.

Rotenberry, J.T., and J. A. Wiens. 1978. Nongame
bird communities in northwestern rangelands.
0. 32-46. In DeGraaf, R.M., technical
coordinator. Nongame bird habitat management
in the coniferous forests of the western
United States: Proceedings of a workshop
[Portland, Oregon, Feb. 7-9, 1977]. USDA
Forest Service General Technical Report
PNW-64, 100 p. Pacific Northwest Forest
Experiment Station, Portland, Ore.

Rotenberry, J.T., and J. A. Wiens. 1980. Habitat
structure, patchiness, and avian communities
in North American steppe vegetation: a
multivariate analysis. Ecology 61:1228-1250.

Shugart, H.H., and B.C. Patten. 1972. Niche
quantification and the concept of niche
pattern. p. 284-327. ln_ Patten, B.C.,
editor. Systems analysis and simulation in
ecology. Volume II. Academic Press, New
York, N.Y.

Smith, K.G. 1977. Distribution of summer birds
along a forest moisture gradient in an Ozark
watershed. Ecology 58:810-819.

Terborgh, J. 1971. Distribution on environmental
gradients: theory and preliminary

interpretation of distributional patterns in
the avifauna of the Cordillera Vilcabamba,
Peru. Ecology 52:23-40.

Tomoff, C.W. 1974. Avian species diversity in
desert scrub. Ecology 55:396-403.

Whittaker, R.H., S.A. Levin, and R.B. Root. 1973.
Niche, habitat, and ecotope. American
Naturalist 107:321-338.

Wiens, J. A. 1969. An approach to the study of
ecological relationships among grassland
birds. 93 p. Ornithological Monographs.

32

THEORETICAL ASPECTS OF HABITAT USE BY BIRDS'
Richard T. Holrnes^^

Abstract. — For the theoretical ecologist interested in
understanding why correlations exist between the occurrence
of bird species and certain habitat variables, it is
necessary to know how birds use their habitats. Such
information also provides the basis for choosing appropriate
habitat variables to measure. These points are illustrated
by observations of birds in a deciduous forest in New
Hampshire, in which different species of trees are found to
be important habitat parameters.

Key words: Bird foraging; factor analysis; Hubbard
Brook; vegetation structure.

INTRODUCTION

A major goal of the field of avian ecology is
to develop an understanding of the factors that
determine the patterns of occurrence,
distribution, and abundance of birds. With
respect to habitat, the avian ecologist is
specifically concerned with why birds occur where
they do, why there are correlations between bird
species distributions and certain habitat
variables, and particularly what are the
ecological and evolutionary causes of these
relationships and patterns. Since it is important
to understand the causal relationship underlying
the observed patterns before devising or
implementing a management plan, such information
is also of direct use to the wildlife
ecologi st/manager who is interested in
manipulating bird habitats.

In this paper, I review briefly some of the
protocols and procedures used in studies of avian
habitats in terrestrial, mainly forested
environments, and recommend that more attention be
focused on the behavior of the animals themselves.
By knowing more about how birds use their
habitats, the ecologist can make better decisions
about which habitat variables to measure.

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Professor, Department of Biological
Sciences, Dartmouth College, Hanover MB 03755.

ASSESSING BIRD HABITATS: A SYNOPSIS AND CRITIQUE

The usual way to examine the relationship
between birds and their habitats has been to
select a set of habitat characteristics and to
relate these, frequently with the use of
multivariate techniques, to the presence of bird
species in those habitats. The habitat variables
to be measured are often chosen rather
arbitrarily, albeit based on the investigator's
biological intuition as to what might or should be
important to the birds.

The variables chosen often include some
measure of plant species composition (usually the
number of tree species) and several features that
describe vegetation structure (often deciduous vs.
coniferous foliage, tree size classes, canopy
height, percent canopy cover, percent ground
cover, etc.). Such measurements are either made
1) systematically through a study plot and then
related to the avifaunal composition of that area
(e.g. MacArthur and MacArthur 1961, James and
Shugart 1970, Willson 1974, Titterington et al .
1979), or 2) on the territories of individual
birds (e.g. James 1971, Whitmore 1975, 1977, Smith
1977). For the latter, the standard procedure has
been to choose a song perch or nest site as the
center of a 0.04 ha circle (11.3 m in radius) in
which the habitat is then measured. One such
circle is taken per territory, and a number of
territories of each species are measured. The
data so obtained are considered to characterize
the places where the birds live, and have led to
biologically meaningful correlations or

33

ordinations (e.g. James 1971, Whitmore 1975, 1977,
Smith 1977). Furthermore, with appropriate
statistical techniques (see other papers in these
proceedings) the habitat variables that are best
correlated with the presence of a particular
species can be identified.

Putting aside for a moment the problems of
arbitrarily chosen variables, there are a number
of difficulties with these sampling techniques.
For instance, since an area of O.OM ha represents
only a relatively small portion of the territories
of most passerine birds, can habitat measurements
made in a 22 m diameter circle around a single
song^ perch adequately represent the habitat
occupied by that bird? I contend, for reasons
given below, that they may not. Also, how many
territories must be measured to provide a
statistically valid sample for representing the
habitat occupied by the species? Most studies so
far have taken measurements from an average of 10
to 20 territories per species, but none has yet
evaluated how sufficient these sample sizes are.

Finally and more pertinent to the subject of
this paper, these techniques provide little
information, if any, on the reasons why the
measured habitat variables might be important to
the birds. Nor do they provide a basis for
evaluating whether the most appropriate variables
have been chosen in the first place. Thus, not
only do the existing sampling methods need further
development and testing, but the basic rationale
for deciding what habitat components to measure in
the first place requires re-evaluation.

HOW DO BIRDS USE THEIR HABITATS?

What then should we measure? Iflnat habitat
variables are important to birds? To answer such
questions, it is necessary to understand how birds
utilize their habitats and what habitat components
ultimately influence the survivorship and
reproductive success of these birds. Although
such information must eventually come from
intensive population studies, considerable insight
can be obtained from detailed observations of
birds in their habitats.

By occupying a particular habitat, birds gain
more than just a place to live (Hilden 1965).
They obtain places to hide from predators, to
escape vagaries of weather, to display, roost,
nest and forage. The relative importance of these
various functions differs from species to species
and habitat to habitat. For example, suitable
nest sites may be particularly important for
desert birds (Tomoff 197^*) or for cavity-nesters
whose presence, abundance and reproductive success
in a habitat will be determined in large part by
the availability of suitable nesting sites (Conner
1978).

Perhaps the way in which habitats are used
most intensively and extensively by birds.

especially those inhabiting forests, is for
foraging. With a relatively large proportion of
their daily activities spent searching for food,
birds move frequently from place to place and
constantly scan plant surfaces, including tops and
bottoms of leaves, twigs, branches and tree boles,
for prey. Many of these birds move long distances
during each foraging bout and at least in
temperate forests, range widely from the forest
floor to the upper parts of the canopy.^

To illustrate how observations on bird
foraging behavior can provide information about
what habitat variables might be important, I
summarize here briefly some results from a study
of birds in the Hubbard Brook Experimental Forest,
West Thornton, N.H. The methods and analytical
procedures have been described in detail by Holmes
et al . (1979a). Basically, data were collected on
the microhabitat use by birds foraging in this
northern hardwoods forest during the breeding
season, late May through mid July 197^-1976.
Information was recorded on the height of
foraging, the tree species on which foraging
occurred, substrates to which foraging maneuvers
were directed, and the kinds of foraging maneuvers
employed. For purposes here, only the 11 bird
species that forage primarily among the foliage of
the forest canopy and the 20 appropriate
foraging-related characters (table 1; Holmes et
al. 1979a) are considered. The 20 x 1 1 "species"
matrix (Q-technique , Sneath and Sokal 1973) was
used to calculate the Euclidean distances between
all combinations of the 11 species in the
multi-dimensional space defined by the 20 foraging
characters. This distance matrix was then
subjected to hierarchical cluster analysis for
purposes of illustrating species relationships
(fig. 1). The transposed 11 x 20 'character'
matrix (Q-technique, Sneath and Sokal 1973) was
used for a varimax rotated factor analysis, as
described by Holmes et al . ( 1979a).

The dendrogram in figure 1 groups the 11 bird
species on the basis of their similarities or
differences in microhabitat use/foraging behavior.
There are clearly two major groups of
foliage-foraging species in this community, each
with two to several subgroupings. The relative
importance of the characters that account for this
pattern can be determined from the factor
analysis, which weights the variables by their
relative contributions to the total community
pattern and reduces a large number of variables to
a smaller number of identifiable factors (^Cooley
and Lohnes 1971, Bhattacharyya 1981). In this
case, the first four factors account for 79% of
the community variance (table 1), and illustrate
how the birds differentially utilize foraging
location, substrate and tree species. Thus,
factor I largely accounts for the first major
subdivision in the dendrogram. It separates those
species that primarily hover for prey on leaves,
hawk insects from the air and associate their

'Holmes, R.T., unpublished data on file,
Dartmouth College.

34

Table 1. The rotated factor pattern showing the most heavily weighted
factors, either positive or negative, for each of 20 foraging
characters (see Holmes et al . 1979a for further details).

Factors

II

III

IV

Eigen roots

7.24

4.27

2.

47

1.71

Factor contribution to

community variance (%)

36.22

21.36

12.

39

8.56

Cumulative %

36.22

57.58

69.

97

78.53

1 Hover leaf

0. 889

2 Glean leaf

0.728

3 Hover branch

4 Glean branch

-0.673

5 Hover twig

-0.951

6 Glean twig

-0. 929

7 Hawk (air)

0.589

8 Hover trunk

0.806

9 Glean trunk

-0.805

10 Proximal

0.946

11 Distal

-0.971

12 Beech

0. 792

13 Maple

0.743

14 Birch

-0.920

15 Ash

-0.499

16 Shrub

0.799

17 Conifer

-0.

809

18 Height (x)

-0.616

19 Height SD

0.

893

20 Body size

foraging with sugar maple (Acer saccharum) and
beech ( Fagus grandifolia) from those that glean
prey, largely from twigs, branches and boles of
trees and often forage on yellow birch (Betula
allegheniensis ) and white ash (Fraxinus
americanus ) . The other factors illustrate further
how species differently utilize the foraging
environment by segregating those that glean prey
from leaves, hover at tree boles and forage
primarily in the inner (proximal) portions of the
trees from those that forage distally along the
branches, usually high in the canopy.

The finding that these birds are differ-
entially using tree species and foraging
substrates suggests that they are responding to,
or are influenced in some way by, these habitat
components. Indeed, it is very likely that these
elements are closely linked, in that different
kinds of foraging behavior may be required to
exploit prey occurring on particular types of
substrates which may vary in color, texture or
frequency on different species of trees. This is
supported by a more detailed study of bird

foraging behavior which indicates that these
insectivorous birds in the northern hardwood
forests at Hubbard Brook have distinct preferences
and/or aversions for certain species of trees
while foraging (Holmes and Robinson 1981). This
is attributed at least in part to differences
among the tree species in arthropod abundances and
in foliage arrangements that influence how birds
search for and capture prey from plant surfaces.
To determine the former, we utilized detailed
information on insect abundances obtained from an
intensive sampling of bird food resources (Holmes
and Robinson 1981, Holmes et al. 1979b). For the
latter, we found that bird species which typically
glean arthropods have the strongest tree species
preferences, in this case for yellow birch and for
conifers (Holmes and Robinson 1981). On these
plants, the leaves are arranged close to the
branches so that a standing bird can reach prey
situated on leaf surfaces. In contrast, in maple
or beech, leaves are arranged either on flat
sprays or at the ends of long petioles, and birds
must fly and snatch their prey or hover for them
at leaf surfaces. In addition, there is

35

Euclidean Distance

10.0 8.0 6.0 A.O

J I I I I I I I

i=6.47 ^

j Solitary Vireo

I Block-copped Chickodee

I I Blockburnion Worbler

i Block-throoted Green Worbler

I I Red-eyed Vireo

I Philodelphio Vireo

I I Rose-breosted Grosbeok

I I Scorlet Tonoqer

I I Least Flycotcher

j ^ American Redstart

j Block-throoted Blue Warbler

Figure 1. Dendrogram indicating the similarities
and differences in microhabitat use among the 11
bird species at Hubbard Brook that forage for
arthropod prey among foliage (for detail of
technique, see text and Holmes et al. 1979a).

preliminary evidence that abundance and
distribution of some of these bird species may be
linked to particular tree species or at least to
trees with similar physiognomic characteristics
(Holmes and Robinson 1981).

The important point here is that observation
of foraging birds has led to the realization that
different species of broad-leaved trees in a
primarily deciduous forest can be important
habitat components. Although the potential
importance of deciduous vs. coniferous foliage has
been previously recognized (Balda 1969, Franzreb
1978) and occasionally incorporated into habitat
analyses (e.g. Titterington et al. 1979), most
studies of bird habitats have not taken into
account the occurrence of particular tree species.
It must be realized, however, that even if tree
species were to be included and if correlations
were found, there would still be no way of
understanding why these tree species are important
without detailed observations of what the birds
were doing in the different tree species and
without measurements of the resources available
there .

CONCLUSIONS

A thorough knowledge of the natural history
of bird species and their habitat requisites is
essential for understanding the relations between
birds and their habitats and for determining which
habitat components are important to birds. To
elucidate further the causal links between birds
and their habitats, however, will require
comparisons of behavior and habitat responses of
the same bird species in habitats that differ in
particular ways or that have been manipulated in a

manner that will test the importance of some
specific habitat parameter (s) . Only through such
carefully planned observations and experiments can
knowledge be obtained that will allow us
eventually to predict from habitat data which bird
species will occur or not occur in particular
habitats. Such a goal is needed if habitats are
to be managed scientifically.

LITERATURE CITED

Balda, R.P. 1959. Foliage use by birds of the
oak-juniper woodland and ponderosa pine
forest in south-eastern Arizona. Condor
71: 399_ij12.

Bhattacharyya , H. 1981. Theory and application
of factor analysis and principal components.
In Capen, D.E., editor. The use of multi-
variate statistics in studies of wildlife
habitat: Proceedings of a workshop
[Burlington, Vt., April 23-25, 1980]. USDA
Forest Service General Technical Report.
Rocky Mountain Forest and Range Experiment
Station, Fort Collins, Colo. (In press).

Cooley, W.W., and P.R. Lohnes. 1971. Multi-
variate data analysis. 36^4 p. John V'/iley
and Sons, New York, N.Y.

Connor, R.N. 1978. Snag management for cavity
nesting birds, p. 120-138. In R.M. DeGraaf,
techincal coordinator. Management of
southern forests for nongame birds:
Proceedings of a workshop [Atlanta, Ga., Jan.
24-26, 1978]. USDA Forest Service General
Technical Report SE-14, 176 p. Southeastern
Forest Experiment Station, Ashville, N.C.

Franzreb, K.E. 1978. Tree species used by birds
in logged and unlogged mixed-coniferous
forests. Wilson Bulletin 90:221-238.

Hilden, 0. 1965. Habitat selection in birds.
Annales Zoologici Fennici 2:53-75.

Holmes, R.T., R.E. Bonney, Jr., and S.W. Pacala.
1979a. Guild structure of the Hubbard Brook
bird community: a multivariate approach.
Ecology 60:512-520.

Holmes, R.T., J.C. Schultz, and P. Nothnagle.
1979b. Bird predation on forest insects: an
exclosure experiment. Science 206:462-463-

Holmes, R.T., and S.K. Robinson. 1981. Tree
species preferences of foraging insectivorous
birds in a northern hardwoods forest.
Oecologia 48:31-35.

James, F.C. 1971. Ordinations of habitat
relationships among breeding birds. Wilson
Bulletin 83:215-236.

James, F.C, and H.H. Shugart, Jr. 1970. A
quantitative method of habitat description.
Audubon Field-Notes 24:727-736.

MacArthur, R.H., and J.W. MacArthur. 1961. On
bird species diversity. Ecology 42:594-598.

Smith, K.G. 1977. Distribution of summer birds
along a forest moisture gradient in an ozark
watershed. Ecology 58:810-819.

Sneath, P.H.A., and R.R. Sokal. 1973. Numerical
taxonomy. 573 p. W.H. Freeman, San
Francisco, Calif.

36

Titterington , R.W., H,S. Crawford, and B.N.
Burgason. 1979. Songbird responses
to commercial clear-cutting in Maine
spruce-fir forests. Journal of Wildlife
Management 43:602-609.

Whitmore, R.C. 1975. Habitat ordination of
passerine birds of the Virgin River Valley,
Southwestern Utah. Wilson Bulletin 87:65-74.

Whitmore, R.C. 1977. Habitat partitioning in a
community of passerine birds. Wilson
Bulletin 89:253-265.

Willson, M.F. 1974. Avian community organization
and habitat structure. Ecology 55:1017-1029.

37

APPLIED ASPECTS OF CHOOSING VARIABLES IN STUDIES
OF BIRD HABITATS'
Robert C. Whitmore^

Abstract. — This paper considers the applied aspects of
choosing and using habitat variables; aspects that deal with
management decisions rather than evolutionary events that
shaped observed community patterns. Vegetation structure is
one of the central parameters used in the explanation of
observed habitat use patterns of birds. When choosing
variables one should consider the range of habitats being
studied, the nature of the bird species, the practicality of
measurement, the time (cost) needed to measure and the
biological importance of each variable. Once the variables
are chosen, experimental designs by Martinka (1972) and
Whitmore (1981) could be used to gather information needed in
making management decisions. The designs basically involve
comparing bird territories with adjacent areas that are not
occupied ( nonterritories ) .

Key words: Blue grouse; discriminant analysis;
grasshopper sparrow; habitat management; variables.

INTRODUCTION

Criteria for determining which habitat
variables to measure generally fall into two broad
categories, theoretical considerations and applied
considerations. The former deal with studies that
ask why certain species of birds inhabit a given
locale and what events shape this process, while
the latter deal primarily with management
decisions. The objective of this paper is to
discuss applied aspects of the question.

When trying to quantify observed habitat use
patterns of birds, variables most often measured

'Paper No. 1643 of the West Virginia
University Agriculture and Forestry Experiment
Station. Paper presented at The use of
multivariate statistics in studies of wildlife
habitat: a workshop, April 23-25, 1980,
Burlington, Vt.

^Associate Professor, Division of Forestry,
Wildlife Biology Section, West Virginia
University, Morgantown, WV 26506.

reflect vegetation structure. There are two
readily apparent reasons for this. First,
vegetation structure is a central factor
determining the habitat that a bird selects (fig.
1) and, although only a proximate factor (Hilden
1965), does evoke the settling response of a bird
arriving in spring. Second, between-site
differences in vegetation structure often are
obvious and while data sets may be voluminous they
are relatively easy and inexpensive to obtain,
especially when compared to other parameters such
as food availability, microclimate and biological
interactions. With increasing use of multivariate
statistics in habitat studies, the number and
complexity of variables measured have increased
(James 1971, Anderson and Shugart 1974, Whitmore
1975, 1977). Moreover, it is likely that the
number of these types of studies will increase
and, in fact, the call for more has already gone
out: "Detailed investigation of the relationships
among numerous habitat variables and bird
populations ... should be encouraged. Such
investigations will likely provide widely
applicable results in a minimum of time" (Verner
1975).

38

Temperature
and
Moisture

Biological
Interactions

Vegetation
Structure

Habitat

Use
Pattern

Food
Resource

Figure 1. Schematic representation of the
relationship between vegetation structure, other
variables and observed habitat use patterns in
birds. Dashed lines represent minor effects
while solid lines represent major effects.
Adapted from Karr (1980).

Although the ecologist may not be seeing or
measuring what the bird is actually selecting
(i.e., "the ecologist may be recognizing distinct
habitats or positions along environmental
gradients, but the bird species present may not be
capable of the same distinctions or their
distinctions may not be equivalent to those of the
observer", Whitmore 1977), detailed multivariate
analysis of species distributions along complex
habitat gradients may have predictive value in
determining which species will occupy a given
site, and may be useful in making management
decisions (James 1971, Martinka 1972, Whitmore
1981). However, as pointed out in numerous
studies, the choices and number of variables
(James 1971) as well as time of year in which they
were measured (lifhitmore 1979) affects results.
Therefore, any multivariate study is only as good
as the input variables. The following sections of
this paper will first examine criteria for
selecting variables and then give an example as to
how they might be used in an applied situation.

CRITERIA FOR SELECTING VARIABLES

The plethora of avian habitat studies can be
placed along a continuum of "quantitativeness"
ranging from subjective, visual analysis
(extensive), such as the Missouri Plan for habitat
evaluation, to microhabitat analysis (intensive)
which may be as detailed as counting the number of
grass stems in a 5 ha field. As an example of an
extremely extensive methodology I am reminded of
the waterfowl manager who developed a three
category system for analyzing marsh vegetation:

0)

c

.Q

o

c
o

(0

E
o
c

Time Taken in Measurement-

Figure 2. A curve representing the relationship
between the level of information obtained and
the time taken in measurement. Dashed zone
indicates an optimal area where the ratio
produces the most benefit per unit expended.

If a duck can swim through it in a straight line —
open; weaving line — moderate; can't make it
through — dense. ^ In the middle of the continuum
of quantitativeness lies what I term "quick and
dirty" methods of habitat analysis. One of the
most" widely used of such techniques was proposed
by James and Shugart (1970) and has been used in a
variety of studies (see Whitmore 1975, 1977).
More detailed analysis, the intensive methods, can
be seen in works of Cody ( 1968) and Wiens ( 1969)
for their analyses of grassland habitats.

A fundamental criterion for picking a
sampling technique is the time needed and relative
cost of each method. When plotting the time taken
in measurements versus the level of information
obtained a Gaussian curve emerges (fig. 2).
Somewhere in the middle of the curve is a "zone of
optimal ity" where a maximum return in information
for amount of time (money) expended is obtained.
An effective sampling scheme should fall within
this zone. For the scope of many studies the
James and Shugart (1970) technique fits the bill.
It remains for others to develop techniques that
will shift the curve to the left on the abscissa
thereby increasing the ratio of information/time
spent .

The nature of variables measured depend on
the range of habitats being considered.
Quantifying habitat use of birds in habitats
ranging from grassland to forest may require less
intensive variables than are necessary to separate
four Panamanian forest plots. Another
consideration is what kind of species are you

'Personal communication with E.D.
Professor, West Virginia University.

Michael ,

39

dealing with? Would variables used to quantify
habitats of an acorn woodpecker (Melanerpes
f ormi ci vor ous ) also be appropriate for a
grasshopper sparrow ( Ammodramus savannarum)?

Some practical aspects \-ihen deciding which
variables to pick might fall into three general
categories. First, variables should be measurable
to the desired level of precision. Second,
variables should be biologically meaningful. For
example, it may be possible to measure tree roots
to 60 cm away from the trunk below the surface of
the ground, but what possible meaning could this
have to a canopy foraging bird such as a cerulean
warbler ( Dendroica cerulea)? Third, variables
should be relevant to the species in question.
Would percent grass cover or the ratio of grasses
to forbs be important to a bark forager such as a
brown creeper (Certhia familiaris)?

As a preface to the next section I would like
to make two comments that seem to have a bearing
on many of the papers in these proceedings.
First, we scientists are often guilty of measuring
everything there is to measure, trusting stepwise
discriminant analysis or some other technique to
sort things out, simply because we don't know what
is important. This may be viewed simply as a
substitutive for thinking. And second, if you
miss the key parameters you can go out and measure
everything else, use the most sophisticated
multivariate techniques and still have nothing of
biological importance. However, if one is careful
in picking variables and follows, at least
generally, the concepts listed above, then
multivariate statistics may be useful in sorting
out habitats and making management decisions.

USING MULTIVARIATE STATISTICS IN
WILDLIFE MANAGEMENT

Assuming correct variables are measured, this
section deals with application of multi-variable
data sets to wildlife management. Standard
management questions might include:

1. What effect will habitat alteration have
on the bird species in question (BSIQ)?

2. What structural characteristics are found
in habitat type A but not in type B that
allow the BSIQ to live in the former but
not the latter?

3. How can I make habitat type B suitable
foY the BSIQ?

4. How can I get more of the BSIQ into type
A?

5. Can I introduce the BSIQ into habitat
type C?

From reading papers in the Journal of
Wildlife Management it is apparent that
multivariate statistics as a tool in making
management decisions have not had widespread use.
The only multivariate paper of interest that I
could find in recent volumes of the Journal of
Wildlife Management presented a technique for
habitat analysis that I feel has broad application

in wildlife management. However, it has
apparently been overlooked. Martinka (1972)
published a study comparing habitats of blue
grouse (Dendragapus obscurus) with surrounding
habitat that appeared similar but did not have
blue grouse using it. He termed the latter sites
" nonter r i tor i es . " This type of analysis,
territory vs. nonterritory , seems appropriate in
addressing several questions listed above.
Basically, Martinka found that by using
discriminant analysis he could correctly classify
96% of his plots as either territories or
nonterritories and by so doing develop a model for
management recommendations.

This same line of reasoning may be of
fundamental use in documenting avian habitat use
in general. Most studies are designed to compare
different bird species or different communities.
Perhaps it would be more fruitful to look at a
single species, comparing sites in which it is
found with similar habitat where it is not found.
As an example of this method I summarize a paper
on grasshopper sparrows (Whitmore 1981). This
species is a common breeder on reclaimed surface
mine grasslands in northern West Virginia, yet not
all sites have grasshopper sparrows and on those
that do it is rare for the entire area to be used.
Questions asked were 1) are there vegetational
structure characteristics that are identifiably
different between used and unused areas, and 2) if
there are, can the unused areas be managed to make
them usable? On one reclaimed site in Preston
County, West Virginia all grasshopper sparrow
territories were located and mapped using the
territory flush technique (Wiens 1969). Once
mapped, vegetation structure variables were
measured in each territory. All areas on the same
mine that were not used by grasshopper sparrows
(nonterritories) were located and sampled. By
using stepwise discriminant analysis it was found
that territories could be separated from
nonterritories with 100? accuracy on a vegetation
density gradient. Nonterritories had

significantly greater (P < 0.01) values for most
grass, shrub and litter cover variables and
significantly lower (P < 0.01) percent bare
ground. A change in the habitat from unusable to
usuable could be manifested by any of a variety of
range management techniques that would reduce
vegetation density and litter build up; the
easiest being fire. Manipulation experiments are
currently being designed in an attempt to shift
unused habitat to usable.

Further extension of the above two examples
may be necessary to fit different species or
groups of species. However, the basic concept
could be used in a variety of wildlife studies
ranging over many habitat types.

SUMMARY

Many studies presented in the current body of
literature are based on a set of variables that
seem to be picked with little regard for the
habitats or species in question and more often

40

than not are based on either convenience or "what
someone else has done." This paper presents a set
of ideas that should be considered before
initiating a bird habitat study. Two examples of
an experimental design based on vegetation
analysis of territories vs. nonterritories , one
with a game bird the other with a songbird, show
great potential for future use in making
management decisions. Scientists should remember
that the choice of vegetation variables, the
number of variables, and the time required for
measurement can greatly affect results in a
multivariate study. Perhaps the greatest amount
of time should be spent in the design and variable
selection stages of the experiment rather than in
the collection of data.

LITERATURE CITED

Anderson, S.H., and H.H. Shugart. igT^t. Habitat
selection of breeding birds in an east
Tennessee deciduous forest. Ecology
55:828-837.

Cody, M.L. 1968. On the methods of resource

division in grassland bird communities.

American Naturalist 102:107-1^7.
Hild^n, 0. 1965. Habitat selection in birds.

Annales Zoological Fennici 2:53-75.
James, F.C. 1971. Ordinations of habitat

relationships among breeding birds. Wilson

Bulletin 83:215-236.
James, F.C, and H.H. Shugart. 1970. A

quantitative method of habitat description.

Audubon Field-Notes 24:727-736.
Karr, J.R. 1980. Geographical variation in the

avifaunas of tropical forest undergrowth.

Auk 97:283-298.
Martinka, R.P. 1972. Structural characteristics

of blue grouse territories in southwestern

Montana. Journal of Wildlife Management

36:498-510.

Verner , J. 1975. Avian behavior and habitat
management. p. 39-58. In Smith, D.R.,
technical coordinator. Management of forest
and range habitats for nongame birds:
Proceedings of a symposium [Tucson, Ariz.,
May 6-9, 19751. USDA Forest Service General
Technical Report WO-1 . 343 p. Washington,
D.C.

Whitmore, R.C, 1975. Habitat ordination of the
passerine birds of the Virgin River Valley,
southwestern Utah. Wilson Bulletin 87:65-74.

Whitmore, R.C. 1977. Habitat partitioning in a
community of passerine birds. Wilson
Bulletin 89:253-265.

Whitmore, R.C. 1979. Temporal variation in the
selected habitats of a guild of grassland
sparrows. Wilson Bulletin 91:592-598.

Whitmore, R.C. 1981. Structural characteristics
of grasshopper sparrow habitat. Journal of
Wildlife Management 45 (In press).

Wiens, J. A. 1969. An approach to the study of
ecological relationships among grassland
birds. 93p. Ornithological Monographs No.
8, American Ornithologists Union.

DISCUSSION

JAMES DUNN: How did you decide on the locations
of the non-territories?

BOB WHITMORE: We had the entire surface mine
mapped as to territory location. All areas that
did not have territories were sampled, subject to
the constraint that an entire 50 m radius circle
would fit within it.

JIM WOEHR: Do you agree that vegetation
patchiness is an important habitat variable, and
if so, how should we measure it and how can we
statistically test for differences in patchiness?

BOB WHITMORE: I agree it is important, but I do
not have a real handle on how to measure it.

JIM WOEHR: In comparing territories with non-
territories, do you think it is critical to study
a habitat over several years to look at
fluctuations in bird density in different
habitats? It seems to me that the "best" habitat
will always have more birds, whereas marginal
habitats will have fewer birds, or none at all, in
years of low populations.

BOB WHITMORE: I totally agree that in most
natural systems yearly changes in environmental
factors such as climate will affect avian
densities and habitat use patterns, often forcing
birds into "suboptimal" habitats. Although it is
difficult to determine quantitatively, I feel that
the strip-mine birds, as yet, are not habitat
limited and, therefore, are taking the best
available. Newly reclaimed surface mine habitat
is being created faster than the birds are capable
of using it. We definitely need some bad years to
study.

41

TECHNIQUES FOR SAMPLING AVIAN HABITATS'
Barry R. Noon^

Abstract. — Standardized methodologies for the sampling
of bird-related vegetation structure in forest and non-forest
habitats are proposed. For forest habitats, the methodology
is based largely on techniques proposed by James and Shugart
(I97O). For non-forest habitats, the methodology is a
synthesis of many previously published techniques,
particularly those of Wiens (1969). For each habitat type a
detailed sampling protocol and sample field data sheet are
provided. In addition, statistical and biological
considerations for the location of sampling points are
discussed. The paper ends with an argument in favor of
standardized methods of sampling avian habitats.

Key words: Avian; habitat structure; sampling
techniques; standardized procedures.

INTRODUCTION

There are two common goals when sampling
avian habitat structure. The first is to measure
features of the habitat that will allow an
accurate determination of the species' habitat
requirements. Habitat parameters believed to be
at least proximally related to a species'
survivorship and reproductive success in that
habitat are selected for measurement. The second
is the ability to make accurate predictions of a
species' response to habitat change and to
anticipate possible detrimental effects to a
species population from various land-use
practices. This second goal is contingent upon
having achieved the first.

To date, there have been a variety of
approaches to describing the habitat associations
of bird species (Niemi and Pfannmuler 1979).
These include the non-quantitative successional

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Research Biologist, Migratory Bird and
Habitat Research Laboratory, U.S. Fish and
Wildlife Service, Laurel MD 20811.

stage (Kendeigh 19^*5, Johnson and Odum 1 955,
Martin I960, Haapanen 1965, Holt 1974) and life
form (Thomas et al . 1975, 1976) approaches, as
well as quantitative approaches based on
statistically defined species-habitat associations
(e.g., Rotenberry and Wiens 1978, Pf annmullei*
1979) or the multidimensional habitat-niche
approach adopted from Hutchinson (1957). The
statistically based, quantitative approaches are
currently favored because they are generally less
subjective and yield more information on specific
niche requirements than do non-quantitative
approaches .

A holistic, community based approach to
species-habitat associations has been outlined by
Niemi and Pfannmuller (1979). The procedure is
based on a cluster algorithm which groups avian
communities, representing various points along a
habitat gradient, into hierarchial clusters on the
basis of their similarity in species composition.
The researcher investigates whether the groups
formed by the clustering algorithm(s) suggest any
structural features of the habitat gradient
recognized as important by the birds. The same
community census data can be subjected to an
inverse clustering algorithm which groups together
bird species that show similar distributions
across the habitat gradient (Pfannmuller 1979).

42

The distribution of species associations and the
community clusters may yield insights into the
habitat requirements of individual species and
also indicate which habitats are recognized as
distinct types.

The avian community approach does not yield
information on specific habitat niche requirements
of individual species. However, it is a valuable
preliminary to more detailed niche analysis
because it accurately delimits the range of
habitats to be sampled for a particular species or
species association.

To clarify the relationship between birds and
habitat structure, we must develop techniques of
habitat measurement which fulfill the following
criteria

1. Yield efficient, precise, and accurate
estimates of the habitat parameters of
interest .

2. Yield data amenable to various
statistical tests, especially in
multivariate analyses on which predictive
models would subsequently be based.

3. Quantify attributes of the habitat that
are biologically relevant to the species
(and biologically interpretable) and that
can be unambiguously communicated to
other researchers.

4. Yield data whose predictive capabilities
can be tested by simulated tests.

5. Are applicable in different structural
habitat types.

The sampling protocol that follows is based
on a multivariate approach to habitat selection,
specifically on the niche concept as formalized by
Hutchinson (1957). The rationale for employing
these techniques is the belief that a species'
response to habitat structure is not univariate.
That is, the suitability of a habitat patch to an
individual bird is a function of several
interrelated habitat parameters whose combined
effect (in a mathematical sense) determines the
habitat's suitability.

When initiating a habitat-based niche
analysis of one or more species, three major
aspects of the experimental design must be
resolved before sampling begins. I will address
each of these below.

How Finely Should Habitat Be Measured?

Obviously, the habitat structure must be
measured in enough detail so that factors believed
important to the species being studied are
accurately estimated. In addition, habitat
structure should be measured so as to bring into
focus differences that may allow two similar bird

^Personal communication with Frances James,
Associate Professor, Department of Biological
Science, Florida State University.

species to coexist. These data will also give
valuable insights into community organization.

As a general rule, the more apparently
homogeneous the habitat, the finer it will need to
be sampled in order to detect its inherent
heterogeneity. Thus, grassland habitats require
sampling vegetation structure at a much finer
level than would forest habitats (cf. Wiens 1969
with Whitmore 1975).

Often the researcher is uncertain how finely
species are discriminating habitat and, as a
consequence, feels the need to sample both micro-
and macrohabitat gradients. Macrohabitat
descriptions require sampling a relatively large
area, while sampling microhabitat gradients over
the same area would result in prohibitive time
constraints. A workable solution is to use nested
samples with microhabitat variables estimated from
sampling areas contained within the larger
sampling units used to estimate macrohabitat
variables. Titterington et al . (1979) provides a
good example of the use of nested vegetation
sampling plots in the study of avian populations.

How Many Samples Should Be Taken?

This topic is covered in more detail by
Johnson (1981); however, a brief point follows. A
sampling criterion such as being within 10 percent
of the population mean 90 percent of the time
should be established for each habitat parameter
estimated. Sample size formulas to meet these and
other criteria are given in Cochran (1963) and
Steel and Torrie (I960). Unfortunately, for some
parameters of interest the appropriate sample size
to meet these criteria is impossible to attain
(Noon, unpublished data). This is particularly
true for habitat variables or gradients that are
very patchily distributed in the study area (e.g.,
the percent of coniferous vegetation in different
vertical strata in a primarily deciduous woods).
For example, using the above criteria, a mature
deciduous forest plot in New York State (Noon
1979) required over 100, 0.04 ha circular plots to
estimate most structural habitat parameters.
Changing the criteria so as to be within 10
percent of the mean 80 percent of the time lowered
the requisite sample size for the same variables
to approximately 60 circular plots.

Where Should Samples Be Taken?

A major assumption of most statistical models
is that the statistics themselves have been
estimated from a random sample of the population
under study. When estimating structural
parameters of the vegetation relative to the bird
community, stratified random sampling may be more
appropriate than pure random sampling.
Stratification will result in a more uniform
sampling of the study area. I suggest that the
stratification be based on the following criteria:

1. Along obvious lines of habitat

43

Figure 1. Point-count census technique with
variable censusing radius. Five, 0.04-ha
circular plots for sampling vegetation structure
are clustered about the censusing point.

heterogeneity within the study area. It
is very important that patches of
structurally distinct habitat be
represented in data from which estimates
of available habitat are made.
2. By the location of individual birds in
the habitat; that is, let the bird define
the sampling location.

The importance of bird-defined sampling
locations is clarified by examining how easily
erroneous inferences about a species' habitat
requirements arise from improper sampling. I will
illustrate the problem with bird census data
collected with the point count technique (Ferry
and Frochot 1970) and habitat data subsequently
gathered from 0.04-ha (0.1 acre) circular plots
(James and Shugart 1970) centered around the
censusing point.

Figure 1 illustrates a sampling design with
five, 0.04-ha circles clustered about the
censusing point. A hypothetical study using this
experimental design would conduct many point-
counts along a habitat gradient and associate all
the species recorded at a particular censusing
location with the mean habitat vector (averaged
across the five circles) at that location. A
two-group discriminant function analysis may then
be conducted, for each species, to distinguish
points where the species was found from points
where it was not detected. Discrimination of
these "present" and "absent" groups would be
interpreted in terms of differences in habitat
structure .

Figure 2. As in fig. 1 but with actual tufted
titmouse territories (determined by territory
mapping) detectable from the point-center
positioned relative to the vegetation samples.

Figure 2 illustrates the actual location of
tufted titmouse territories around one of these
census points. Note that habitat data from
non-utilized areas are associated with the
species. If the habitat sampled from non-utilized
areas is substantially different from the utilized
areas, then it will be quite difficult to discern
the true habitat preferences of a species with
this experimental design. The problem is not
unique to this sampling situation but may arise
anytime habitat data associated with a particular
individual represent areas not actually utilized
by that individual.

METHODS

I have outlined a sampling methodology that
combines random, or statified random, sampling
with bird defined sampling locations as follows
(see figs. 3 and 4): 1) Samples of vegetation
structure are taken at randomly selected locations
within the study plot boundaries. Vegetational
characteristics of individual territories are
determined by superimposing a map of the
territory boundaries over a map of the numbered
sampling sites and summing the data from all
sample sites included within the boundaries (Wiens
1959). Only samples falling totally within
territory boundaries are assigned to that species.
2) Territories that by chance were not sampled

44

Figure 3. Line transect system with randomly
located sampling points indicated. Territories
are superimposed over sample locations.

are sampled in some random fashion.

Figure 4. Areal plot system with randomly located
0. 04-ha circular plots indicated. Territories
are superimposed over sample locations. The 25-m
grid system illustrated here would only be
necessary in very closed habitats.

Given sufficient data, this sampling scheme
allows direct comparison between species'
structural habitats as well as a comparison of
each species against a random sample of the
available habitat. The sampling protocol is
amenable to studies in both non-forest (j< 25
percent cover by trees) and forest habitats. I
suggest that non-forest habitats be sampled by
line transect methods and forest habitats by a
modification of the James-Shugart (1970) 0.04-ha
circular plot technique. Transects should be
placed across contour lines (Costing 1956), and
evenly spaced across the study plot with the
location of the initial transect line determined
by some random process (fig. 3). For forest
habitats, sampling points can be determined by a
random (or stratified random) selection of grid
points used in the territory mapping procedure
(fig. 4).

Sampling Protocol
Non-forest Habitats

a sampling protocol that combines line-intercept
and point-centered quarter techniques. In
developing this protocol I have borrowed
extensively from the work of others, particularly
Wiens (1969).

Line-intercept Variables. The line is considered
to be a belt 1 cm wide running along one side of
the transect. The transect is divided

(stratified) into 10-m intervals. Within each
interval the distance on the transect line
interrupted by specific life forms or habitat
features is recorded (appendix lA ) .

For shrub and tree life forms, coverage is
estimated by the downward vertical projection of
their foliage lying above the line. Within each
10-m segment the minimum total amount of coverage
summed over all habitat features is equal to the
length of that segment; however, total coverage is
usually higher. Data collected in this way allow
calculation of the frequency, density, and
dominance of each habitat feature (Smith 1974).

Avian habitat studies in "non-forest"
environments (<^ 25 percent cover by trees) have
used primarily transect sampling procedures to
establish sampling locations. Along the transect,
or at randomly selected points, vegetation
structure has been estimated by fixed quadrat,
line intercept, or point-centered quarter
techniques (refer to Smith 1974 for a general
description of these procedures). I propose

If the ground-level vegetation is not arrayed
as discrete patches then line-intercept techniques
are very difficult to use. When the ground cover
is an intricate mosaic, line-intercept techniques
should be replaced by point-intercept methods.
The procedure is to uniformly select numerous
points within each 10-m interval and record the
presence and absence of specific ground cover
life forms intercepted at these points.

45

statistical treatment of the results and
cautionary notes on methods of point sampling are
found in Goodall (1952).

Point-quarter Variables. Each 10-m interval along
the transect will correspond to a sampling unit.
Within each interval the sampling point is
determined by selecting three random digits to
indicate 1) the linear distance in meters along
the transect interval given by the first digit, 2)
the side of the line to be sampled (odd digit =
left; even digit = right) given by the second
digit, and 3) the number of 0.5-m intervals to be
marked off perpendicular to the transect given by
the third digit. At the sampling point, quarters
are established by placing two 1-m sticks on top
of each other oriented in the cardinal directions
to form a '+'. Within each quarter the following
variables are estimated:

A. Species, distance to, and height of the
nearest shrub (woody vegetation > 1 m
tall and < 3 cm dbh) .

B. Species, distance to, dbh, and height of
the nearest sapling (3 cm <^ dbh < 8 cm)
and tree (>; 8 cm dbh).

C. Vertical vegetation density. At each of
the four ends of the meter sticks,
vertically lower into the vegetation a

1- cm diameter rod graduated into the
following intervals: 0-0.3 m, 0.3-1 m,
and 1-2 m. Vertical vegetation density
is estimated by recording the number of
contacts of vegetation falling within
each interval as well as visually
estimating the number of contacts from

2- 9 m and > 9 m.

D. "Effective vegetation height". At the
approximate intersection point of the two
meter sticks record "effective vegetation
height" as detailed by Wiens (1969).

Methods to calculate density, dominance, and

frequency estimates from point-quarter data are

given in Smith (197^). A sample field data sheet
is given in appendix II.

Forest Habitats

The sampling protocol for forest habitats (>
25 percent cover by trees) is based on the James
and Shugart (1970) O.O^-ha (0.1 acre) circular
plot techniques. I have modified the techniques
to include additional data and to clarify existing
ambiguities (cf. James 1978); however, the core of
the technique remains unaltered. A sample field
data sheet for recording the estimates outlined
below is given in appendix III.

The following habitat features are measured within
the O.OU-ha (radius = 11.3 m) circle:

1. The diameter at breast height (dbh),
1.3 m above the ground, of all saplings
and all standing trees. The dbh values
will be recorded by tree species within
nine size classes (appendix IB),
Standing, dead trees are recorded

N

S

Figure 5. Circular plot (r = 11.3 m) used to
estimate vegetation structure in forest
habitats. Rectangular plots used to estimate
shrub density, transects establishing quarters,
point-quarter tree and log variables, and sample
points for estimating canopy and ground cover
are all indicated.

separately by size class. The size
classes are almost identical to those
proposed by James and Shugart (rounded to
nearest cm) but with the addition of a
smaller dbh size class, S (sapling: 3-8
cm dbh) ,

2. Shrub density at breast height is
estimated along two transects running in
the cardinal directions and centered
within the 0. 04-ha circle (fig. 5). The
observer proceeds along the transect
lines counting the number of woody stems
< 3 cm dbh intersected with his body and
outstretched arms at breast height.
Counted stems only include the main stem
and those stems branching from the main
stem below breast height. The total
number of contacts made in two transects
(each 22.6 m long) times 125 is used to
give an estimate of the number of shrub
stems per ha. The contribution of
deciduous and coniferous shrubs is
recorded separately.

3. Canopy cover and ground cover are
estimated by sighting through an ocular
tube, made from a cardboard cylinder with
cross hairs at one end. The observer
walks along the transect lines used to
estimate shrub density sighting up to the
canopy and recording a total of 20 (10
each transect) plus or minus readings
indicating the presence or absence.

46

respectively, of green vegetation at the
intersection point of the cross hairs.
Percent of the canopy cover contributed
by coniferous foliage is recorded in
addition to total canopy cover. Green
vegetation within a meter of the ground
is recorded in an identical manner except
that the observer sights downward through
the tube held at waist height
(approximately 1 m above the ground) .
Canopy and ground cover are recorded as
percents (i.e., number of hits/20 x 100).

4. A qualitative plant dispersion index is
recorded for ground (0-1 m tall) and
shrub (> 1 m tall and < 3 cm dbh) strata
plants. The index is identical to that
proposed by Emlen (1956). The categories
are

E - Even matrix (more or less randomly
dispersed)

I - Irregular or uneven (indistinct
clumps)

SC - Small clumps
LC - Large clumps

SR - Small distinct rows or hedges
LR - Large distinct rows or strips.

5. Canopy height should express the average
height (m) of the canopy within the
0.04-ha circle. The observer should make
several measurements (with a clinometer,
range finder, Abney level, or similar
instrument), average these measurements,
and record this average. Also, the
maximum and minimum estimates of canopy
height are recorded.

6. Slope is estimated with the aid of a
clinometer. This estimate is the maximum
slope within the circular plot.

7. Indices of tree and log dispersion are
gathered from point-quarter techniques.
The point is centered in the circle and
the quarters are established by the
transects used to estimate shrub density
(fig. 5). Within each quarter the
distance to, and dbh size-class of the
nearest tree are recorded. In addition,
the distance to, total length of, and dbh
size-class of the largest (by diameter)
fallen log (>^ 1.5 m in length and >^ 8 cm
dbh) are recorded (fig. 5). The dbh
size-class of the log is determined by
the maximum dbh attained throughout its
length whether lying totally within the
plot or not.

8. Understory foliage volume is estimated
with a density board (Wight 1938, DeVos
and Mosby 1969). The density board, or
drop cloth, (fig. 6) is divided into four
height intervals, 0-0.3 m, 0.3-1 m, 1-2
m, and 2-3 m, corresponding to low
ground, high ground, and low and high
shrub levels, respectively. Foliage
volume contributed by the sapling level
is indirectly assessed by the number of
trees falling in dbh size-class S. The
drop cloth is placed at each of the four
points where the transect lines intersect
the edge of the circle. Four readings

2-3 m (50 squares)

I - 2 m (50 squares)

0.3 - I m (35 squares)

0 - 0.3 m (15 squares)

I-- -0.3 m- H

Figure 6. Density board ("drop cloth") used to
estimate foliage volume from 0-3 m.

are made from the center of the circle
(i.e., 11.3 m distance) sighting along
each of the transects (i.e., N,S,E,W),
The observer counts the number of squares
within each height interval at least 50
percent obscured by foliage and records
this number. To minimize parallax
problems, foliage volume in the first two
height intervals is estimated from a
crouching position, and from a standing
position for the upper two intervals.
Problems may arise if dense, low
vegetation lies within the immediate
vicinity of the center of the circle (<^
1.5 m). In this situation the observer
should move to the side the minimum
distance necessary to give an
unobstructed view within the first 1.5 m.

47

9. Dominant shrub species and ground cover
(ground to 1 m tall) life forms are
recorded in rank order with the most
common species, or life form, listed
first. The ranking is estimated only
within the 0. 04-ha circle. A list of
ground cover life forms to discriminate
is given in appendix IC.

Equipment needed for estimating density,
basal area, and frequency of trees, canopy height,
shrub density, and percent ground and canopy cover
is given in James and Shugart (1970). Sapling
trees are measured with a forester's diameter
tape. Density boards (drop cloths) can be made
from a variety of materials but those made with
either oil cloth or vinyl prove to be both
resilient and portable. The gradations are scored
on the material with an indelible marking pen.
The drop cloth is extended to its full height with
the aid of a "telescoping" aluminum pole such as
is used by painters and window washers. When not
in use, the drop cloth may be rolled around the
wooden dowel used to attach the top of the cloth
to the aluminum pole.

15

/

/

/

/

/

/ /
/ /

/ /
/ /

✓

/

/

/ y
/ /
/ /
/ /
/ /

/ /
/ /
/ /
/ /
/

/

/

/

/

r /
/

✓

/

/ / ^

/ / ,

/ / ^
1 / /
1 / /

// ;/' s

/ /(SAPUNGS)
^ ^

A

(LOW SUBCANOPY>

B

{HIGH SUeCANOPY)

C

(CANOPY) (

So iSo 2^0 5ao

DIAMETER IN CENTIMETERS

Figure 7. Regression of tree height on diameter
at breast height (modified from Schreuder and
Hafley 1977).

DISCUSSION

Many of the papers presented in these
proceedings have used a habitat sampling protocol
similar to that outlined here and illustrate a
variety of approaches to data analysis and
interpretation. Additional sources of references
are Anderson and Shugart 1974; Bertin 1977;
Bourgeois 1977; Cody 1968, 1978; Cody and V/alter
1976; Conner and Adkisson 1976; James 1971; Karr
1968, 1971, 1976; Karr and Roth 1971; Noon 1981;
Noon and Able 1978; Rabe 1977; Rice 1978; Roth
1976; Smith 1977; Sturman 1968; Titterington et
al. 1979; Whitmore 1975, 1977, 1979; and Wiens
1968, 1973, 1974.

Ecologically meaningful information may be
derived from understory foliage volume and tree
size-class data. Vertical foliage volume is
directly estimated up to 3 m by the drop cloth.
Further, indirect estimates of foliage volume may
be extracted from dbh size-class data for specific
species or types of trees (e.g., hardwoods and
conifers; Harris et al. 1 973, Weinstein'* , Smith^).
This results from the predictable relationship
between tree height and dbh for the lower size
classes (S-C) of most species of trees (fig. 7;
e.g., Curtis 1967; Schreuder and Hafley 1977).
Coupling these two sources of information by tree
species or tree type allows estimates of foliage
volume by vertical strata.

As an example, consider the data collected by

"Manuscript in preparation, D.A. Wienstein,
Environmental Science Division, Oak Ridge National
Laboratory .

* Personal communication with Thomas R. Smith,
Environmental Science Division, Oak Ridge National
Laboratory.

the modified James-Shugart techniques outlined
here. Estimates of foliage volume from the drop
cloth are stratified as 0-.3 m (low ground), 0.3-1
m (high ground), 1-2 m (low shrub), 2-3 m (high
shrub), and for the tree strata (fig. 7) as 4-9 m
(saplings), 9-14 m (low subcanopy) , 14-17 m (high
subcanopy) , and > 17 m (canopy). Estimates of
foliage volume at different vertical strata allow
calculation of total foliage volume as well as
both vertical and horizontal foliage distribution
and heterogeneity.

Several horizontal habitat heterogeneity
indices have been proposed (Wiens 1974, Roth 1976,
Anderson et al . 1979). However, the technique
used by Anderson et al . , based on the variance in
foliage volume both within and across vertical
strata, is most amenable to data of the type
considered here. Insights into the relationship
between total foliage volume and bird species
abundance, and between vertical and horizontal
foliage heterogeneity and bird species diversity
may arise when detailed vegetation data are
combined with information on the spatial
distribution of birds in the habitat.

CONCLUDING REMARKS

The standardized procedures outlined above
are not meant to be a constraint on additional or
new ways of looking at avian habitat structure.
However, I believe that several points can be made
in favor of some degree of standardization in
methodology. First, standardized methods will
clarify communication among researchers and avoid
ambiguities in the interpretation of avian-habitat
interrelations. Second, standardized data will
allow researchers to address questions requiring
comparable data sets. For example, two

48

researchers who had studied the habitat relations
of a species in different parts of its range could
collaborate to examine the extent of geographical
variation in habitat use by that species. The
Breeding Bird Census data, with associated
James-Shugart structural vegetation data, have
already permitted geographical comparisons
(Robbins 1978, Noon et al . 1980, James and Warner,
ms) because of standardization in sampling
methodology. Finally, once researchers and land
managers have reached a common interpretation of
the habitat parameters, research findings can be
more directly incorporated into land management
practices for avian species.

ACKNOWLEDGMENTS

Stanley Anderson, Deanna Dawson, Douglas
Inkley, Frances James, and Chandler Robbins have
contributed extensively to the ideas presented
here. In addition many biologists at the
Migratory Bird and Habitat Research Laboratory
have field tested these techniques and offered
valuable criticism. However, I did not always
heed the advice of my colleagues and much of the
methodology and emphasis on "important" variables
reflects my personal biases.

LITERATURE CITED

Anderson, B.W., R.D. Ohmart, and J. Disano. 1979.

Revegetating the riparian floodplain for
wildlife. p. 318-331. In Johnson, R.R., and
J.R. McCormack, technical coordinators.
Strategies for protection and management of
floodplain wetlands and other riparian
ecosystems: Proceedings of a symposium
[Callaway Gardens, Ga . , December 11-13,
1978]. USDA Forest Service General Technical
Report WO-12, 410 p. Washington, D.C.
Anderson, S.H., and H.H. Shugart. 1974. Habitat
selection of breeding birds in an east
Tennessee deciduous forest. Ecology
55:828-837.

Bertin R.I. 1977. Breeding habitats of the wood
thrush and veery. Condor 79:303-311.

Bourgeois, A. 1977. Quantitative analysis of
American woodcock nest and brood habitat.
Proceedings Woodcock Symposium 6:109-118.

Cochran, W.G. 1963. Sampling techniques. 413 p.
John Wiley and Sons, New York, N.Y.

Cody, M.L. 1968. On methods of resource division
in grassland bird communities. American
Naturalist 102:107-147.

Cody, M.L. 1978. Habitat selection and inter-
specific territoriality among the sylviid
warblers of England and Sweden. Ecological
Monographs 48:351-386.

Cody, M.L., and H. Walter. 1976. Habitat
selection and interspecific interactions
among Mediterranean sylviid warblers. Oikos
27:210-238.

Conner, R.N., and C.S. Adkisson. 1976.
Discriminant function analysis: a possible
aid in determining the impact of forest
management on woodpecker nesting habitat.
Forest Science 22:122-127.

Curtis, R.O. 1967. Height-diameter and height-
diameter-age equations for second-growth
Douglas Fir. Forest Science 13:365-375.

DeVos, A., and H.S. Mosby. 1969. Habitat
analysis and evaluation. p. 135-172. Iji
Giles, R.H., Jr., editor. Wildlife manage-
ment techniques. The Wildlife Society,
Washington, D.C.

Emlen, J.T. 1956. A method for describing and
comparing avian habitats. Ibis 98:565-576.

Ferry, C. , and B. Frochot. 1970. L' avifaune
indif icatrice d'une foret de chenes
pedoncules en bourgogne etude de deux
successions ecologique. La Terre et la Vie
24: 153-250.

Goodall, D.W. 1952. Some considerations in the
use of point quadrats for the analysis of
vegetation. Australian Journal of Scientific
Research, Series B 5:1-41.

Haapanen, A. 1965. Bird fauna of the Finnish
forests in relation to forest succession. I.
Annales Zoologici Fennici 2:153-196.

Harris, W.F., R.A. Goldstein, and P. Sollins.
1973. Net above ground production and
estimates of standing biomass on Walker
Branch Watershed, p. 41-64. In Young, H.E.,
editor. lUFRO Biomass Studies. University of
Maine Press, Orono, Me.

Holt, J. 1974. Bird populations in the hemlock
sere on the Highlands plateau. North
Carolina, 1946 to 1972. Wilson Bulletin
86: 397-406.

Hutchinson, G.E. 1957. Concluding remarks. Cold

Spring Harbor Symposium on Quantitative

Biology 22:415-427.
James, F.C. 1971. Ordination of habitat

relationships among breeding birds. Wilson

Bulletin 83:215-236.
James, F.C. 1978. On understanding quantitative

surveys of vegetation. American Birds

32: 18-21 .

James, F.C, and H.H. Shugart. 1970. A

quantitative method of habitat description.

Audubon Field-Notes 24:727-736.
James, F.C, and N.O. Wamer. ms. Forest bird

communities and vegetation structure. (In

review) .

Johnson, D.H. 1981. How to measure habitat — a
statistical perspective. In Capen, D.E.,
editor. The use of multivariate statistics
in studies of wildlife habitat: Proceedings
of a workshop [Burlington, Vt . , April 23-25,
1980]. USDA Forest Service General Techincal
.Report. Rocky Mountain Forest and Range
Experiment Station, Fort Collins, Colo. (In
press) .

Johnston, D.W., and E.P. Odum. 1956. Breeding
bird populations in relation to plant
succession on the Piedmont of Georgia.
Ecology 37:50-62.
Karr, J.R. 1968. Habitat and avian diversity on
strip-mined land in east-central Illinois.
Condor 70:348-357.
Karr, J.R. 1971. Structure of avian communities
in selected Panama and Illinois habitats.
Ecological Monographs 41:207-233.
Karr, J.R. 1976. Within- and between-habitat
avian diversity in African and neotropical
lowland habitats. Ecological Monographs
46:457-481.

49

Karr, J.R., and R.R. Roth. 1971. Vegetation
structure and avian diversity in several new
world areas. American Naturalist 105: 423-
435.

Kendeigh, S.C. 1945. Community selection by
birds on the Helderberg Plateau of New York.
Auk 62:418-436.

Martin, N.D. I960. An analysis of bird
populations in relation to forest succession
in Algonquin Provincial Park, Ontario.
Ecology 41: 126-140.

Niemi, G.J., and L.A Pfannmuller. 1979. Avian
communities: approaches to describing their
habitat associations. p. 154-178. I_n
DeGraaf, R.M., and K.E. Evans,
compilers. Management of north central and
northeastern forests for nongame birds:
Proceedings of a workshop [Minneapolis,
Minn., January 23-25, 1979]. USDA Forest
Service General Technical Report NC-51,
268 p. North Central Forest Experiment
Station, St. Paul, Minn.

Noon, B.R. 1979. Climax maple-birch-beech
forest. American Birds 33:58.

Noon, B.R. 1981. The distribution of an avian
guild along a temperate elevational gradient:
the importance and expression of competition.
Ecological Monographs. 51:105-124.

Noon, B.R. and K.P. Able. 1978. A comparison of
avian community structure in the northern and
southern Appalachian Mountains. p. 98-117.
In DeGraaf, R.M., technical coordinator.
Management of southern forests for nongame
birds: Proceedings of a workshop [Atlanta,
Ga., January 24-26, 1978] USDA Forest Service
General Technical Report SE-14, 176p. South-
eastern Forest Experiment Station, Asheville,
N.C.

Noon, B.R., D.K. Dawson, D.B. Inkley, C.S.
Robbins, and S.H. Anderson. 1980.
Consistency in habitat preference of forest
bird species. Transactions North American
Wildlife and Natural Resources Conference
45:226-244.

Costing, H. 1956. The study of plant commun-
ities. 440 p. W. H. Freeman, San Francisco,
Calif.

Pfannmuller, L.A. 1979. Bird communities in
northeastern Minnesota. M.S. Thesis, 75 p.
University of Minnesota, Minneapolis.

Rabe, D. 1977. Structural analysis of woodcocck
diurnal habitat in northern Michigan.
Proceedings Woodcock Symposium 6: 125-134.

Rice, J. 1978. Ecological relationships of two
interspecif ically territorial vireos.
Ecology 59:526-538.

Robbins, C.B. 1978. Determining habitat require-
ments of nongame species. Transactions North
American Wildlife and Natural Resources
Conference 43:57-68.

Rotenberry, J.T., and J. A. Wiens. 1978. Nongame
bird communities in northwestern rangelands.
p. 59-86. In DeGraaf, R.M., technical
coordinator. Nongame bird habitat management
in the coniferous forests of the western
United States: Proceedings of a workshop
[Portland, Ore., February 7-9, 1977]. USDA
Forest Service General Technical Report
PNW-46, 100 p. Pacific Northwest Forest and
Range Experiment Station, Portland, Ore.

Roth, R.R. 1976. Spatial heterogeneity and bird

species diversity. Ecology 57:773-782.
Schreuder, H.T. , and W.L. Hafley. 1977. A useful

bivariate distribution for describing stand

structure of tree heights and diameters.

Biometrics 33:471-478.
Smith, K.G. 1977. Distribution of summer birds

along a forest moisture gradient in an Ozark

watershed. Ecology 58:810-819.
Smith, R.L. 1974. Ecology and field biology.

849 p. Harper and Row, New York, N.Y.
Steel, R.G.D., and J.H. Torrie. I960. Principles

and procedures of statistics. 481 p.

McGraw-Hill, New York, NY.
Sturman, W.A. 1968. Description and analysis of

breeding habitats of the chickadees, Parus

atricapillus and ?_. ruf escens . Ecology

49:418-431.

Thomas, J.W. , G.L. Crouch, R.S. Bumstead, and L.D.
Bryant. 1975. Silvicultural options and
habitat values in coniferous forests. p.
59-86. In Smith, D.R., technical

coordinator. Management of forest and range
habitats for nongame birds: Proceedings of a
symposium [Tuscon, Ariz., May 6-9, 1975].
USDA Forest Service General Technical Report
WO-1, 343 p. Washington, D.C.

Thomas, J.W., R.J. Miller, H. Black, J.E. Rodiek,
and C. Maser. 1976. Guidelines for
maintaining and enhancing wildlife habitat in
forest management in the Blue Mountains of
Oregon and Washington. Transactions North
American Wildlife and Natural Resources
Conference 41:452-476.

Titterington , R.W., H.S. Crawford, and B.N.
Burgason. 1979. Songbird responses to
commerical clear-cutting in Maine spruce-fir
forests. Journal of Wildlife Management
43:602-609.

Whitmore, R.C. 1975. Habitat ordination of
passerine birds of the Virgin River Valley,
southwestern Utah. Wilson Bulletin 87:65-74.

Whitmore, R.C. 1977. Habitat partitioning in a
community of passerine birds. Wilson
Bulletin 89:253-265.

Whitmore, R.C. 1979. Temporal variation in the
selected habitats of a guild of grassland
sparrows. Wilson Bulletin 91:592-598.

Wiens, J. A. 1969. An approach to the study of
ecological relationships among grassland
birds. 93 p. Ornithological Monographs 8.

Wiens, J. A. 1973. Pattern and process in
grassland bird communities. Ecological
Monographs 43:237-270.

Wiens, J. A. 1974. Habitat heterogeneity and
avian community structure in North American
grasslands. American Midland Naturalist
91: 195-213.

Wight, H.M. 1938. Field and laboratory technique
in wildlife management. 105 p. University
of Michigan Press, Ann Arbor, Mich.

50

Appendix I

Life forms and habitat features

A. Life forms and habitat features to be discrim-
inated as line-intercept variables.

Grasses - Narrow-leafed herbaceous plants

Forbs - Broad-leafed herbaceous plants

Woody ground cover - Woody vegetation < 1 m
tall

Shrubs - Woody vegetation > 1 m tall and < 3
cm dbh

Saplings - Woody vegetation > 1 m tall and 3
cm _< dbh < 8 cm

Trees - Woody vegetation 2 8 cm dbh

Litter - Dead plant material excluding downed
logs

Water

Bare ground
Rocks

Downed logs - Woody vegetation >^ 8 cm dbh and
>^ 1.5 m long

B. Tree size classes based on diameter at breast
height (dbh) (modified from James and Shugart
1970).

Class Label Dbh Range (cm)

s

3

<

dbh

<

8

A

8

<

dbh

<

15

B

15

_<

dbh

<

23

C

23

_<

dbh

<

38

D

38

<

dbh

<

53

E

53

<

dbh

<

69

F

69

<

dbh

<

84

G

84

<

dbh

<

102

H

102

<

dbh

C. Ground cover life forms to be discriminated in
circular forest plots.

Mosses
Ferns

Grasses and sedges - Narrow leafed herbaceous
plants

Forbs - Broad leafed herbaceous plants

Woody ground cover - Woody vegetation <^ 1 m
tall

Seedlings - Regeneration from overstory trees,
saplings, or shrubs

Litter - Dead plant material excluding slash
and logs

Slash and logs - Unrooted woody vegetation
(usually dead) lying
prostrate

Rocks

Bare ground

51

Appendix II. Field data sheet for non-forest habitat (< 25 percent cover by trees)

Bird sp.

Quad

Plot

Transect

State

County

Latitude

Lon;!

itude

Elevation

Date

Observers

LINE INTERCEPT VARIABLES

TRANSECT
INTERVAL

Bare
ground

Rocks

Water

Litter

Downed

lOfiS

Mosses

Grasses

Forbs

Woody veg.
(< Im tall)

Shrubs Im
tall^Scm dbh)

Saplings
OsdbhcS)

Trees
Cdbh2 8)

1

2

3

u

5

NEAREST TREE IN EACH QUARTER

NE

SE

SW

NW

POINT

Species

Dlst.

DBH

Hgt.

Species

Dlst.

DBH

Hft.

Species

Dist.

DBH

Hgt.

Species

Dlst.

DBH

Hgt.

1

2

3

4

5

NEAREST SHRUB IN EACH QUARTER

SW

Species Dlst. Hgt

Species Dist

Species Dlst

H£t.

Species Dlst

_H£ti.

0

-C.3m

0

3-1 m

1-2 m

2-9 m

>9 n

POINT

N

E

S

W

T

N

E

S

W

T

N

E

S

W

T

N

E

S

W

T

N

E

S

w

T

1

2

3

k

5

Appendix III. Field data sheet for forest habitats (> 25 percent cover by trees),

Bird sp.

Quad

18-20

County

;3-?* ?i-;fe

Longitude

Distai'ice
to edge

1-8 TREE

Genus species

3-8 cm

3-15

IS- 17 B

15-23

18-20 C

23-38

2I-J3 D

38-53

!<-!6 E

53-69

27-:9F
69-84

30-33G

84-102

33-«H' -"

>102

SHRUBS Number of stems intersected

N

S

E

W

Decid.

3-4

S-6

Conlf.

9

NEAREST TREE i LARGEST LOG IN EACH QUARTER

Tree

Lo);(>l 5

m lona >!

cm diom.)

Dlst.

DBH

Dlst.

Dlam.

Length

NE

17-10

3b

27-19

SE

30

SU

*3

NW

DISPERSION INDEX:

69-70 ;

Shrubs Ground cover

CANOPY COVER

GROUND COVER

% coniferous

DENSITY BOARD Count squares at least

N

E

S

w

0-. 3m(HH-JJ)

I6-"

11-13

16-17

. 3-lm(AA-GG)

>8

l-2in (K-T)

it

2-3in (A-J)

Dominant shrubs(>lin tall,<3cm dbh):

1.

2. '_

3. ^^

2J

4.

33

5.

Dominant ground cover (ground-lm)
life forms:

CANOPY HEIGHT(m)

'-T2-75 1 46-49 [60-53

SLOPE ( ° )

52

HOW TO MEASURE HABIT AT-A STATISTICAL PERSPECTIVE'
Douglas H. Johnson^

Abstract. — The present workshop reflects the increasing
interest in the use of sophisticated statistical analysis for
relating wildlife to their habitats. Multivariate methods
are useful tools and their results to date have been
promising, but closer attention to scientific principles and
statistical requirements will prove beneficial, particularly
when wildlife-habitat studies are used as the basis for
prediction and management of wildlife populations.

This paper discusses several points, some well known in
theory but often not fully recognized in practice, others
less well known, but ideas that should guide the researcher
toward an improved study design that yields more credible
results. The major points are 1 ) The objectives of a study
must be clearly thought out and precisely stated in order to
design the study properly. 2) Correlation, including its
analogs regression and discrimination, is not necessarily
causation. 3) If habitat variables are not measured
accurately, a host of analytic problems can arise. 4) The
reliability and repeatability of habitat measurements are
important both statistically and biologically. 5) The
question of how many observations are necessary is an open
one, but a few methods for addressing the question are
offered.

Key words: Correlation; errors in variables; habitat
studies; reliability; research design; sample size;
scientific method.

INTRODUCTION

The topic I was asked to address can be
viewed quite broadly. I chose to focus on some of
the problems that have been encountered in my own
consultations or have been identified as
potentially important in published studies. The
following ideas are not new, but, to judge from my
experience, they are still worthy of
consideration. Nor are the suggestions

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Statistician, U.S. Fish and Wildlife
Service, Jamestown, ND 58401.

comprehensive, but attention to them will probably
result in improved research designs.

"THE CHOICE OF A SAMPLING STRATEGY
DEPENDS ON THE STUDY OBJECTIVES"

This statement is obvious, but, nonetheless,
it seems too frequently disregarded, particularly
in studies intended by investigators to "learn all
we can." I will illustrate the point in a
simplified example. Suppose we are interested in
bobolinks (Dolichonyx oryzi vorus) in grasslands of
North Dakota. If I hold as the prime objective an
accurate estimate of bobolink breeding density, I
might proceed as follows. Stratify the state into
regions based on physiography and/or prior

53

Figure 1. Hypothetical curve indicating the
response of a species to an environmental
gradient X. Solid line indicates average
density; dashed lines denote range in density.

knowledge of bobolink densities. Select sample

units of land within each stratum, the sample size

proportional to the area of the stratum and to the

anticipated standard deviation of bobolink counts.

Suppose the bobolink responds to some habitat

gradient X, which might be an east-west gradient

in precipitation, as shown in figure 1. The

average density is indicated by the solid line;

the dashed lines denote ranges an4 reflect

variability about the average. To estimate most

accurately the mean bobolink density, I would

sample most intensively those strata containing

land units with values of the habitat gradient

between X, and X„, where bobolink densities, and
A B

their standard deviations, are greatest. I would

sample at a low rate the strata of marginal

habitats, in which X < X. or X > X„. Random

A B

sampling is necessary here to insure valid
estimators of the precision of the mean. As a
sideline, I might also measure vegetational
features associated with each sample unit.

If, on the other hand, I am not so interested
in estimating density as I am in determining
relationships between bobolinks and grassland
habitat features, perhaps for predictive or
management purposes, I would proceed differently.
I would then sample the units as much throughout
the feasible region as possible, trying to get
points all along the gradient. I would be
particularly interested in marginal habitats for
bobolinks, because some features there are
presumably limiting the bird, and those features
merit detailed study. Random sampling in this
situation may help eliminate misleading results
caused by selecting nonrepresentative units.

In general, when the objective of estimating
the mean is foremost, we should sample most
intensively where birds are common. If we are
interested in determining relationships, we should
sample more evenly along the gradient of habitat
features. This is but a simple example of how
specific objectives dictate the design of research

and analysis of data that result.

"CORRELATION IS NOT NECESSARILY CAUSATION"

This caveat needs to be remembered as we
proceed with linear or nonlinear models involving
numerous and often highly intercorrelated
variables, all of which are uncontrolled by the
investigator. Educational statistics provide a
classic example. Among grade school children,
performance on scholastic achievement tests is
positively correlated with body weight. Yet
parents, desirous of improving their childrens'
scores, would be ill-advised to fatten them in
preparation for testing, because the correlation
is spurious: older children tend to perform
better in the tests than younger ones, and they
also are likely to be heavier. This example may
be so obvious as to appear trite, but the
analogous situation can occur readily and less
overtly in wildlife-habitat studies, in which the
plethora of variables and paucity of knowledge
about their true relationships can promote
misconceptions. Given enough variables and access
to a high-speed computer, nearly anyone can find a
"significant" association among some of the
variables .

"ERRORS IN VARIABLES CAN BE TROUBLESOME"

Consider the linear model

Y = 3q + B^X^ + B^X^ + ... + Bj^Xj^ + error. (1)

This form ordinarily represents a regression model
but discriminant analysis can be viewed in the
same manner if Y is a dummy-variable indicator of
group membership (Lachenbruch 1975).

In the usual formulation, the X's are assumed
fixed and known quantities, measured without
error. In actual practice, however, particularly
in ecological work in which the X's are habitat
measurements, errors of unknown magnitude often
creep in. I personally suspect them to be large
more often than not.

Errors in the X's lead to biases in the
regression coefficients. A bias is induced
whether errors are systematic, reflecting a bias,
or simply random, reflecting added variability.
Suppose the true relationship between, say, bird
density Y and a vector of habitat features X is
given by equation (1), but the X's are not
measured exactly; instead !_ is measured, where
= )( + 6^ and _6 is an error of measurement. Many
sources of error may contribute to _6. For
example, sampling variability is usually present;
the entire habitat of the animal is not measured,
only a sample of it. There is often instrument
error; the measured variable may not be exact.
Temporal variability may contribute (Whitmore
1979); habitat measurements may not reflect
conditions at the time the animals selected the
habitat. More generally, the wrong variable may
be measured because the correct one is not known.

54

Suppose that Z is unbiased for X, that is
E (6) =0, and the variance-covariance matrix of 6_
is D, a diagonal matrix. The diagonality implies
that errors associated with measurements of
different habitat features are uncorrelated .

The regression coefficients are biased
(Davies and Hutton 1975, Seber 1977), and the bias

-1

vector can be estimated by n(Z'Z) DB . Z is the
matrix containing the Z vectors for all
observations and n is the sample size. Note that
the biases tend to increase in absolute magnitude
as D becomes large and as Z'Z becomes less well
conditioned. That is, the biases increase when
measurement errors increase and when highly
correlated variables are included. These two
conditions are, I believe, very common in
ecological practice.

Errors in measured variables affect not only
the estimated coefficients, but also their
standard errors. The biases here might be either
positive or negative, depending on values of the
coefficients and the variances (Bloch 1978).
Generally, however, the magnitude of the biases
will not be unduly large if errors of measurement
are reasonably small (Hodges and Moore 1972,
Davies and Hutton 1975). Regardless of whether
the standard error is biased positively or
negatively, in the case of one explanatory
variable, the ";t" statistic for assessing the
significance of that variable will be biased low
(Bloch 1978). This feature could cause a truly
important habitat variable to be eliminated as
nonsignificant. Also, the multiple correlation
coefficient may be diminished (Cochran 1970),
possibly to the point that the entire set of
explanatory variables is nonsignificant.

Errors in measured variables tend to deflate
regression coefficients associated with those
variables and to lead an ecologist to claim
unjustly that their effects are insignificant.
The impact of this problem may be particularly
severe when the resulting models are used for
prediction and/or management. Hodges and Moore
(1972) noted that any bias in the regression
coefficients will be transmitted into a biased
forecast. They also reported several studies in
which predictions were available based either on
accurately measured predictor variables or on
inaccurate (preliminary) values of those
variables. The increased error resulting from the
use of inaccurate variables was often striking.

The problem of errors in variables is often
glossed over by use of a conditional argument.
The relationship between Y and X is explored,
given that values of X were observed values Z.
This line of reasoning is not clearly stated,
perhaps not clearly understood, and certainly of
minimal value in prediction and management; the
manager is not interested only in habitats
possessing those exact values.

The general problem of errors in variables
seems to have been ignored in ecology; most

applications have been in economics. Moreover,
most attention has been given to multiple
regression, although limited work has been done
with factor analysis (Lawley and Maxwell 1973,
Chan 1977).

Davies and Hutton (1975) and Seber (1977:159)
presented some working rules for evaluating the
effect of errors in variables. If an independent
batch of data is available, it is instructive to
calculate the reliability of each measurement:

g. = Var(X. )/Var(Z. )

= Var(X. )/[Var(X. ) + Var(6.)].

Reliability values appreciably below 1 should
raise some suspicions about the role of the
corresponding variable in the analysis.

At least until the problem is more fully
explored, it seems prudent to design ecological
studies so that habitat features are accurately
measured and relatively independent of one
another .

"VARIABLES SHOULD BE MEASURED RELIABLY"

The argument just offered suggests that
habitat variables should be measured accurately,
in order to reduce D as much as possible and
minimize bias in the regression coefficients.
There is a further reason for improved
reliability: measurements should be repeatable.
This feature will become increasingly valuable as
studies evolve from their local orientation
involving single study areas and are replicated by
different researchers in different locations. To
truly define the niche of a species requires
studies beyond a single woodland; the species must
be studied in many parts of its range. To that
end, it is mandatory that measurements be reliable
and without serious variability due to observer,
season, occasion, or other causes.

Precious little is known about intra-observer
and inter-observer sources of variability in
habitat measurements, and how they compare with
differences that are of interest. Ecologists
either do not like to make these comparisons or,
if they do, prefer not to discuss them.

THE FINAL QUESTION - "HOW LARGE A SAMPLE?"

As a consulting statistician, the first
question I am usually asked is how many
observations are necessary. I generally respond
with one of three numbers: 1, 50, or "great gobs."
The former answer, 1, is given occasionally when I
believe the hypothesis is stated incorrectly, or
in too much generality; a single observation is
likely to refute it. The latter answer, great
gobs, is given when no hypothesis is at hand, or,
if there is one, it is so slippery as to evade
capture and possible rejection. Neither answer
satisfies the biologist, of course, but either

55

serves as a necessary prelude to sitting down and
thinking about an appropriate hypothesis. The
middle answer, 50, or possibly 100 or 10 or 30, is
given when the hypothesis is well stated and the
experimental procedures thoughtfully described,

I can only give here some general guidelines
for determining sample size. First, more
observations are needed when the number of
variables is large. Many published studies have
only slightly more observations than variables, or
sometimes even fewer. An appropriate minimum
sample size might be 20 observations, plus 3 to 5
more for each variable in the analysis. Larger
sample sizes help to overcome difficulties caused
by violations of the assumptions of multivariate
methods (Green 1979:165).

Second, examine the stability of the
estimates, both means and variances. This can be
done in several ways, for example, by sampling
sequentially, until the mean and variance
stabilize. Once the data have been gathered, the
stability of the results can be assessed by
subsampling, jackknif ing , leaving-one-out methods,
etc. If the data set is split randomly into two
halves, does each half yield conclusions
consistent with the other? Better yet, apply the
results to another area, or a different year.
What predictive value do they have?

Third, investigate the sources of
variability, and how they compare in magnitude.
Observer variability, temporal variability,
variability in measurement method, and others all
add up to cloud the variability between features
that animals may be responding to. Calls for
larger samples are the "knee-jerk" reaction when
variability is excessive, but it may be far more
advantageous to try instead to reduce this
variability by better design.

CONCLUSIONS

It is useful to distinguish two kinds of
research, exploratory and confirmatory. In
exploratory investigations, the researcher is
simply trying to "see what's going on" with
respect to a system. This preliminary

reconnaissance is for the purpose of hypothesis
generation and will likely entail at least a
cursory examination of many variables, in order to
determine those that might be influential. In
exploratory research, a variety of stepwise and ad
hoc procedures are acceptable. Results of an
exploratory investigation are hypotheses for
further testing, not well founded conclusions on
which management practices can credibly be based.
For the more definitive answers necessary for
prediction and management, confirmatory research
is needed.

A confirmatory investigation is more in the
mold of a classic scientific experiment, in which
a hypothesis is stated, an experiment conducted to
test that hypothesis, and the outcome used either
to reject the hypothesis or to retain it. It is

clear that this research, unlike the exploratory
investigation, demands a precise hypothesis,
clearly stated and unambiguous. The design also
needs to be rigorous and the analysis must be
statistically correct. A fresh set of data must
be brought up; the data used in the exploratory
stage to generate a hypothesis cannot be
resurrected in the confirmatory stage to test it.
Confirmatory research is more difficult to apply
to ecological problems than is an exploratory
investigation, but management of ecological
systems requires the more definitive methods if it
is to prove successful. The following suggestions
are offered for an investigator planning a
confirmatory study of habitat in relation to
wildlife. [Green (1979) presented "ten

principles" for environmental studies, many of
which are equally applicable to wildlife-habitat
studies. ]

1 . Think carefully about the objectives of
the study and ask yourself, and other qualified
scientists, if the procedures are truly designed
to meet those objectives.

2. Remember that correlation (as well as
regression and discrimination) is not necessarily
causation .

3. Try to obtain habitat measurements that
are relatively uncorrelated , to avoid the bias
associated with the errors-in-var iables problem,
and for other good reasons as well. This approach
appears far preferable to many ex post facto
methods for reducing the number of variables, such
as stepwise procedures and principal components
analysis,

4. Learn about the kinds of variability in
the measurements. Would the same values be
obtained tomorrow as today? Would another
qualified investigator record the same values?
How different would another randomly selected
point be? Answers to questions such as these will
not only help assess the bias due to the
errors-in-var iables problem, they will also
facilitate cooperative and comparative studies.

5. Sample sizes should be large, especially
in relation to the number of variables involved.
Samples should be large enough to yield stable and
reliable estimates, but methods of reducing the
inherent variability of measurements may be more
fruitful than simply increasing the sample size.

6. And finally, for the ecologist, consult
your friendly neighborhood statistician. Consult
him early, in the project design phase. Consult
him often, as data-gathering proceeds. Then, when
you consult with him about analysis and
interpretation, he will be of much better humor
and of far greater benefit to you,

ACKNOWLEDGMENTS

I am grateful to Drs. L.M. Cowardin and S.G.
Machado for comments on an earlier draft of this
report .

56

LITERATURE CITED

Bloch, F.E. 1978. Measurement error and
statistical significance of an independent
variable. American Statistician 32(1):26-27.

Chan, N.N. 1977. On an unbiased predictor in
factor analysis. Biometrika 64:642-644.

Cochran, W.G. 1970. Some effects of errors of
measurement on multiple correlation. Journal
of the American Statistical Association 65:
22-34.

Davies, R.B. , and B. Hutton. 1975. The effect of
errors in the independent variables in linear
regression. Biometrika 62:383-391.

Green, R.H. 1979. Sampling design and
statistical methods for environmental
biologists. 257 p. John Wiley and Sons, New
York, N.Y.

Hodges, S.D., and P.O. Moore. 1972. Data

uncertainties and least squares regression.

Applied Statistics 21:185-195.
Lachenbruch, P. A. 1975. Discriminant analysis.

128 p. Hafner Press, New York.
Lawley, D.N., and A.E. Maxwell. 1973. Regression

and factor analysis. Biometrika 60:331-338.
Seber, G.A.F. 1977. Linear regression analysis.

465 p. John Wiley and Sons, New York, N.Y.
Whitmore, R.C. 1979. Temporal variation in the

selected habitats of a guild of grassland

sparrows. Wilson Bulletin 91:592-598.

DISCUSSION

JIM WOEHR: Is a distribution of plots system-
atically with respect to space a random sample of
plants?

DOUGLAS JOHNSON: No, but it may be adequately
represented by a model of randomness. The crucial
issue is to define the population of which the
sample is representative. Random sampling insures
that condition if the sample is infinitely large ,
although finite ones can certainly be misrepre-
sentative. The Bayesian concept of exhangeability
is analogous.

The immediate question is whether the plants
might conceivably vary sysmatically in space. In
North Dakota, for example, plots located 1 mile
apart might not give a representative portrayal of
the plants. If the initial transect was along a
roadside, most subsequent ones would be also, and
smooth brome ( Bromis inermis) , for instance, would
appear much more commonly in the plots than in the
state as a whole. An interval of a different
length between plots could yield rather accurate
results, however.

57

Multivariate Methods

DISCRIMINANT ANALYSIS IN WILDLIFE RESEARCH:
THEORY AND APPLICATIONS'
Byron Kenneth Williams^

Abstract. — Discriminant analysis, a method of analyzing
grouped multivariate data, is often used in ecological
investigations. It has both a predictive and an explanatory
function, the former aiming at classification of individuals
of unknovm group membership. The goal of the latter function
is to exhibit group separation by means of linear transforms,
and the corresponding method is called canonical analysis.
This discussion focuses on the application of canonical
analysis in ecology. In order to clarify its meaning, a
parametric approach is taken instead of the usual data-based
formulation. For certain assumptions the data-based
canonical variates are shown to result from maximum
likelihood estimation, thus insuring consistency and
asymptotic efficiency.

The distorting effects of covariance heterogeneity are
examined, as are certain difficulties which arise in
interpreting the canonical functions. A "distortion metric"
is defined, by means of which distortions resulting from the
canonical transformation can be assessed. Several sampling
problems which arise in ecological applications are
considered. It is concluded that the method may prove
valuable for data exploration, but is of limited value as an
inferential procedure.

Key words: Canonical analysis; covariance hetero-
geneity; discriminant analysis; eigenvector.

INTRODUCTION

Discriminant analysis is a technique which
has come to be much used in ecological
investigations. It is applicable to the study of
niche breadth, niche overlap, resource
partitioning, habitat selection, community
structure, and many other topics. In fact the

'Paper presented at The Use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Biometrician , U.S. Fish and Wildlife
Service, Migratory Bird and Habitat Research
Laboratory, Laurel, MD 20811.

methodology is potentially useful for any
ecological situation in which an association is
desired between well defined groups and a set of
ecologically meaningful measurements.

The data for a discriminant analysis comes to
the investigator in the form of a categorical
"response" variate and a corresponding set of
(usually continuous) "predictor" variates. One of
the objectives of the analysis is to predict the
category to which an observation belongs, based on
values of the predictor variates and an
appropriate underlying statistical model. Such a
formulation is essentially classif icatory , and the
prediction equations which result are called
classification functions. Alternatively, the

59

Table 1. Examples of discriminant analysis from the ecological literature. A
number of measurements are made on each sample, and samples are aggregated into
groups according to a grouping index.

Groups defined by Observation measurements Authors

Faunal species

habitat structural
characteristics

Green (1971, 1974); James
(1971); Bertin (1977);
Cody (1978); Dueser and
Shugart (1978, 1979)

Vegetation species

faunal species abundances

Ricklefs (1977)

Species presence/
absence

habitat structural
characteristics

Anderson and Shugart
(1974); Conner and
Adkisson (1976)

Animal behavior

habitat structure, climate

Conroy et al . (1979)

Season

photosynthetic rates

Kowal et al . (1972)

Species and sex

behavioral measurements

Conley (1976)

Geographic area

vegetation densities

Norris and Barkham (1970)

Abiotic categories

habitat factors

Smith (1977)

Artificial classes

vegetation densities

Grigal and Goldstein
(1971 ); Goldstein and
Grigal (1972)

Faunal species

meristic and morphometric
characteristics

nont-anucci ly/o^

Faunal species

song and feeding behaviors
1978c)

Rice (1978a. 1978b,

Geographic area

faunal abundance

Buzas (1967)

Soil groups

chemical concentrations

Horton et al . (1978)

Socially defined
breeding demes

body measurements

Buechner and Roth (1974)

objective of the discriminant analysis may be to
establish optimal "separation" of groups, based on
certain linear transforms of the predictor
variates. This latter approach aims at
interpretation as well as prediction, and the
linear functions used to explain group separation
are called canonical variates. As indicated
below, under certain distribution assumptions the
classification approach to discriminant analysis
is logically consequent to the canonical analysis.
It can be shown (Williams, publication submitted)
that the canonical variates themselves may be used
to develop a classification procedure entirely
equivalent to that produced by the predictive
methodology.

Applications of discriminant analysis in the
ecological literature are many and varied. A
substantial proportion of these, though by no

means all, concern the assessment of
species-habitat associations. There is a
preponderance of applications to avifauna and, to
a lesser degree, to small mammals. Many are
single-species studies in which groups are
determined by presence or absence of an individual
species. Table 1 displays some examples of the
use of discriminant analysis reported in the
literature. As indicated in the table, the
grouping index can range over many different
attributes, from vegetation types to faunal
species to artificial classes built by clustering
procedures. The corresponding measurement
variables can range over a variety of habitat
measurements, such as plant densities or
vegetation structure characters, and faunal
measurements such as species abundance or
behavioral and morphological characteristics.

60

H

»1

«2

f^(x)

f2(x)

^2

h

^3(21)

^3

Ibl

Figure 1. Partition of an environment by species utilization. (a) The
unpartitioned environment, at each point of which there corresponds a
vector jc. (b) The environment partitioned into species-specific
habitats.

Most of these studies involve the use of
canonical variates at some point in the
investigation. Observations are plotted in a
"canonical space" of reduced dimensionality, and
the patterns displayed are interpreted. Attempts
generally are made to interpret the canonical
functions themselves, often by an examination of
coefficients. These efforts are hampered by
certain statistical and conceptual difficulties
which arise in applications with ecological data.

In view of the increasing use of canonical
variates in ecological investigations, this paper
focuses on the theory and applications of
canonical analysis. For purposes of elucidation
the general probability model for discriminant
analysis is introduced below, along with a brief
description of linear classification based on it.
Then canonical analysis is investigated from the
point of view of vector transformations. My goal
is to specify precisely the model, the estimation
procedures and the geometric logic of canonical
analysis. A geometric approach, by clarifying the
structure of canonical analysis, reveals the
importance of its statistical assumptions and the
possible consequence of their violation.

PROBABILITY STRUCTURE

The probability structure to which
discriminant analysis applies is specified as
follows. Consider a mixture of g populations
Tr^,...,iTg with mixing proportions q^,...qg,

represented by the vector Associated with each
individual in the mixture is a vector [i.jc']' of
p+1 components, the first of which (i) specifies
population membership. The remaining p components
in X are a set of measurement values associated
with the individual. Random selection from the
mixture defines a random variable I for population
membership and a random vector X of measurement
values. The probability structure of the mixture
is defined by a set of distributions:

(i) P (I = i) = q.

the probability that an individual chosen at
random from the mixture is in population ir. (the
values in ^ are often called prior probabilities);

(ii) fj^ I j(x I i) = f^U) ,

the conditional distribution of X over the

population ir. . If the set of conditional

distributions f ^ (jc) , . . . , f ^ (jc ) is given as the

vector £, then the mixture is represented by an
ordered pair (f ,£) .

For the ensuing discussion it may be of value
to think of the sampling universe, denoted by H,
as a habitat which is partitioned by way of
species utilization. A set of habitat variables
is measured on each sample plot, for example
canopy height, ground cover and shrub density.
These variables have a frequency distribution f(_x)
over the habitat H with mean _u and covariance l^.

In addition, each plot has associated with it a
variable which indicates which species utilizes
it. This variable partitions H into specific
habitats, each with its own frequency distribution
f . of habitat variables. The proportion of H
w^iich is included in the habitat of species i is
q. . Such a partitioning is shown in figure 1 for
t^ree groups.

Figure 1(a) indicates the sampling universe
H, over which can be measured vector values of jc.
The distribution of _x over H is given by f (jc) ,
with mean v_ and covariance l,^. No partitioning

principle is involved in this distribution. In
Figure 1(b) H has been partitioned into three
subsets. The values that _x can take in subset H.
define the conditional distribution fj^(jc) witA
mean jj. and covariance I. . Mixing proportions q.
specify the proportion of'' H constituted of H., an^
the relationship of moments in partitioned and
unpartitioned spaces is given by

61

i = Qiii + ... + qgig

and
where

I = q^l^ + ... -^qglg

and

g

A = Z q - (jj - VI.) ' .
i = 1

It is, of course, not required of the
specific habitats that their corresponding
measurements _x be grouped into disjoint sets. H.
and Hj may both contain plots characterized by the

same measurements x, and in fact the range of
measurements may be identical over and Hy It

is necessary, however, that the statistical
distribution of habitat values differs across
species. Discriminant analysis seeks to highlight
these among-group statistical differences.

CLASSIFICATION PROCEDURES

The classification problem may be stated in
terms of the structure exhibited above: to
predict population membership for an individual
chosen from (£,£) , given that X = jc. Stated
differently, the problem is to classify
individuals into one of the g populations based on
observed values _x. Classification is determined
by

P[I = i |X = j<] ,

the probability that a sampling unit with observed
values X is a member of population tt . . A
procedure which minimizes classification error is
to assign an individual to tt^ if

P[I = i|X = x] = max {P[I = j|X = x] : j = 1 g}.

Since

q.f.(x)

P[I = i |X = x] = g

I qS Ax)
j = 1

this procedure is equivalent to classification
based on

q^f^(x) = max {qjfjCx): j=l g).

The earliest and best developed discriminant
methodology assumes a multivariate normal
distribution. That is, the conditional

distribution of predictor variates is given by

f.(x) =
1 —

(2ti)~P^^ 1^-1"^^^ exp[-l/2(x-y. ) ' l."^ (x-u. )] .

'—1 ' 1 — 1 — — 1 '

where i indexes population it. and p., L_. are the
corresponding mean vector and covariance matrix.
Linear discrimination results for the assumption
of covariance homogeneity:

= Z, i = 1 q.

In this case the conditional distributions can be
rewritten in group)-specif ic terms:

f.(x) = (2ir)"P^^|zr^^^exp[1/2x'z"\] •

exp[(x - 1/2 i^)

= w(x)exp[(x - 1/2 u^)'l~\^],

where w(jc) is free of group-specific parameters
Jii-

For these distributions the optimal
classification rule may be simplified by noting
that

q.f . (x) > q .f .(x)
if and only if

In q^ + In T^ix) > In + In fj^l).

Since

In f.(x) = In w(x) + (x - 1/2 y.)'s"V

1 — — — — 1 -

and In v{x) has no discriminating power, the
classification rule can be specified with the
linear function

L.(x) = (x - 1/2 u.)'J:"''u. + Inq..
1 — — —1 — -^1 1

The procedure is to classify an observation into
group i, based on the observed values x, if

L^(_x) = max {L ^Cjc) | j = 1 , . . . ,g} .

This is simply Fisher's linear discriminant
function (Fisher 1936). When ji^^ , I, and q^ are
unknown, the maximum likelihood estimates may be
used in the expression for L^(jc). A substantial
body of literature exists on the statistical
consequences, especially in error rate analysis,
of this replacement (see, e.g., Toussaint ^9^^).

SAMPLE-BASED CANONICAL ANALYSIS

A different approach to discrimination, which
sometimes yields additional information about
group differences, is based on the use of
canonical transformations. The usual data-based
methods of canonical analysis involve the
determination of linear transforms

62

I

i

!

which maximize

g _ -2^"i -2
E n. (z. - z) / I Z (z. . - z. ) ,

i=1 ^ ^ i=1 j=1 '

statistic by which to interpret sample structure,
a geometric focus emphasizes the parametric
structure which then is estimated with the data.
The intended result from this orientation is a
sharpening of the assumptions for canonical
analysis and a clarification of its utility for
inferences in ecological studies.

where

and

Consider the group means jj. , i = 1,...,g and
common dispersion z^. Me seek in the canonical
analysis to describe the separation of these means
in a statistically meaningful way. The global
mean of the population mixture (f , is given by

z = a'(1/n 2: n.x.),
i=1

In these expressions n. is the sample size for
g

population tt. and n= i n. . If the data for a
i--1 ^

sample from the mixture (.f_,^) are aggregated by

T = I 2: ( X . . - x) ( X. . - x) '
i = 1 j = 1

g "i _

I z;(x..-x.)(x..-x.)'

i=i

■ I n. (x. - x) (x. - x)
i=1 - -

V = 2 ^i-H-i'
i=1

and the population deviations v_- - v are defined
by the difference between group and overall means.
Let L be any arbitrary line through v with
direction cosines a. Then the projection of
ji. - _U onto L is a vector with squared length
given by

d^ = [a' (u.
1 — —1

= a'(y. -
— —1

v) - p) 'a.

The average of squared lengths is

g 2
1 = 1

i = 1

y)'a

W + B,

= a' [ Z q. (u.
- i=1^-^

u) (y - u) ' ]a

then the procedure above is formally equivalent to
finding eigenvectors corresponding to the g-1
non-zero eigenvalues from

[B - X W] a = 0.

These eigenvectors define g-1 canonical variates

= li'j^.. i = 1 g-l

which are often described as linear transforms
which "maximize among-group variation relative to
within-group variation."

PARAMETRIC CANONICAL ANALYSIS

Under certain probability conditions this
intuitive and essentially non-parametric method
can be generated by a parametric procedure which
considers linear transformations of population
means. The goal once again is to establish
separation of group means, but in this instance
the focus initially is on population differences
and the parameters characterizing them. Data
enters the analysis by way of parameter
estimation, at which point the appropriate
computing forms can be developed. This approach
in some sense reverses the foregoing data based
analysis: rather than manipulating data into a

= a'A a,

which is maximum for some particular direction a*
of the projection line. It can be shown that a*
is the dominant eigenvector of A^. The optimal
direction which is orthogonal to a* is given by
the second dominant eigenvector of A, and so on.
Since _A is of rank g-1 there exist g-1 orthogonal
directions by which to separate means,
corresponding to the non-zero eigenvalues of A.

The use of a* (and the remaining eigen-
vectors) to separate group means is optimal in the
sense that for no other projection line is the
average of squared distances between means as
large. This method is suboptimal , however, in
that it does not account for variances and
covariances within each population. The failure
to accomodate the covariance structure can lead to
two highly undesirable results. First, variates
with high variation (and therefore low information
content) have the same influence on the analysis
as do variates with low variation (and high
information content). Second, the effect of
weighting highly correlated variables equally is
to base the analysis less on statistical content
than merely on the number of variates included in
it. In such a situation one could force group
separations to reflect any arbitrarily chosen
variate merely by including additional positively

63

correlated variates in the analysis. Such
distortions and abritrariness are precisely the

kinds of effects which motivate the use of jT'^ in
the Mahalanobis distance formula.

Canonical analysis includes the same
adjustment, the effect of which is to eliminate
covariances and unequal variances. This
adjustment utilizes a square-root factorization

1/2

of _I to define new variables
_-1/2

1 = L _

-1 /2

with conditional mean _2 _u. and identity
covariance (Graybill 1976). Thus the original
variates are transformed into unit variance,
uncorrelated variates, eliminating ambiguities and
distortions .

The same projection argument as above may be

-1 /2

used on the transformed means i u.. The

— —1

corresponding deviations are
-1/2 , ,

and the least squares lines fitting these
deviations are based on a spectral decomposition
of

In most situations none of the parameters for
the population mixture is known with certainty,
and they must be estimated from the data. Assume
that a random sample of size n from the population
mixture is obtained, n. of which are from
population tt.. Based on the multivariate normal
assumption \he following maximum likelihood
estimates result:

q . : n . /n
^1 1

u. : X. = 1/n. r X
-1 -1 -iJ

8 "i _
£ : S = 1/(n-g) I z (x. . - X. )(x. . - x. )'
- - i = i j = i

g

g n.
= 1/n I z x.

i = 1

i=1 j=1

t : 1/n B = 1/n z n. (x. - x) (x.

i = 1 ^ -

x)

When these maximum likelihood estimates are used
in equation (1) in place of the population
parameters, the equation becomes

-1/2 , -1/2

i = 1

^-1/2 , -1/2*
= Z A Z

The corresponding matrix equation is

which may be written as

-1/2 r. T -1/2'

Z [A-XzJz v=0

- X ?] u = 0

(3)

[1/n B - X S] u

= 1/n [B - n/(n-g) X W] u

= 0 •

Equation (3) has the same eigenstructure as does
[B - X W] u = 0.

[A - X Z] u = 0,

(1 )

The effect of n/(n-g) is to scale the eigenvalues,
leaving eigenvectors unchanged. This can be seen
by

where

-1/2 '
u = Z V .

[B - X n/(n-g) W ] u

= W^^^ [W"''^^ B W~''^^'-X n/(n-g) I] W^''^'u.

Canonical variates are then given by

' -1/2
= V z X

There are g-1 such variates, corresponding to the
non-zero eigenvalues of equation (1). They may be
represented by the canonical transform

^ = IJ x ,

(2)

where the columns of U are vector solutions of
equation (1).

from which it follows that n/(n-g) X is an
1 /2 '

eigenvalue and W ^ the corresponding

-1/2 -1 /2 '

eigenvector of W B M • Therefore

eigenvectors of

[fi; - X _|] u = 0

and

[B - X W] u = 0
are identical. This demonstrates the equivalence

64

of parametric and data-based approaches to
canonical analysis, given the following three
assumptions :

(i) the data consist of a random sample from
a population mixture of multivariate normal
populations;

(ii) covariances are homogeneous across all
populations ;

(iii) maximum likelihood estimates are used
in place of parametric values.

In ecological investigations it is hoped that
an intelligent sampling plan can be combined with
the central limit theorem to approximately satisfy
condition (i). Also, the many advantages of
maximum likelihood estimation make condition (iii)
a reasonable procedure. Therefore the

applicability of canonical analysis in ecological
investigations seems to hinge on condition (ii),
the homogeneity of covariance.

It is noted parenthetically that the

canonical variates in equation (2) are

uncorrelated and have a variance of unity.
Furthermore, it can be shown that

!

u. A u. = X. , i=1 , . . . ,g-1

-1 1 1

(Williams, publication submitted). Two extremely
useful invariance properties follow (let the
transformed population means be represented by
Ri = U'lLi' i=l....g):

First, relative distances are maintained in
canonical space. That is, if Mahalanobis
distances in observation and canonical space are
defined by

D. (x) = (x - p. ) • rVx - u. )
1 — — —1 — — -^1

and

D.(z) = (z - - _n.)

respectively, then

D.(x) - Dj(x) = D.(z) - D.iz)

(Williams, publication submitted). Second, group
mean differences are also maintained:

- Mj)' (iii -

= (Hi - nj)-(ni - nj).

This last result is of obvious importance in the
study of resource partitioning and niche overlap
(MacArthur and Levins 1967, Harner and Whitmore
1977).

INEQUALITY OF COVARIANCES

It remains to rationalize the procedures of
canonical analysis under the assumption of unequal
conditional covariance matrices. Given conditions
(i) - (iii) above, the canonical analysis

represents a statistically meaningful attempt to
optimally separate population means. Its meaning
when condition (ii) is violated becomes more
problematic, and obviously depends on what matrix
Z_ is used in equation (1). It should be noted
that the square root transform, invoked to
eliminate covariances, will most certainly fail to
have its intended effect when group-specific
covariances are unequal. Indeed, no matter what
"covariance" matrix l_ is used in the analysis, the
resulting conditional covariances are still
heterogeneous and have the form

In effect nothing of value for covariance
stabilization has been gained by the transform,
and the meaningfulness of the canonical procedure
is thrown into question.

An often used approach is simply to use the
overall mixture covariance l^, or to use an

"average" covariance defined by

I = qi Ii - ... - qg ig. (4)

That these two matrices yield the same eigen-
structure can be easily shown:

Since

It " ^ -
we have

[A - X Z_^] u

= [A - X (|; + A)] u

= [(i-x)A-xr]u

= (1 - X) [A - X/(1 - X) I] u
= O-

Therefore 1^ and Y yield the same eigenvectors

when used in equation (1). Note that when
conditional covariances are identical, I in
equation (4) is simply their common value, and the
equation

[A - X Z^^] u = 0

is equivalent in its eigenvectors to equation (1).
When covariances are unequal, however, both i_ and
are defined only over the population mixture,

and neither generates an eigenstructure equivalent
to that based on individual dispersions

One result of heterogeneous covariances is
that the canonical variates are uncorrelated over
the mixture (f , q) , but not over the conditional
distributions. For any given conditional
population they may be highly correlated, and this
greatly complicates their interpretation.
Furthermore, the representation of conditional

65

Figure 2. Population representatives in observation and canonical space,
(a) The dispersion of two populations in observation space. Canonical
axis are superimposed by dotted lines. (b) The sample population
dispersions in canonical space.

populations in the g-1 dimensional space defined
by a canonical analysis may severely distort their
geometric configurations. An example is shown in
figure 2.

Figure 2(a) shows two conditional distribu-
tions in observation space, one (ir^) with its

dominant axis nearly orthongonal to the canonical
axes and z^. Representation of these popula-
tions in canonical space is shown in figure 2(b).
Such group specific distortions result from the
projection of p-variate distributions into g-1
dimensions, and they greatly complicate the
interpretation of areas in canonical space. It
follows that one must exercise caution in
analyzing conditional population dispersions with
canonical variates. This is not unexpected, since
the fundamental purpose of discrimination is the
assessment of group differences, rather than the
analysis of dispersion.

INTERPRETATION OF CANONICAL VARIATES

The canonical analysis assumes its most
obvious value for ecologists by providing
interpretable transforms of data. As mentioned
above, these transforms are specifically chosen to
separate group means in an optimal manner. The
canonical variates which result are uncorrelated
linear combinations of observation variables, the
coefficients in which may be given ecological
interpretations. At this point in the analysis
biological insight is of fundamental importance.
It is, however, important to recognize certain
limitations on interpretability with these
transforms.

Both magnitude and sign of the canonical
coefficients often are used for interpretation.
Under certain conditions, e.g., when one
coefficient dominates all others in the canonical
transform and the correlation structure is simple,
this is an acceptable practice. In such a

situation it is fairly straightforward to assess
the "meaning" of the canonical variate, i.e., to
specify what it represents biologically. When the
correlation structure is complex and there are
several coefficients of significant size, however,
interpretation is not so direct. The difficulty
arises from the fact that the observation
variables in the canonical transforms are
correlated, some of them perhaps highly
correlated. Individual canonical coefficients in
this case reflect not only the influence of their
corresponding observation variables, but also the
influence of other variables as reflected through
the correlation structure of the data. Most
people who have worked with discriminant analysis
have probably seen cases in which positively
correlated variates have canonical coefficients
with different signs. These apparently

inconsistent coefficients indicate that the two
corresponding variables include information from
the remaining observation variates which
influences the canonical variate in opposing ways.
It can be shown (Williams, in prep.) that the
correlation of the canonical variate z^ and an
observation variate x^ is

corr ( z. ,x .) = 1 //X . (a* . . + t a* . , p, . ) , (5 )

where a*. . is the "standardized" canonical
1 J

coefficient of x . in the i^*^ canonical transform.
J

This expression reveals that the effect of an
observation variable on the canonical variate is
only partially given by the numerical value of its
corresponding coefficient. Terms involving the
remaining coefficients and the correlation
structure between variables also influence the
association between z^ and Xj, and sometimes this

latter influence is predominant. It follows that
one cannot safely interpret the coefficients
singly. A similar argument can be made against
interpreting pairs, triples, or any subset of

66

coefficients in a canoncal function. As one might
expect, this same problem also arises in ordinary
least squares regression: the values of
individual parameters reflect the correlation
structure of the data. For complex structures,
magnitudes and even signs of coefficients are
dependent on what additional variables are
included in the model (Weiner and Dunn 1966). It
makes little sense in that case to base one' s
interpretations on individual coefficients, A
safer technique is to examine the correlation of
the canonical variate either with individual
observation variables included in the canonical
analysis (5), or with ancillary information not
included in it (Green 1971, James 1971, Dueser and
Shugart 1978). High correlations may then provide
an interpr etable "meaning" of the canonical
variate .

There is in the literature one other method
of interpretation, about which some words of
caution should be voiced. This involves the use
in canonical space of "equal frequency ellipses",
defined in multivariate populations by
hyperellipses of constant probability density
(Harner and Whitmore 1977). For a given
distribution with moments , _z . ) , a point in
observation space will be on only one such
ellipse. When g populations are defined in a
discrimination analysis, the observation lies on
ellipses which are specific to each population.
It is noted that when populations are normally
distributed these ellipses are defined by
Mahalanobis distances. They may be used to
generate a classification methodology identical
with the usual classification procedure as
outlined above.

Since the canonical variates are uncorrelated with
unit variance, the covariance y_ is in fact_lj.he
identity matrix I^. The terms (jf-p.. )l (Jl-M.- )
define equal frequency ellipses on which
classification may be based, and the corresponding
ellipses in canonical space are given by

(z-n.)'V ^(z-n.)
identical rankii

Equal covariances assure
.ngs of observations in both
observation and canonical space.

Unfortunately none of these characteristics
obtains when covariances are unequal. In
particular, the identity of rankings in canonical
and observation space no longer holds. This lack
of invariance may be indicated by

Ranking

Observation Space

Canonical

Space

1

__ 1

(Z-lL-i ) 'li'^ (l-lLi^

(z-ni)'VT^

(z-rL.)

2

_-|

(z-nj)'V-^

(z-nj)

3

Unequal covariances in observation space are
indicated by Z^. , i = 1,2,3. Covariances of the
canonical varia''tes are also unequal, since the
canonical transform generates variates which are
globally uncorrelated but conditionally
correlated. expresses these conditional

correlations. Note that the rankings of
Mahalanobis distances in canonical space are not
given a priori by their rankings in observation
space. Since the canonical transform

Ux

Several results concerning equal frequency
ellipses follow from the equal covariance
assumption. First, they intersect along straight
lines in the observation space. Second, the
optimal classification procedure based on them is
linear. These two properties are of course
equivalent, since the linear classification
functions correspond to a linear partitioning of
the observation space. Third, the relative
distances of points in canonical space as measured
by D^(z^), are identical to those in observation
space (Williams, publication submitted). This
effectively means that equal frequency ellipses in
canonical space can be used to assess the position
of an observation relative to the group means:
there is, in essence, no important "loss of
information" about conditional distributions when
canonical variates are used. Thus ecologists are
justified in using the canonical variates rather
than the original data to investigate ecological
distributions. The invariance property is
indicated below with Mahalanobis distances, using
three group means and an observation vector x:

Ranking

Observation Space

Canonical

Space

1

(jC-JJ ) ' _Z ^ (ji-ji]^ )

(l-iii)'v"^

(z-il )

2

(2^-1^2) '-"^ (1-112)

(2-112)

(2-112)

3

(jc-u.3)'l (21-^13)

(2-113) 'V"^

(2-113)

maintains rankings when covariances are equal,
this shift in rankings is a direct result of
covariance heterogeneity. This is shown by noting
that conditional covariances for transformed and
untransformed variates are related by

V. = U L. U' ,

—1 1 —

where U^ is the (g-1) x p transform matrix
generated in equation (1). Then Mahalanobis
distances are given by

,-1

D. (z) = (z - n. )' V (z - n- )

1 — — —1 — 1 - —1

= [U ( X - M^)]' (U I. U)"'' [U (x - jj.)]

(6)

The influence of U on D^(z) - Dj(z) clearly

depends on the difference between U i: . U' and U I.

1 - J

L[' , which in turn depends on the structures of

and i_y Therefore the relative magnitudes of

Mahalanobis distances in canonical space are
priori indeterminate.

A measure of the distortion induced by
covariance heterogeneity may be defined by means
of the function

67

h. .(x) = 1 if D. (x)=D .(x) or D. (z)=D .(z)
ij- i-j- i-j-

= I (D. (x)-D .(x))/(D. (z)-D .(z)) I
otherwi se ,

where D^(^) is given by equation (6) and

D. (x) = (x - u. )' X

1 — — 1

-1

(X - u.),

This function assumes only positive values, and
when covarlances are equal its value is unity for
all ^ e H. Also, as covarlances become more
heterogeneous the dispersion of its values
increases correspondingly. Now let h*(_x) be the
maximum value of 1/2(h. .(x) + 1/h. .(x)) over all i
and j: - "

demonstrably unequal should be cautiously offered,
and cautiously accepted. V/hile heterogeneity of
covariance may in fact result in no major
misinterpretations, this is by no means a
certainty. The point of this discussion is that
the canonical analysis requires specific
well-defined assumptions, violations of which may
have unforeseen and potentially serious distorting
effects. The canonical analysis then becomes an
ad hoc procedure for generating linear data
transforms which may or may not focus the
researcher's attention on meaningful
relationships. The hope that it will suggests
that there may be value in the methodology, even
when its assumptions fail. The same claim of
course can be made of any data analysis procedure,
or indeed of any activity whatever.

h»(x) =

max[1/2(h. .(x) + 1/h. .(x)):i = 1 g,

ij - ij -

j = 1...g, Uj].

Then the canonical analysis is defined to be
distortion-free (within an e-tolerance) if

1 - e < h»(x) < 1 + e

over some appropriate proportion of the sampling
universe H. Note that this condition will be met
as long as relative Mahalanobis distances are
approximately maintained in canonical space.
Since relative distances are exactly maintained
when covarlances are equal, h*(_x) = 1 for all
^ e H and the transform is completely free from
distortion ,

It should be emphasized that a canonical
analysis is distortion-free only for certain
combinations of covariance matrices , i=1...,g,
and that the property cannot be automatically
assumed. When the canonical procedure is not
distortion-free, statistical relationships between
distances in observation space and their canonical
representations are complex and non-intuitive.
Under such conditions one cannot base inferences
about observation data on statistical
characterizations in canonical space. Before the
canonical analysis can be interpreted with equal
frequency ellipses, distortion induced by
covariance heterogeneity must be assessed.

A practical consequence for ecologists is
that, contrary to the case with equal covarlances,
one cannot safely use equal frequency ellipses in
canonical space for interpreting group
differences. The inferences drawn from the
canonical analysis are not translatable back to
the observation space, because distance measures
(and the corresponding probability measures) are
distortted by the canonical transform. One
conclusion seems inescapable: the canonical
analysis, which possesses so many positive
geometric and statistical properties for
homogeneous covarlances, is fraught with problems
and ambiguities otherwise. Any recommendation to
use this methodology when covarlances are

CONCLUSIONS

This discussion has focused on a parametric
development of canonical analysis and its
relationship to the usual data-based approach
suggested by Fisher (1936). Notwithstanding the
problems associated with covariance heterogeneity,
it is fortuitous that maximum likelihood
estimation for normal populations yields the
familiar computing forms. This insures

consistent, asymptotically efficient estimators.
Nevertheless, there remain a number of problems
concerning the effects of sampling. For example,
nothing has been said about small sample
variability, rates of convergence, and possible
effects of stratified sampling designs.

Results from simulation studies, further
theoretical investigations and many different data
analyses provide a fairly bleak picture for the
use of canonical analysis as an inferential
procedure in ecology. The statistical assumptions
which insure that the canonical analysis
corresponds to posterior classification are almost
never met by ecological data. Frequency
distributions are almost always non-normal,
usually highly skewed, often bimodal , in a great
many cases discrete, and covarlances are almost
universally heterogeneous. The separate and
combined effects of these violations on the
canonical analysis are almost totally unknown.

Even when the assumptions are met, small
sample stability problems arise. Preliminary
simulation results by the author indicate
considerable instability of the canonical
coefficients, due solely to sampling variability.
This instability increases rapidly with increases
in numbers of variables and groups, and with
decreases in sample size and distances between
group means. What this means in practice is that
any pattern exhibited by the canonical
coefficients may be accidental and therefore of no
ecological consequence. The effects of sampling
variability on the canonical coefficients remain
largely unexplored, though the work of Anderson
(1963), Rao (1965, 1966) and others provides a
theoretical starting point.

68

other problems arise when the canonical
analysis is based on data gathered by a stratified
sampling design. The canonical analysis is
sensitive, to a largely unknown degree, to
relative as well as absolute sample sizes. This
means that decisions concerning relative sampling
intensities of groups may have a major impact on
the structure of the canonical functions
irrespective of the underlying probability
structure of the mixture. Such an inherent
indeterminancy can undermine any interpretation of
the analysis.

There are, in short, a number of more or less
serious problms in the application of canonical
analysis. That they remain unresolveed at the
present time is not an argument to abandon the
technique. It does suggest, however, the need for
careful planning of studies utilizing canonical
analysis, and a healthy scientific skepticism in
both the interpretation and the reporting of
results.

LITERATURE CITED

Anderson, S.H., and H.H. Shugart, Jr. 1974.
Habitat selection of breeding birds in an
east Tennessee deciduous forest. Ecology
55:828-337.

Anderson, T.W. 1963. Asymptotic theory for

principal component analysis. Annals of

Mathematical Statistics 34:122-148.
Bertin, R.I. 1977. Breeding habitats of the wood

thrush and veery. Condor 79:303-311.
Buechner, H.K., and H.D. Roth. 1974. The lek

system in Uganda kob antelope. American

Zoologist 14:145-162.
Buzas, M.A. 1967. An application of canonical

analysis as a method for comparing faunal

areas. Journal of Animal Ecology 36:563-577.
Cody, M.L. 1978. Habitat selection and

interspecific territoriality among the

sylviid warblers of England and Sweden.

Ecological Monographs 48:351-396.
Conley, W. 1976. Competition between Microtus:

a behavioral hypothesis. Ecology 57:224-237.
Conner, R.N., and C.S. Adkisson. 1976.

Discriminant function analysis: a possible

aid in determining the impact of forest

management on woodpecker nesting habitat.

Forest Science 22:122-127.
Conroy, M.J., L.W. Gysel, and E.K. Dudderer.

1979. Habitat components of clear-cut areas

for snowshoe hares in Michigan. Journal of

Wildlife Management 43:680-690.
Dueser, R.D., and H.H. Shugart, Jr. 1978.

Microhabitats in a forest-floor small-mammal

fauna. Ecology 59:89-98.
Dueser, R.D., and H.H. Shugart, Jr. 1979. Niche

pattern in forest-floor small-mammal fauna.

Ecology 60:108-118.
Fisher, R.A. 1936. The use of multiple

measurements in taxonomic problems. Annuals

Eugenics 7: 179-188.
Goldstein, R.A., and D.F. Grigal. 1972.

Definition of vegetation structure by

canonical analysis. Journal of Ecology

60:277-284.

Graybill, F.A. 1976. Theory and application of
the linear model. 704 p. Duxbury Press, N.
Scituate, Mass.

Green, R.H. 1971. A multivariate statistical
approach to the Hutchinsonian niche: bivalve
molluscs of central Canada. Ecology
52:543-556.

Green, R.H. 1974. Multivariate niche analysis
with temporally varying environmental
factors. Ecology 55:73-83.

Grigal, D.F., and R.A. Goldstein. 1971. An
integrated ordination-classification analysis
of an intensively sampled oak-hickory forest.
Journal of Ecology 59:431-492.

Harner, E.J., and R.C. Whitmore. 1977.
Multivariate measures of niche overlap using
discriminant analysis. Theoretical
Population Biology 12:21-36.

Horton, I.F., J.S. Russell, and A.W. Moore. 1968.
Multivariate-covariance and canonical
analysis: a method for selecting the most
effective discriminators in a multivariate
situation. Biometrics 24:845-858.

James, F.C. 1971. Ordinations of habitat
relationships among breeding birds. Wilson
Bulletin 83:215-236.

Kowal , R.R., M.J. Lechowicz, and M.G. Adams.
1972. The use of canonical analysis to
compare response curves in physiological
ecology. Flora 165:29-46.

MacArthur, R.H., and R. Levins. 1967. The
limiting similarity, convergence, and
divergence of coexisting species. American
Naturalist 101:377-385.

Montanucci , R.R. 1978. Discriminant analysis of
hybridization between leopard lizards,
Gambelia (Reptilia, Lacertilia, Iguanidae) .
Journal of Herptology 12:299-307.

Norris, J.M., and J. P. Barkham. 1970. A
comparison of some Cotswold beechwoods using
multiple-discriminant analysis. Journal of
Ecology 58:603-619.

Rao, C.R. 1965. Linear statistical inference and
its applications. ■ 522 p. John Wiley and
Sons , New York, N. Y.

Rao, C.R. 1966. Inference on discriminant
function coefficients, p. 587-602. In Bose,
R.C, I.M. Chakravarti, P.C. Mahalanobis,
C.R. Rao, and K.J.C. Smith, editors. Essays
in probability and statistics. University of
North Carolina Press, Chapel Hill, N.C.

Rice, J.C. 1978a. Behavioral interactions of
interspeci f ically territorial vireos. I.
Song discrimination and natural interactions.
Animal Behaviour 26:527-549.

Rice, J.C. 1978b. Behavioral interactions of
interspecif ically territorial vireos. II.
Seasonal variation in response intensity.
Animal Behaviour 26:550-561.

Rice, J.C. 1978c. Ecological relationships of
two interspeci f ically territorial vireos.
Ecology 59:526-538.

Ricklefs. R.E. 1977. A discriminant function
analysis of assemblages of fruit-eating birds
in Central America. Condor 79:228-231.

Smith, K.G. 1977. Distribution of summer birds
along a forest moisture gradient in an Ozark
watershed. Ecology 58:810-819.

69

Toussaint, G.T. 1974. Bibliography on estimation
of misclassif ication . IEEE Transactions
Inferential Theory. IT-20: 472-479.

Weiner, J.M., and O.J. Dunn. 1966. Elimination
of variates in linear discrimination
problems. Biometrics 22:268-275.

DISCUSSION

[T - Xl]v = 0

or

[R - XX] V = 0,
where is the correlation matrix based on T, and
T = B + W.

LESLIE MARCUS: Your approach requires that each
sample of the habitat correspond to a unique
species utilizing it. In some restricted cases
this is appropriate, but in a great many others it
is not. More generally there is some probability
of use by each of the species, so that a technique
like canonical correlation is more appropriate.

Clearly the relationship of axes for PCA and
DA depends on differences in structure for T, B
and W. To see how these differences may affect
interpretation of the data, consider a partition
of data into groups such that the group means lie
along the dominant axes of dispersion. In two
dimensions this situation appears as:

KEN WILLIAMS: I agree. The conceptual framework
of DA requires the ^existence of distinct
statistical populations and an unambiguous
association of sampling units to them. There is a
very large number of problems which fit this
framework, as indicated in my paper. However,
multiple use of sampling units by different
species does not. Problems of species-habitat or
community-habitat relationships for such cases
could better be handled by something like
canonical correlation. Another possibility might
be to combine correspondence analysis and multi-
variate multiple regression.

BARRY NOON: There are instances in the literature
where principal components analysis (PCA) and
discriminant analysis (DA) have been used with the
same set of data. How does one interpret
differences in ordination loadings?

KEN WILLIAMS: These two procedures focus on
different structural features of the data, and
address different questions. It is important to
recognize this, in order to avoid misinter-
pretation of the different results. Some
confusion about the relationship of PCA and DA
probably arises from their mathematical similar-
ities. Both, for example, involve an eigen-
structure analysis, and both can be used to assess
differences among groups of observations in a
space of reduced dimensionality. In the case of
PCA the space is defined by the dominant eigen-
vectors of a covariance (or correlation) matrix,
whereas for DA the space is generated from eigen-
vectors of a matrix involving group means
themselves. Nonetheless, both techniques
represent a coordinate rotation of multivariate
measurements and a reduction of their
dimensionality.

The basic difference between them is in the
way the group structure is accounted for. DA
requires well defined groups, and utilizes this
feature by means of the equation

[B - XW]u = 0.

PCA, on the other hand, uses the data matrix
without reference to structure:

Then provided \./z X. is close to unity (i.e., the
i

first eigenvector dominates the eigenstructure of
T) the principal component = £^ 21 and canonical

variate -l^ = Jl will be approximately the same

linear function. Under these conditions the
clustering of groups will be displayed in
component as well as discriminat space.

Now consider a partition which looks like:

The dominant principal component will not display
clustering of groups in this situation, since
groups overlap extensively along the dominant
axis. DA, however, displays almost total
separation of groups along the dominant canoni-
cal axis, reflecting the data partition. This
difference between principal components and the
canonical functions demonstrates that group
structure, though highlighted by appropriate
linear transforms (the canonical variates), may be
completely obscured by others (the principal
components) .

70

Thus there is no guarantee that PCA will
display important data structures; nor can one
conclude a lack of structure based on projections
of data in component space. This is not
unexpected, since PCA is designed specifically for
variance maximization rather than maximum group
separation. In fact, every linear transform of
observation data gives us a different look at the
multivariate system. What one sees depends in

large measure on the manner in which he looks,
i.e., the transform he uses. PCA provides insight
into the components of overall variation for a
system, but not necessarily its group structure.
DA gives a look at group separation, but not at
overall variation. That these two procedures
often ordinate in quite different ways therefore
should come as no surprise.

71

THEORY AND METHODS OF FACTOR ANALYSIS AND
PRINCIPAL COMPONENTS'
Helen Bhattacharyya^

Abstract. — A brief discussion is given on the historical
development of factor analysis, the kind of data for which
factor analysis is applicable, and the kind of result one may
obtain. The distinction between principal components
analysis and factor analysis is clarified, and the
relationship between the unique factors and the communalities
in the reduced correlation matrix described. A derivation of
the principal components is given, followed by a description
of the principal factor solution and interated principal
factor solution. Without mathematical details, the concept
of maximum likelihood solution is introduced. Once a direct
solution is obtained possible rotations, orthogonal and
oblique, are discussed as attempts at deriving more
conceptually meaningful common factors. The nonuniqueness of
factorization of the correlation matrix is shown. The
concept of simple structure is introduced and a graphical
representation is given illustrating the meaning of rotation
of factors. Factor scores and scoring coefficient matrix are
described .

Key words: Characteristic equation; characteristic
root; communalities; correlation matrix; factor analysis;
least squares; maximum likelihood; principal components.

INTRODUCTION

In understanding factor analysis, it helps to
know a few things that the procedure does not do.
Unlike discriminant analysis, factor analysis does
not attempt to distinguish two or more distinct
populations. There is no classification problem.
Unlike multiple regression, factor analysis does
not estimate any relationship between a set of
independent variables and one or more dependent
variables. The kind of data for which factor
analysis is applicable usually consists simply of
one sample of multivariate observations. With a

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Mathematical Statistician, USDA Forest
Service, Southeastern Forest Experiment Station,
Research Triangle Park, NC 27709.

sample size N and n variables, the raw data for
factor analysis would be of the following form:

Obs.

1
2

where ly Z^, Z^ denote the n variables

measured for each observation. An example of the

variables Z. , i = 1 n, follows:

1

Eight Physical Measurements (Harman 1968)

= Height
- Arm span

= Length of forearm

72

= Length of lower leg

= Weight
Zg = Bitrochanteric diameter
Z = Chest girth
ZL = Chest width.

The primary purpose of factor analysis is to
bring about a reduction in dimensionality, to
explore relationships among large numbers of
observed variables in an effort to find "factors"
that reduce the complexity of the situation. In
the example, the goal is to find a relatively few
underlying common factors such that the eight
measurements may be described as a linear function
of these common factors.

HISTORICAL BACKGROUND

factor pattern matrix, and an element a^^ is

called the factor loading of variable Z. on factor
Fy The unique factors are assumed to be

independent of the common factors and independent
of each other. For simplicity of exposition, the
common factors may be assumed orthogonal to each
other, although this is not strictly necessary as
will be seen in the discussion of oblique factor
rotation. A further assumption is that Z^, Fy

are all standardized to have mean 0 and unit
variance .

Referring to the above example, Z. would
denote the observed variables height, arm span,

,m,

chest width, so that n=8. The F., j=1

Spearman ^^90>^) first introduced the concept
of factor analysis in an article titled "General
intelligence, objectively determined and
measured." Early researchers in the field were
primarily concerned with finding one common
underlying factor, intelligence for example, that
would help explain an individual's performance on
a variety of different tests. Spearman called
this the two-factor theory, one common factor and
another specific to a particular test. By the
early 1930's it became evident, through the
writings of Thur stone (1931) and others, that the
two-factor theory was not sufficient to describe a
battery of psychology tests. In the 1930's, there
developed the concept of multif actor theory; that
is, there can be more than just one common factor.
The subject of factor analysis, then, is to find
the common factors and the relationship between
the observed variables and the common factors.

are the common factors. Prior to performing the
factor analysis the common factors and the value
of m are assumed unknown. Letting R denote the
observed correlation matrix corresponding to the N
n-variate observations, the problem is to find a
numerical solution for the elements of the pattern
matrix ((a^j)) which would, in some sense, best

reproduce R.

PRINCIPAL COMPONENT ANALYSIS

The use of principal components as a data
reduction technique was introduced by Pearson
(1901) and further developed by Hotelling (1933).
The principal component solution will be described
first as it avoids some of the inherent
difficulties in factor analysis.

FACTOR ANALYSIS MODEL

Principal Component Model

Let the observed variables be denoted by Z^,

Zg, .... Z^ and the common factors by F^, F^, ....

F where m < n. With the inclusion of the unique
m

factors, U^, U^,

U , the basic linear
n

relationship between variables and factors may be
written:

Let the principal components be denoted as

P P

The linear relationship between

Z. , the observed variables, and P. may be written:

^ = ^1^ ^ ^2^2 ^ ••• * ^n^n
h = ^21^ ^22^2 * ^ ^2n^n

^1 = ^l^'l * ^2^2 *

a, F + d.U,
1m m 11

.11 I
'I'

^2 = ^21^1 * ^22^2 *

+ a„ F„ + d^U^
2m m 2 2

Z =
n

^nl^l

" ^2^2 ^

+ a P
nm n

The principal components P j=1,...,m, where m=n.

Z = a ,F, + a -F„ +
n n1 1 n2 2

where a . . and d . , i= 1 ,
ij i'

+ a F + d U
nm m n n

,n, j = 1 ,

,m, are

constants. The matrix ((a^^)) is called the

are assumed to be orthogonal to each other and to

have variances Var(P^)

> Var(P2) >

> Var(P ),
— n

Note that this model differs from the factor
analysis model in that m = n and that there are no
unique factors. As in factor analysis, the
objective is to find the matrix of coefficients
((a..)).

73

Principal Component Solution
Consider the derivation for , the first

principal component. Let P^ = a'Z = a^Z^ + Si2^2 *

. . . + a Z , then
n n

Var(P^ ) = a'Ra, (1 )

where R is the covariance matrix of the Z. 's, or
equivalently the correlation matrix of the \.'s if
all Z.'s are standardized to mean 0 and unit
variance. Although standardizing the Z.'s is not
necessary in principal component analysis, this is
usually recommended especially when the variables
are measured in different scales.

dimensionality has been achieved if all principal
components are kept. However, in defining the
principal components, it is hoped that the sum of
the m characteristic roots, m < n, would nearly
equal n. The sum of the variances of Z. is
n

I Var(Z^) = n X 1 = n and the sum of the
i =1 n
variances of the pi-incipal components is I Var(P.)

n m i = 1

= Z = n. If z X. is very near n, then the

i = l i=l
components corresponding to the smaller X., i = m
+ 1, n, contribute very little to the

variation in the Z. and, hence, may be neglected.

The next step is to maximize Var(P^), subject

FACTOR ANALYSIS

to a normalizing restraint a^'a^ = 1. Using
Lagrange multiplier X and setting to 0 the partial
derivatives with respect to a^,

3 [a'Ra + X(1 - a' a)] = 0

3a

Recall the factor analysis model given
earlier ,

Z. = a. -F + a.„F + ... + a. F + d.U.,

1 ill i22 imm ii

i = 1 n.

2(R - Xl)a = 0.

(2)

For a nontrivial solution to (2) X must satisfy
the characteristic equation

R - XII = 0

(3)

The left hand side of (3) is a polynomial in X of
degree n, hence there are n solutions X^, X2.

X . These are called the characteristic roots (or
n

eigenvalues), and corresponding to each X. there
is a characteristic vector (or eigenvector) a.
such that

(R - X^Da^ = 0.

Premultiplying (2) by a' gives a'Ra - a'Xl£ = 0,
or a'Ra = X. But, a'Ra = Var(P.|) from (1).

Therefore, Var(P^) = X.^, where x.^ denotes the

largest characteristic root; and P.| = a^'I, where

a^ is the characteristic vector corresponding to

Similarly, the second principal component
^2 " — '— derived using restriction b'b = 1

and the further restriction a_^'b = 0. However,

this is not necessary. If the characteristic
roots X .| >. X2 >. ••• 2. '^n ^""^ ordered, then the ith

principal component is P. = a.'l. and Var(P.) =
X,.

One may ask what has been accomplished. The
n variables have resulted in n principal
components. No apparent reduction in

where F^ are orthogonal to each other and
independent of U^,

m

Var(Z.) = 1 = Var( I a. .F.) + Var(d.U.).
j-.l J

Let h.^ = 1 - Var(d.U.). The quantity of h.^,
1 11 ^ 1

i=1,...,n, is called the communality of Z. because

it represents the variance of Z. due to tfie common
factors Fy j=1,...,m.

In solving for the factor pattern matrix, the
object is to find that set of loadings a^j which

best reproduces the correlation matrix. Since the

variance of Z. due to the common factors F.,
1 J

2

j = 1,...,m, is h^ rather than unity, it is

2

reasonable then to substitute h. for unity in the
diagonal elements of the correlation matrix. The
result is called the reduced correlation matrix
and computationally this constitutes the basic
difference between the principal component
solution and the common factor solution. Note
that in principal component analysis the variance
of Z. due to the components is necessarily unity
since there is no provision for a unique factor.

Two commonly used methods for estimating the

2

communalities are 1) take h. to be the square of
the multiple correlation coefficient between Z^

2

and the other n-1 Z's and 2) take h^ to be the
largest (absolute value) correlation in each row.

Principal Factor Solution
Once the numerical value of each element

74

including the communalities in the reduced
correlation matrix is obtained, the solution for
the factor pattern matrix may proceed as with the
principal components solution. The characteristic
equation is formed and solved for the character-
istic roots and characteristic vectors. The
reduced correlation matrix for the example is

Z.

1

1

3 4
( symmetric)

'7

0.846 h.

^4
Z5

h

h
Z8

0.805 0.881 h.

0.859 0.826 0.801 h.

0.473 0.376 0.380 0.436 h^.

0.398 0.326 0.319 0.329 0.762 h^^

0.301 0.277 0.237 0.327 0.730 0.583 h^^

0.382 0.415 0.345 0.365 0.629 0.577 0.539 h

8J

At this point, the question that usually
arises is how many factors should be retained.
Although under certain normality conditions the
statistical significance for the number of factors
can be tested, the usual rule of thumb is to keep
only those characteristic roots of value greater
than unity. This concept is intuitively
reasonable since Var(Z^) = 1, and any common
factor which accounts for less than unit variance
would not be very helpful in achieving parsimony.

A number of other factorization methods are
available. Two other methods will be described
briefly, the iterated principal factor solution
and the maximum likelihood solution.

Iterated Principal Factor Solution

This method may be considered an improved
principal factor solution. Beginning with initial
estimates of communalities, factor loadings are
obtained by the principal factor method. Using
the factors extracted, new estimates of
communalities are computed and new factor loadings
obtained. This iterative procedure is continued
until the new communalities differ by less than a
predetermined small amount from the previous
communalities.

concepts involved are those of standard maximum
likelihood estimation.

The variables Z. , i = 1,...,n, are assumed to
1 ' ' ' '

follow an n-variate normal distribution, hence the

sample covariance matrix follows a Wishart
distribution. (Recall in the principal factor
solution no assumptions were made on the
underlying distribution of the variables Z..) The
goal is to find the expression for the loadings
which would maximize the Wishart density. No
assumptions are made about the communalities, but
it is necessary to assume the number of common
factors. This number, however, may be tested
subsequently for goodness of fit using a
likelihood ratio test. The maximum likelihood
solution for factor loadings has the usual
desirable properties of maximum likelihood
estimators, such as consistency, asymptotic
efficiency, and asymptotic normality. One
disadvantage of the method at the present time is
that it does not always converge properly
depending on the particular data set and the
computer package used.

ROTATION OF FACTORS

The ability to describe a set of n variables
in terms of m factors implies that the observed
data points essentially lie in an m-dimensional
space imbedded in the original n-dimensional
space. However, there is not a unique set of
axes, or factors, for describing this
m-dimensional space. It can be shown

algebraically that the factorization of the
correlation matrix is not unique.

The factor analysis model presented earlier
may be written as Z^ = AF if we omit the unique
factors and use A to denote the n x m pattern
matrix. Assuming the common factors to be
orthogonal and standardized, the variances and
covariances of the standardized variables

Z.

1

+ a. F , i=1 , . . . ,n , may

im m

be denoted as

Var(Z. ) = r . . = Z a. . and

Cov(Z. Z, ) = r. , = Z a. .a, .
1 k' Ik ij kj

Maximum Likelihood Solution

Although the principal factor solution was
well accepted by psychologists and social
scientists, there were mathematical statisticians
who continued trying to find a solution within the
rigorous framework of statistical inference. The
breakthrough came when Lawley (1940) developed
equations for the maximum likelihood estimation of
factor loadings. Although the derivation and
algorithms are extremely difficult, the basic

so that the correlation matrix of Z. may be
written R = AA'. If T is an orthonormal matrix
and the transformation B = AT is made, then BB' =
(AT)(AT)' = ATT'A' = AIA' = R. Factorization of R
is not unique since it may be factored into AA' as
well as BB'. The question, then, is which factor
pattern matrix should be chosen, A or B.

Intuitively, it is obvious that the factor
pattern chosen should be the one which leads to
conceptually meaningful factors. Thurstone (1935,
1917) gave the following rules which he called

75

simple structure principles for choosing the
factor pattern matrix.

1. Each row of the factor matrix should have
at least one zero.

2. If there are m common factors, each
column of the factor matrix should have
at least m zeros.

3. For every pair of columns of the factor
matrix there should be several variables
whose entries vanish in one column but
not in the other.

4. For every pair of columns of the factor
matrix, a large proportion of the
variables should have vanishing entries
in both columns when there are four or
more factors.

5. For every pair of columns of the factor
matrix, there should be only a small
number of variables with non-vanishing
entries in both columns.

The ideas embodied in these rules, simplification
of the rows and simplification of the columns,
were formalized by Carrol (1952), Kaiser (1958),
and others and led to the orthogonal and oblique
rotations commonly used today.

Quartimax Rotation

side is a constant, maximizing the first term is
equivalent to minimizing the second term.

Varimax Rotation

This orthogonal transformation attempts to
simplify the columns of the pattern matrix. If a
simple factor is defined as one with only O's and
1's in the column, then the varimax rotation
attempts to bring about this simplicity by
maximizing V^.

n 2 2 2 " 2 2

V. = 1/n I (a. . ) - l/n"" ( £ a. . ) ,

J i=l i=l

j=l, 2, m, the variance of the squared

m

loadings in each column. Maximizing I V. results

j=1 ^

in a rotation overly influenced by the size of the
communalities of the variables. In practice, this
is taken into account and the quantity actually
maximized in the varimax rotations is

P mn ^ m n

n V = n E I (a. Vh. ) - Z ( Z (a. ./h. ) ) .
J = 1 1 = 1 J = 1 1 = 1

This is an orthogonal transformation which
attempts to simplify the rows of the pattern
matrix. The quantity maximized is

Q = I la...

i=l j=l

ij

The name quartimax is applied because a^^ is

raised to the ^th power. This procedure is
equivalent to minimizing Q' where

Oblique Rotation

The criterion of simple structure remains as
with orthogonal rotations, but the method is no
longer limited to orthogonal transformations of
the factors. The use of oblique factors is
becoming increasingly popular because it affords
more flexibility in defining the underlying
factors and because computational difficulties are
no longer an obstacle with the availability of
computer packages. Harman (1968) gives an
excellent discussion of several oblique rotation
methods .

1' = Z I (a. .a. „ )

ij Hi

i = l j<s, = l

(5)

Graphical Representation

It is evident from (5) that minimizing Q'
simplifies the rows of the pattern matrix. It
will be shown now that maximizing Q is equivalent
to minimizing Q' .

Since communalities, and hence the squares of
communalities, remain constant under orthogonal
transformations, we have

m 2 2 4 ^ 2

(E a. . ) = z a. . +2 z (a. .a. ) = constant.

j = i j=i ja = i

Summing over the n variables gives

n m ^ n m 2

Z z a. . + 2 z Z (a. .a.) = constant.

i = l j = 1 i = l j<Jl = l

Since the sum of the two terms on the left hand

Thurstone's ( 1 935, 1 9^*7) simple structure
principles may be interpreted graphically as
requiring the data points to lie, to the extent
possible, on or near the reference axes defined by
the factors. Factor rotation then is an effort to
rotate the axes to achieve this goal. Figure 1
gives an illustration of orthogonal and oblique
rotations for two factors. Data points are
denoted by X's. The original factor solution is
represented by F^, F^, the rotated orthogonal

solution by G .| , G2, and the rotated oblique

solution by G.^ and G2'.

Numerical Example

The factor patterns under different rotations
for the example is given in table 1. Two factors
were retained in the initial solution. Under all

76

'rotations, variables to loaded heavily on

and variables Z^ to Z„ loaded heavily on F . An
bo

examination of these variables leads us to
conclude that Factor 1 is a measure of lankiness
and Factor 2 is a measure of stockiness.

FACTOR SCORES

Often the interest in
not end in simply being able
pattern and verbalize the
Sometimes one may wish to go
find the "observed" values,
factors; that is, to describe
of the observed variables,
for example, in carrying out
using m factors instead
variables as the independent

factor analysis does
to obtain the factor
factors extracted,
one step further and
or scores, of the
the factors in terms
This may be useful,

a regression analysis

of the original n

variables .

In principal component analysis, the
computation of the scoring coefficient matrix is
straightforward. In the model !_ = AP, the matrix
A is square and usually full rank, and the

solution for P is simply P = A~^Z^.

For the factor analysis model,

Z = AF + DU, (6)

where the matrix A is n x m and D denotes the
diagonal matrix containing the coefficient d^ of
the unique factors, a least squares regression
approach is generally used. If DIJ is the error
term, then (6) may be considered a linear
regression of Z^ on A where F are the coefficients.
From standard regression theory the least squares

solution for F is F = (A'A)~^A'Z.

Table 1. Factor patterns for example on eight physical measurements.

Figure 1. Graphical representation of factor
rotation illustrating original factor solution
(.F ,F^); rotated orthogonal solution (G^,G2);
and rotated oblique solution (G^,G'2).

COMPUTER PACKAGES

Computer programs for factor analysis may be
found in SAS (Statistical Analysis Systems 1979),
SPSS [Statistical Package for the Social Sciences
(Kim 1975)], and other statistical packages.
Typically, the programs are easy to use and only
require a few lines of code. In SAS, for example,
the following code would produce a factor analysis
using the iterated principal factor method and the
varimax rotation on variables Z^, Z^, Zg.

PROC FACTOR METHOD=PRINIT ROTATErVARIMAX SCORE;
VAR Z1 - Z8;

Factor pattern

Initial
solution

Quartimax
rotation

Varimax
rotation

Oblique
rotation

^1

0.86

-0.33

0.

90

0

20

0.

88

0.27

0.78

0

05

h

0

85

-0.41

0.

93

0

13

0

92

0

21

0.84

-0

03

s

0

81

-0.41

0.

90

0 .

10

0.

89

0.

18

0.81

-0.

05

h

0

83

-0. 34

0.

88

0

17

0.

86

0.

25

0.77

0.

02

h

0

75

0.56

0.

32

0.

88

0.

24

0.

90

-0.00

0.

83

h

0

64

0.51

0.

25

0

77

0

18

0.

79

-0. 01

0

71

0

56

0. 49

0.

20

0,

72

0.

13

0.

73

-0.06

0.

70

h

0

62

0.37

0.

31

0.66

0

25

0.

68

0.09

0

57

77

Output would include the correlation matrix,
characteristic roots, the factor pattern matrix,
the rotated factor pattern matrix, and the scoring
coefficient matrix. To create a new data set
containing the factor scores for the original
observations, the above code may be modified.

PROC FACTOR METHOD=PRINIT ROTATE=VARIMAX
SCORE OUT=COEFF; VAR Z1 - Z8;

PROC SCORE DATArOLD SCORErCOEFF OUT=NEW;

In this example OLD is the name of the original
data set containing the N observations on
variables Z^, Z^, .... Zg, and NEVJ is the name of

the newly created data set containing the N factor

scores on the factors F,, F„, .... F .

12 m

LITERATURE CITED

Carrol. J.B. 1953. An analytical solution for
approximating simple structure in factor
analysis. Psychometrika 18:23-28.

Harman, H.H. 1958. Modern factor analysis.
Second edition. 469 p. University of
Chicago Press, Chicago.

Hotelling, H. 1933. Analysis of a complex of
statistical variables into principal
components. Journal of Education Psychology
24:417-41, 498-520.

Kaiser, H.F. 1958. The varimax criterion for
analytic rotation in factor analysis.
Psychometrika 23:187-200.

Kirn, J. 1975. Factor analysis, p. 469-514. In
Nie, N.H., et al . , editors, SPSS:
statistical package for social sciences. 675
p. McGraw-Hill Co., New York. N.Y.

Lawley, D.N. 1940. The estimation of factor
loadings by the method of maximum likelihood.
Proceedings of the Royal Society of Edinburgh
60:64-82.

Pearson, K. 1901. On lines and points of closest

fit to systems of points in space.

Philosophical Magazine 6:559-72.
SAS Institute, Inc. 1979. SAS User's Guide. 494 p.

SAS Institute, Inc., Raleigh, N.C.
Spearman, C. 1904. General intelligence,

objectively determined and measured.

American Journal of Psychology 15:201-93.
Thurstone, L.L. 1931. Multiple factor analysis.

Psychology Review 38:406-27.
Thurstone, L.L. 1935. The vectors of mind.

256p. University of Chicago Press, Chicago.
Thurstone, L.L. 1947. Multiple factor analysis.

535 p. University of Chicago Press, Chicago.

DISCUSSION

JAMES DUNN: I question the statement that PCA
does not assume anything. Certainly the existence
of second-order moments, i.e., a covariance
matrix, is assumed. It seems nonsense to apply
PCA to this if the variances and covariances
depend on the mean values, i.e., as in sampling
from multivariate Poisson distribtution . But the

only distribution in which the covariance matrix
is not functionally related to the mean is
multinormal .

HELEN BHATTACHARYYA: Principal components and
factor analysis are performed on the sample
covariance or sample correlation matrix, which
always exists with any reasonable data regardless
of the existence of the population moments.
Classical factor analysis is a method of
determining the dimensionality of the data space
and does not attempt at statistical inference or
tests of hypotheses, procedures for which
assumptions on underlying distributions are
necessary. There is no disagreement on the
assumption of multivariate normal distribution
when the maximum likelihood method of
factorization is used.

As to the functional dependence of the mean
and the variance, surely the variance of a uniform
distribution on (c-a, c+a) is not functionally
dependent on the mean c.

JAMES DUNN: What is your impression of the
usefulness of oblique factor solution in habitat
analysis?

HELEN BHATTACHARYYA: Since our interest is in
extracting factors that are conceptually
meaningful rather than mathematically expedient,
there is no reason to limit ourselves only to
orthogonal factors.

KEN MORRISON: Is there a reason why you did not
mention equamax rotation?

HELEN BHATTACHARYYA: Equamax rotation is somewhat
of a compromise between quartimax rotation and
varimax rotation and may be considered very
desirable. There was no reason why it wasn't
mentioned, other than that a number of other
rotation methods were also not mentioned.

MICHAEL KINGSLEY: Principal components are known
not to be invariant against linear transforms of
data. Is factor analysis free of similar quirks?

HELEN BHATTACHARYYA: It is true that PCA when
performed on the variance covariance matrix is not
invariant under linear transformations of the
variables. Factor analysis is performend on the
correlation matrix and does remain invariant as
correlations remain invariant under linear
transformations .

JAMES SKALEY: When using PCA or factor analysis
there can be non-linear relationships in the data
set along with unequal covariances. The results
have been distorted, and we cannot assume the
underlying linear model. Are there guidelines for
the interpretation of the results from these
analysis?

78

HELEN BHATTACHARYYA: Factor analysis assumes the
observed variables may be expressed as a linear
combination of the factors. If this is not true
then the methodologies described are not
appropriate. If factor analysis is performed, I
do not believe there can be a general guideline on
interpreting the results. For a discussion on
nonlinear reduction of dimensionality see R.
Gnanadesikan (1977. Methods for statistical data
analysis of multivariate observations. John Wiley
& Sons, New York, N.Y.).

B. KEN WILLIAMS: Just a comment: there are both
scale-free and non-scale-free techniques for

deteriming factor loadings. A good description is
found in N.H. Timm (1975. Multivariate analysis
with applications in education and psychology.
Brooks-Cole, Monterey, Calif.).

JAMES HARNER: This is merely a comment. I see a
fundamental difference between principal
components analysis (PCA) and factor analysis.
PCA is simply a rotation of axes to achieve
certain criteria, whereas factor analysis is based
on a model. Also the question of measurement
scale is important in ecology. A PCA can be done
by scaling with other means than just using the
correlation matrix.

79

CANONICAL CORRELATION ANALYSIS AND ITS USE IN
WILDLIFE HABITAT STUDIES'
Kimberly G. Smith^

Abstract. — Canonical correlation is the generalized case
of multiple regression wherein one attempts to examine
interrelationships between two (or more) sets of variables
simultaneously. This is accomplished by maximizing the
correlation between a composite of variables from one set
with a composite of variables from the second set. The
statistical assumptions involved are discussed briefly and a
geometric representation of the procedure is presented.

A major problem with canonical correlation analysis has
been interpretation of results. Cross-validation aids in
detection of sample-specific covariation to which canonical
correlation is quite sensitive. The test of redundancy
offers a method whereby relationships between data sets
themselves can be examined rather than relationships between
composites of two data sets by calculating the variance of
one set that is accounted for by the second set. Selection
of variables is critically important; sample sizes should be
as large as possible; and jackknif ing estimators and use of
robust estimators may be helpful.

Canonical correlation analysis has been used most often
in the fields of education and psychology. It has met with
limited success in phytosociological studies partly because
the requirement of linear relationships between variables is
too strict. In ecological research, studies have sought to
characterize morphological and behavioral data with
environmental data. Most animal community studies that have
used canonical correlation have suffered from small sample
sizes. Canonical correlation analysis seems to hold some
promise for wildlife habitat studies, and its use should
increase as researchers become more familiar with it. The
need for planning and appropriate data collection methods are
extremely important.

Key words: Bartlett's test of significance; canonical
correlation analysis; cross-validation; jackknifed
estimators; ordination; redundancy; robust estimators.

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington Vt .

^Research Ecologist, University of
California, Bodega Marine Laboratory, P. 0. Box
247, Bodega Bay, CA 9^923

INTRODUCTION

Ecological researchers often collect data on
two sets of variables and attempt to elucidate
relationships between the sets. The two data sets
may have a cause-effect (predictor-criterion)
relationship; but, more commonly, one is
interested in relating biotic and abiotic factors,

80

e.g., the response of an animal population to a
suite of environmental factors along some
gradient. Measurements may be from only one area,
or encompass data collected simultaneously in
several areas. The available statistical options
include reporting simple product-moment
correlations (r), principal component analysis
combining the two sets of variables, or canonical
correlation analysis (Barkham and Norris 1970).
In most cases, calculating simple correlations is
not very informative, and complexity (and usually
confusion) increases with the size of the data
set. Combining the two data sets into one
principal component analysis does not maintain the
distinctiveness of the two data sets and is
therefore difficult to interpret although it does
highlight variables that covary. Canonical
correlation analysis, a multivariate statistical
technique that attempts to find maximal
correlations between two data sets, would appear
therefore to hold some promise for wildlife
habitat studies. Canonical correlation can also
be used as a multiple partial correlation
technique (Cooley and Lohnes 1971) and has been
employed with limited success as an ordination
technique (Gauch and Wentworth 1976).

Canonical correlation analysis was developed
by Hotelling (1935, 1936) to examine the
relationships between a set of mental test scores
and measurements of performance on the same test
subjects, although seeds of the technique can be
found in the works of Galton, Edgeworth and
Pearson at the turn of the century (see Bryant and
Atchley [1975] for historical development and
comprehensive bibliography) . Because calculations
involved are extremely laborious to perform by
hand, canonical correlation did not become a
widely used statistical tool until the advent of
computers and publication of a computer program by
Cooley and Lohnes (1962). Other problems that
have been associated with the use of canonical
correlation are the availability of more familiar
techniques; the difficulty of interpreting
results; and the tendency of results to be
situation-specific and not generalizable
(Thorndike and Weiss 1973). Canonical correlation
has been applied widely in psychological and
educational research, but has seen little use in
biological research, despite recommendations
(e.g., Dunn 1972, Pielou 1977).

Another reason canonical correlation may have
been overlooked in biological research is that
often biologists are more interested in explaining
variance associated with individual measurements
(e.g., through the use of principal component
analysis) whereas psychological and educational
workers have been more interested in correlational
relationships (Rao 1955). In some instances,
principal component and canonical correlation
analyses will produce similar results (e.g.,
Cassie and Michael 1968, Gauch and Wentworth
1976); however, differences in results from the
two procedures may also be of interest (e.g.,
Barkham and Norris 1970). Cassie (1969) pointed
out the great potential for establishing important
correlational relationships in biological

research, especially when it is tedious and/or
expensive to obtain one of the desired data sets.
For example, Cassie suggested it would be easier
to monitor water temperature and salinity
automatically than to collect and analyze plankton
samples. Conversely, it would be easier to
collect samples of beach infaunal organisms than
to determine particle size distributions of the
substrate at many sampling sites. The potential
value of establishing correlational relationships
in such decision-making processes as environmental
impact statements or management programs is
obvious .

CANONICAL CORRELATION ANALYSIS

Statistical Assumptions

The usual multivariate statistical
assumptions apply to canonical correlation
analysis: 1) The covariance matrix must be of full
rank (Morrison 1976), i.e., there must be more
samples than number of variables in both data sets
combined. 2) No singularities can exist within
the data matrices, i.e., all rows (or columns)
must be independent. Singularities arise when a
row (or column) of the data matrix is a linear
combination of one or more of the other rows (or
columns) (see Gauch and Wentworth 1976).
Canonical correlation analysis has no solution
when a matrix singularity is encountered. The
possibility of singularities in large sets of
distributional data may limit the usefulness of
canonical correlation as an ordinational technique
(e.g., Barkham and Norris 1970).

Two other statistical conditions are usually
associated with canonical correlation: a
multivariate normal (or multinormal) distribution
and linear relationships between variables (see
Cassie 1969). The constraint of linear
relationships between variables (inherent in any
linear model, e.g., principal component analysis)
limits the usefulness of canonical correlation as
an ordination technique for communities, such as
in phytosociological studies (Gaugh and Wentworth
1976), but should not limit use of canonical
correlation in situations where correlations along
a gradient are of interest. Canonical correlation
analysis does not require a multivariate normal
distribtuion per se (Morrison 1976), but the
closer the data approach normality and linear
relationships between variables, the more
realistic and more easily interpretable (usually)
the relationships between variables will be
(Williams 1981). Elsewhere in this volume, Dunn
(1981) discusses the use of transformations to
arrive at homogeneous variance, an attribute of a
multinormal distribution; see also Smith (1977).

Description

Canonical correlation analysis is a technique
for finding the correlation between one group of
variables, taken as a set, and a second group of
variables, also taken as a set, and in many

81

respects is like principal component analysis with
two data sets instead of one. More precisely, we
are interested in finding linear combinations of
variables in each set that have maximum
correlation. Several linear combinations of the
two sets of original variables are possible and
each pair of functions is so determined as to
maximize the new correlation, subject to the
restriction that new correlations must be
independent of previously defined ones (i.e.,
orthogonal) (Cooley and Lohnes 1962). The
following discussion draws heavily on a readable
discussion of canonical correlation by Thorndike
(1978). For other useful discussions, see Cooley
and Lohnes (1962, 1971), McKeon (1965), Blackith
and Reyment (1971), Bock (1975), Harris (1975),
and Morrison (1976). Gittens (1979) presents an
exhaustive review of the use of canonical
correlation in biological sciences, with an
emphasis on botanical research.

Definitions

An understanding of the following definitions
is important for the discussion of canonical
correlation. In all examples addressed in this
paper, analyses are concerned with two sets of
original variables , such as beach organisms and
substrate characteristics, for example. Through
the use of iterations, new values are found for
the original variables that taken together are
maximally correlated. These new values are called
canonical variates and the canonical correlation
is the maximum Pearson product-moment correlation
between variates. It is important to remember
that the canonical correlation is not between
original variables, but between variates
calculated from the variables. Weights used to
derive variates from variables are called
canonical coefficients and should not be confused
with canonical factors which are correlations
between the original variables and the variates.

Canonical correlations are invariant
statistics since, being product-moment
correlations, they are unaffected by linear
transformations of the variables in either set.
Several authors, e.g., Kettenring (1971), have
generalized canonical correlation analysis to
three or more sets of original variables, but I
know of no biological studies where more than two
data sets have been used.

Geometric model

Geometrically, canonical correlation can be
considered as a measure of the extent to which
entities occupy the same relative position in the
space defined by each of the data sets (Cooley and
Lohnes 1962). For example, Barkham and Norris
(1970) stated that their canonical correlation
study attempted to measure the extent to which
study sites occupied the same relative position in
vegetation space as in soil property space.

A geometric representation of canonical

Figure 1. Geometric representation of canonical
correlational analysis in two planes A and B.
See text for explanation, (from Thorndike 1978)

correlation is presented in figure 1. A set of
original variables, A and B, defining plane 1
comprise data set 1. Data set 2 is composed of
variables C and D and defines plane 2. This
example of two sets each with two original
variables is the simplest canonical correlation
situation; if one data set contained only one
variable, this would be a multiple regression
problem. Canonical correlation analysis will
reveal a composite of A and B, in this example a
line called 1^ (fig. 1), that is correlated with

composite of C and D, line 2^. Remembering that

the objective is to define new values of the

original variables that taken together will be

maximally correlated, if the correlation between

1 and 2 were the highest that could be found for
c c °

the data sets 1 and 2, lines 1 and 2 would be

c c

the first canonical variates. The correlations

between A and B with 1 and C and D with 2 would
c c

be vectors of canonical factors. The canonical
correlation (R) is the cosine of the angle between
1^ and 2^, dotted in figure 1. As the angle

approaches 0°, the canonical correlation
approaches 1.00. The canonical correlation is

2

usually squared and R is the proportion of
variance in the canonical variate of one set that
is accounted for by the variate of the other set.
Again, the relationship being described is between
the variates, not between the original variables
themselves .

2

R is a symmetric index in the sense that the
proportion of variance in one set accounted for in
the composite of the other set is independent of
which set is considered first. In other words,
there is no difference in the statistical
treatment of the two data sets so that one does
not have to make the distinction between a
predictor set and criterion or outcome set (Harris
1975).

Test of Significant Roots

As many canonical correlations can be found
as there are variables in the smaller set of

82

original variables. Early workers assumed that
only the first canonical correlation was
important, although this is not always the case.
More than the first canonical correlation may be
important depending on the question being asked
(Cooley and Lohnes 1962). Bartlett (1941-42)

2

developed a x test that tests the probability
that a particular canonical correlation is
significantly different from that which would have
been expected with random data. The null
hypothesis is that the canonical variates derived
from one data set are unrelated to the variates of
the second data set. Harris (1976) questioned the

2

use of Bartlett's x test, but Mendoza et al .
(1978) challenged Harris's arguments. Lawley
(1959) suggested a slight improvement for

2

Bartlett's x test of significance.

Thorndike (1978) cautioned that statistical
significance does not guarantee meaningful
biological relationships, a point which may
explain many of the interpretational problems
researchers have had with canonical correlation.

2

Also, Bartlett's x test is very sensitive to
departures from normality in data (Barkham and
Norris 1970). Blackith and Reyment (1971) comment
that biologically meaningful combinations may
still be present between data sets that have no
statistically significant canonical correlations.

Some Solutions To Problems Of Interpretation

Cross -Valid at ion

Some controversy (e.g., Barcikowski and
Stevens 1975) has arisen concerning whether one
should investigate relationships between weights
(coefficients) or correlations between original
variables and variates (canonical factors).
Coefficients are usually difficult to interpret,
if not meaningless (Cassie and Michael 1968), so
that most researchers have investigated canonical
factors (e.g., Meredith 1964). As pointed out by
Poore and Mobley (1980), examining canonical
factors has two advantages: 1) species with high
correlations with the same canonical variate are
grouped together, and 2) environmental variables
which have high correlations with the species
groupings are elucidated. Usually one would want
to consider any factor above 0.50 (e.g., Webb et
al . 1973); factors below 0.30 are probably trivial
(Cooley and Lohnes 1971). Bock (1975) has
suggested using correlations of the variates from
one set with original variables of the other set
to characterize between set relationships, but I
can find no studies that have applied this
technique .

The problem of using correlations between
original variables and variates is that
interpretation becomes dependent upon the specific
relationships between measured variables.
Situation-specific covariance, such as that

arising from the manner in which data were
collected, may become a problem so that one cannot
generalize the results to other situations. One
would, of course, like the covariance to reflect
that of the whole population, not just the subset
sampled. Thorndike and Weiss (1973) proposed a
check of canonical correlation analysis to
discover if any abnormalities in the data are a
consequence of site-specific covariance (see also
Thorndike 1978). Using this technique, called
cross-validation, the original data sets are
randomly split into two subsets, canonical
correlation is performed on one subset, and
relationships in the second subset are examined
using the weights derived from the analysis on the
first subset. If the results within both subsets
are similar, then situation-specific covariance
is assumed to be minimal. Also, double
cross-validation may be performed whereby both
subsets are submitted to canonical correlation
analysis and both are cross-checked using weights
obtained from the opposite analysis. To my
knowledge, no one has attempted this procedure
with biological data. One needs a fairly large
data set to afford the luxury of splitting the
data set in half.

Redundancy

From an interpretational point of view,
another major problem with canonical correlation

2

analysis is that R represents variance shared by
canonical variates and not variance shared between
data sets. Thus, canonical correlation cannot be
interpreted as correlations between sets of
original variables. A strong correlation may be
obtained between two canonical variates even
though these variates do not extract significant
proportions of variance from the respective sets
of original variables. To help with this problem
of interpretation, Stewart and Love (1968)
developed a measure they called redundancy,
whereby one can calculate the amount of variance
in one set of original variables that is explained
by the variate of the other data set. By
calculating redundancy for all variates of a data
set and summing the results, the proportion of
variance of one set that is accounted for by the
other set can be calculated. Since the amount of
variance of one set explained by another set does
not necessarily equal the amount of variance of
the second set explained by the first, the measure

2

of redundancy is not symmetrical (as was R ) and
must be calculated for both data sets. Moreover,
the first canonical correlation need not and often
does not have the highest redundancy. This
procedure is thoroughly discussed in Cooley and
Lohnes (1971) and Thorndike (1978).

Several biological investigators have used
the measure of redundancy recently and use should
increase as the technique becomes more well-known.
In an investigation of estuarine diatoms, Mclntire
(1978) found 41% redundancy in species data,
meaning that 41% of variation associated with the

83

26 taxa used could be accounted for by variation
in the six environmental variables measured.
Likewise, Poore and Mobley (1980) found that 48%
of variation in 22 species of marine benthic
animals sampled near a sewage treatment plant
outwash was associated with variation in nine
environmental variables they measured. Note that
in both cases, the reverse redundancy calculation,
i.e., how much of the environmental variation can
be accounted for by the variation in species, is
probably meaningless.

In studying dietary relationships among
shrubsteppe passerine birds, Rotenberry (1980)
found variation in prey size taken could be
accounted for by redundancies of only 24% in
horned lark (Eremophila alpestris) , 28% in sage
sparrow (Amphispiza belli) , and 38% in western
meadowlark (Sturna neglecta) . Grouping species
data together, redundancy was still about 30%,
leading Rotenberry to conclude that the
relationship between diet and morphology is more
general in nature than other authors have
suggested. One would, of course, have expected
high redundancy values if morphology and diet were
tightly coupled.

Selection of Variables

All too often investigators cannot interpret
the factor correlations that appear between
environmental and organismal data. Proper
selection of variables may help reduce this
problem. Variables should be chosen with some
priori knowledge that a relationship exists
between the data sets; the ease with which some
environmental variables are measured is no reason
to assume that they relate to the distribution of
organisms of interest. Also, data should be
collected specifically for use in canonical
correlation analysis rather than using canonical
correlation analysis as an ad hoc or a posteriori
approach.

In most environmental studies that have
applied canonical correlation analysis, the
situation has commonly occurred where
investigators have measured many variables, but
collected few samples. Since the

sample-to-variable ratio must be greater than 1
(the greater the better — see section after next),
which variables to exclude from analysis becomes
important. Thorndike (1978) found that

elimination of some variables in most cases did
not change the magnitude of the first canonical
correlation. He argued that for a given
magnitude, the fewer the variables, the greater
the likelihood that a canonical correlation is
attributable to real population-wide sources of
covariation, rather than to situation-specific
covariance .

Poore and Mobley (1980) tried three schemes
to reduce the number of variables in their marine
benthos study. Initially, they eliminated all
rare species (average capture < 1 per sample),
reducing the total number of species considered

from 246 to 49. For the first analysis, they
further eliminated all species with less than 118
individuals sampled which reduced the number of
species to 22. They found that this procedure
produced results that were very difficult to
interpret. For the second analysis they deleted
all species found in less than half the sampling
stations, reducing the 49 species to 18. This
procedure eliminated species found only at a few
stations, but since they were interested in the
subtle influence of sewage on species
distributions, this elimination defeated the
purpose of the study. For the third analysis,
they deleted species correlated with only 1 or 2
of 13 environmental variables measured and also
deleted any environmental variables correlated
with 6 or less of the 49 species. This left a
total of 22 species and 9 environmental variables.
This procedure gave the most easily interpretable
results since it eliminated both species and
environmental variables which were behaving
independently of the pattern they were
investigating. Care should be used in this
procedure, however, since Cassie (1969) found that
adding and subtracting variables at random
"capriciously" changed values of the corresponding
canonical coefficients. In some situations these
changes in coefficient values may be desirable,
such as in the empirical step-up and step-down
procedures discussed in Thorndike (1978).

Plot of Sample Scores

Interpretation of results of a canonical
correlation analysis can usually be aided by
plotting individual sample canonical scores in a
simple two-dimensional plot. Canonical scores are
the summation of canonical variates (data for
original variables multiplied by canonical
coefficients) and are usually available from any
canonical correlation computer program.
Standardized scores are used in most cases
(Morrison 1976:262). One score will be produced
for each sample in each data set. For example, in
the study of lizard behavior and microclimate
discussed later, James and Porter (1979) plotted
sample scores from the first canonical correlation
with microclimate on one axis and behavior on the
other (see fig. 4). In ecological research, as
pointed out by Poore and Mobley (1980), the
spatial relationship within the sampling stations
(or between data points) may be important.

Sample Size

One goal of canonical correlation analysis is
the generalization of results to other situations.
In order to reach this goal, one must have faith
that canonical coefficients and factors actually
reflect underlying relationships; stability in
estimates of weights and factors should be
expected when sample sizes vary. Monte Carlo
examinations of weights and factors (Mendoza et
al. 1978) demonstrates that the smaller the sample
size, the greater the variability in weight and
factor estimates. This argues for large sample

84

sizes. Thorndike (1978) suggests as a "rule of
thumb" that sample size should be the total number
of variables in both matrices squared, plus 100 or
so for good measure. He suggests minimal sample
size to be 10 times the total number of variables
plus 100 for good measure.

Few ecological studies have approached sample
sizes of the magnitude suggested by Thorndike.
Since individual samples in ecological research
usually have greater meaning than test results of
anonymous individuals in psychological and
educational research (Poore and Mobley 1980), a
sample size of three or four times the number of
variables is probably adequate in most instances.
The inherent problem is that as sample sizes
approach the number of variables, the canonical
correlation approachs 1.00 and statistical
significance of the first canonical correlation is
assured. Biological significance, however, should
be questioned in such situations and results based
only on canonical correlation with equal or near
equal numbers of samples and variables should be
viewed with skepticism. Although it has been
suggested (e.g.. Green 1971) that multivariate
statistics are still useful when the statistical
assumptions are relaxed if the results are
biologically meaningful (e.g., see Dueser and
Shugart 1979), I would still argue for statistical
rigor in the application of multivariate
statistics.

A case in point is the study by Herrera
(1978), in which an attempt was made to depict
differences between non-resident and resident
passerine bird species in Spain using six
measurements of avian community structure to
characterize each group. These measurements were
calculated for each month and canonical
correlation analysis was used to compare monthly
changes in community structure. With only 12
samples (1 per month) and 12 variables (6 for each
bird group), Herrera (1978) understandably found a
high canonical correlation (1.00), resulting in a
linear plot with fall, winter and spring avian
communities at one extreme and the summer
community at the other (fig. 2). A linear
ordination is indicative of a 1.00 canonical
correlation, and the high canonical correlation is
most probably an artifact of the small sample-to-
variable ratio. Herrera (1978) discussed at
length differences between resident and
non-resident canonical factors, but certainly
stability of the canonical factors, as well as the
lengthy discussion of their importance must be
questioned .

i! 0

-09

.07

-08

02,.

10

.06
.04

05

Resident species

Figure 2. Ordination of 12 monthly samples (e.g.,
01 = January) based on canonical correlation
analysis (R = 1.00) of resident and non-resident
bird communities studied in southeastern Spain.
Summer bird communities (April, May, June and
July) are separated from other monthly samples.
Results are questionable (see text) and
linearity of the ordination is due to 1.00
canonical correlation, (from Herrera 1978)

are small, jackknifing estimates is appropriate
(see Dempster 1966, Miller 197^). This is a
procedure (appendix I) whereby the estimates are
recalculated, deleting (usually) one sample from
each run. Thus, if one had 15 samples total, 15
canonical correlation runs would be made with 14
samples, each time deleting a different sample. A
new value for each estimate is calculated based on
the sum of the new runs, and a t-test can
determine if the new estimates are significantly
different from original ones obtained using all
samples (appendix I). If there is no significant
difference between original estimates and
jackknifed estimates, the confidence one places in
the estimates is greatly increased. Confidence in
results of the Herrera ( 1 978) and Aart and
Smeenk-Enseink (1975) studies would have been
greatly improved by jackknifing estimates of
canonical factors and coefficients.

Another example of the sample size problem is
the investigation of Aart and Smeenk-Enseink
(1975) into the distribution of 12 spider species
compared to 17 environmental variables. With only
29 samples, this study also found a canonical
correlation of nearly 1.00 and the plot of data
points was linear.

Several statistical options are now available
for examining stability of coefficient and factor
estimates. To reduce bias that may be associated
with the estimates, especially when sample sizes

Dempster (1966) suggested that for canonical
correlation, deletion of one degree of freedom in
each run may be superior to deleting complete
samples, which have two degrees of freedom. In
the example above, 30 new canonical correlations
would be run deleting one sample from one of the
two data sets each time, and new values for each
estimate would be calculated based on the sum of
30 runs.

To reduce the effect of multivariate outliers
(data points that do not appear to be members of

85

the multivariate population under study),
canonical correlation using robust estimators can
be helpful (Gnanadesikan 1977:127-137). True
multivariate outliers can severely affect
estimates obtained in most results. Robust
estimation decreases, to varying degrees depending
on which type of robust estimator is used, the
contribution of outlying samples to the estimation
of weights (Harner and Whitmore 1981).
Gnanadesikan (1977) cautioned that care should be
used in the application of robust estimators to
small data sets; an apparent "outlier" may in fact
only be due to inadequate sampling of the
population of interest.

APPLICATIONS OF CANONICAL CORRELATION ANALYSIS

Considering the length of time since
development of canonical correlation, the
technique has not Seen applied widely in
biological research (see review in Lee [1969]).
One of the earliest uses was by Hughes and Lindley
(1955) to investigate environmental factors in
vegetation trends. Other researchers have
attempted to use canonical correlation for
phytosociological studies (Austin 1968, Barkham
and Norris 1970) but, as mentioned earlier, the
use of canonical correlation as an ordinational
technique may be limited (Gauch and Wentworth
1976). Webb et al. (1971) found canonical
correlation useful in determining the potential of
forested areas as agricultural land. Sparling and
Williams (1978) suggested the use of canonical
correlation for analyzing bird vocalizations, but
Martindale (1980) questioned that application.

Correlations Between Morphology and Environment

In animal ecology, some of the most promising
applications of canonical correlation have
attempted to correlate morphology with some aspect
of the environment. In a series of investi-
gations, Jameson and his colleagues used canonical
correlation analysis to explore relationships
between environmental and morphological variation
in the ectothermic Pacific tree frog (Hyla
regilla) . Since Pacific tree frogs exhibit a wide
array of morphological variations which are
potentially related to climate and weather through
the need to conserve water and energy, canonical
correlation analysis was chosen as an appropriate
means of depicting the relationships. Calhoon and
Jameson (1970) examined relationships between 10
morphological variables and 16 environmental
variables that all corresponded to weather
patterns the year prior to collection of tree
frogs. Tree frogs were collected during one
spring from seven different populations in
southern California. Discovering that many
weather variables were intercorrelated , they
performed a second canonical correlation analysis
with 8 weather and 10 morphological variables and
discussed only the results of the second analysis.
Only the first canonical correlation was
statistically significant and the results were
inter pretable as a trade-off between maximizing

water conservation by large body size and
minimizing heat absorption by small body size.
Although not statistically significant, the
authors discussed the second canonical correlation
as being biologically significant.

In a similar study, Vogt and Jameson (1970)
investigated phenotypic variation in one
population of Pacific tree frogs sampled monthly
over a 3-year period near San Diego, California.
In this canonical correlation analysis, 7
morphological measurements of male tree frogs were
investigated with 21 environmental (weather)
variables covering the 2 years previous to
collection of specimens. Five canonical
correlations were significant and, taken together,
results generally confirm that milder weather
correlated with rounder frogs, while decreased
rainfall correlated with more elongated animals.
Toe length showed the strongest correlation with
weather variables suggesting that it responded
most rapidly to changes in precipitation. More
importantly, canonical correlational analysis
elucidated two significant periods in the life of
Pacific tree frogs when weather patterns are
important: the amount of rain during growth and
metamorphosis in spring and the extremes of the
high temperatures during fall.

In an expanded version of the two previous
investigations, Jameson et al. (1973) examined the
relationships of weather patterns, both short- and
long-term, to morphological variation in male,
female, and juvenile tree frogs from 14 locations
in western Oregon. As before, 10 morphological
measurements were used and these were correlated
to 19 weather variables that characterized weather
the year previous to tree frog collection. A
second analysis was performed with morphological
data using the long-term average (from available
weather stations) for the 19 weather variables.
The first analysis was termed a "weather"
relationship and the second a "climate"
relationship. The analyses were performed
separately on male, female, and juvenile data
sets. The authors concluded that there were
meaningful parallels between intraspeci f ic
variation and weather and climate. Large males
were found in areas of low summer temperatures,
but this pattern was not evident in females.
Males were more highly correlated with weather
patterns, while females were more highly
correlated with climatic patterns, a pattern
consistent with field observations that males
inhabit a more variable microhabitat (ponds).
Juvenile morphology was more closely related to
weather than climate, presumably because the
developmental period they pass through only lasts
several weeks. However, the juvenile results were
highly variable, suggesting to the authors that
there may be selection for variability of response
in offspring, a strategy that fecundity of the
tree frogs would allow. The major conclusion of
this work was that natural selection appears to
operate on the size and shape of frogs in
different directions during different life history
stages and at different localities.

86

All three tree frog studies illustrate
exemplary uses of canonical correlation analysis.
Another excellent example is by Boyce (1978) who
examined climatic variability and body size in
muskrats (Ondatra zibethicus) . Results of his
analysis are summarized in figure 3. Ten climatic
variables were compared with nine morphological
measurements and the first canonical correlation
was 0.776. (The first six canonical correlations
were significant.) Body length (BODY), total
length (TOT), condylobasal length (CB) and
zygomatic breadth (ZB) were all highly correlated
with the first morphological variate and represent
body size variation. This is not surprising since
the first canonical correlation will almost always
be a size-related axis. Longitude (LONG), annual
range of evapotr anspir ation (Rgy)» s"'^ annual

coefficient of variation of monthly evapo-
transpiration (S^^) had high correlations with the

climatic variate. Boyce interpreted these results
as showing that body size variation is positively
correlated with climatic seasonality. Discovering
this relationship supported the notion that

herbivore body size is positively correlated with
primary productivity. Boyce concluded (in part)
that the larger individuals may be favored in
seasonal environments because they can subsist
longer without food.

Another important study is that of Karr and
James (1975) in which 196 species of birds from
several continents were compared using 17
morphological measurements and 14 ecological
variables, asking the question "What are the
ecological correlates of patterns of morphological
variation?" The first six canonical correlations
were statistically significant and relationships
between morphology and ecology became apparent:
relatively longer thinner bills and smaller body
size (first correlation), relatively longer legs
and narrower bills with ground foraging (second
correlation), etc. (see Karr and James [1975:table
6] for complete listing). The authors chose to
discuss in depth the second canonical correlation
since the first correlation related primarily to
size phenomena, the results of which were somewhat
misleading due to the influcence of the
morphologically extreme hummingbirds and sunbirds.

Karr and James (1975) cautioned that care
should be used in the selection of variables and
generalization of results. They purposely
restricted analysis to primarily forest-inhabiting
bird species and did not generalize their results
beyond that. Addition of, say, water-dependent
species such as ducks and herons would have
changed the results dramatically (due to the
changes in morphology) and the scope of the study
would have changed accordingly. Occasionally
researchers have combined variables which do not
appear to have any apparent relation. For
example, DesGrandes (1978) related seven
morphological measurements of 21 hummingbird
species in Mexico with five ecological variables
in order to support his field observations that
large birds tend to be sedentary and dominant
while smaller species tend to be highly vagile and
subordinate. The canonical factors from the first
canonical correlation that demonstrate this
relationship are presented in table 1. It is
unclear what relationship "altitude" has with
other variables. Such variables should be culled
from canonical correlation analyses and only
variables that clearly relate to the same
phenomenon should be included in the respective
data sets.

Figure 3. Diagram of the first pair of canonical
variates in study of climatic variability and
body size variation in muskrats (from Boyce
1978). Definitions of important climatic
variables and morphological variables are given
in text. is the first climatic canonical

variate, is the first morphological canonical
variate, 0.776 is the canonical correlation, and
the numbers along the arrows are correlations
between original variables and canonical
variate .

Behavioral Applications

A recent paper by James and Porter (1979)
demonstrated the applicability of canonical
correlation to ecological behavioral
investigations. They were interested in relating
behavior of the African rainbow lizard (Agama
agama ) to microclimatic variables. Six
microclimatic variables were measured at hourly
intervals over several days in October, January,
and April and the behavior of a dominant male
lizard was also categorized at the same hourly
intervals. The authors used canonical correlation

87

Table 1. Canonical factors associated with first
canonical root (R = 0.93) showing relationship
between dominance and large body size of
hummingbirds contrasted with subordinance and
small body size. Data from analysis of Mexican
hummingbird communities (DesGrande^ ) .

Ecological
variables

Factors

Morphological
variables

Factors

Dominance

0

92

Length

0.

85

Altitude

-0

08

Culmen

0.

71

Status

-0.

65

Bill

width

0.

85

Territory

0.

40

Bill

depth

0,

89

Food

0.

19

Bill

curve

0.

14

Wing

disc

loading

-0.

23

Dimorphism

0.

65

^Personal communication with J-L. DesGrandes,
Canadian Wildlife Service, Quebec Region, Ste-Foy,
Quebec, Canada.

analysis to investigate relationships between
behavior and microclimate, and obtained a nice
depiction of the effect of heat load on behavior
(fig. 4). The distances between columns is a
function of the relationship between responses to
low (body flattened) and high (body vertical) heat
loads. Obtaining the same pattern in October and
April argued against a seasonal variation in these
relationships .

11-14 JANUARY

•

•

•

•

•

•

•

•

•

1

1

•

c

1

R

•

•

C

I

L

C

1

0

N

u

G

R

•

t

•

T

B

1

1

0

F

N

D

E

G

Y

E
D

0

F

1

R

L
A

T

N

G

F
1

T

G

E

H

N

T

E

0

N

G

• '1

0
• F
-F

G

•r

•0

0°
B
6
I

N

G

N

G

U
P

BEHAVIOR

Figure 4. Representation of the relationship
between microclimate and behavior of the African
rainbow lizard based on canonical correlation
analysis (R = 0.72) of data collected between
11-14 January 1971 in Ghana (from James and
Porter 1979). Ordination depicts behavioral
changes as heat load increases. Each point
represents an hourly measurement of behavior and
microclimate .

Animal Community Ecological Applications

This work is also important because it
employs two methods which improve the usefulness
of a canonical correlation analysis. First,
although only six microclimatic variables were
actually measured in the field, 17 were used in
the analysis. These additional 11 variables were
constructed by defining various interactions
between variables (see James and Porter [19791 for
details) . The technique of building interactions
into the model is used widely in multiple
regression, but few workers have considered
applying it to canonical correlation analysis.

The second improvement concerns the removal
of singularities. Since behavioral data are
mutually exclusive, i.e., a lizard cannot be doing
two behaviors at the same time, all behavioral
rows add up to the same value (1 since all data
were categorized). Therefore, the behavior
"resting" was eliminated from the analysis so that
the sum of the rows was now different. The
elimination of one column of data does not affect
the conclusions, and in this case, removed the
singularity .

Using canonical correlation analysis, Webb et
al. (1973) were able to present a land-use scheme
for nature conservation in Australia based on
vegetational characteristics and bird communities
of tropical forests. Few other studies have used
canonical correlation analysis in terrestrial
community ecology, and most that have, suffered
from small sample-to-variable ratios (e.g.,
Herrera 1978).

Use of canonical correlation analysis is
increasing in marine ecological studies, which is
understandable since specific environmental
factors enter prominently into the distribution of
many marine invertebrates. One of the first
applications was by Cassie and Michael (1968) in a
study of invertebrate distribution and sediment
size on an intertidal mud flat near Auckland, New
Zealand. They were interested in the relationship
between eight invertebrate species and nine
sediment variables sampled at 21 locations. The
analysis suffered from a small sample to variable
ratio, and the authors obtained much better
results with principal component analysis for

88

which they had 40 samples. But, realizing the
sample-to-variable ratio problem, they suggested
that canonical correlation analysis may still be
useful in larger studies.

Two recent studies have been more successful
with canonical correlation analysis. Mclntire
(1978) investigated distribution of 26
non-planktonic diatom taxa based on six
environmental variables in an estuary on the
Oregon coast. The first canonical correlation
(R=0.98) ordinated the species along a salinity
gradient. The second canonical correlation
(R=0.95) highlighted mean daily salinity range and
mean temperature, which Mclntire interpreted as a
gradient to salt tolerance (stenohaline to
euryhaline species). In Victoria, Australia,
Poore and Mobly (1980) used canonical correlation
analysis to ordinate 22 benthic invertebrate
species based on nine environmental variables
measured at 36 stations at the outwash of a sewage
treatment plant. Although the sample-to-variable
ratio was small, the authors still found useful
results. The first canonical correlation (R=0.99)
reflected a depth gradient based primarily on
amount of fine sand associated with shallow
waters. The second canonical correlation (R=0.98)
separated the middle samples from the shallow and
deep water samples (figure 5). Such a "horseshoe-
shaped" ordination is to be expected when
attempting to ordinate species distributions along
a steep gradient (e.g., see Phillips [1978]).

CV2

Figure 5. Ordination of benthic marine samples
from outwash of sewage treatment plant based on
first two canonical variates (CV1 and CV2) (from
Poore and Mobley 1980). Separation is along
depth gradient: A' = very deep samples, A =
deep samples, B = intermediate depths, and C =
shallow water samples (ellipses drawn by eye).
Placement of some sample stations outside of
ellipses (e.g., 14, 72, and 63) is due to
unusual environmental conditions at those sites
(see Poore and Mobley 1980 for details).

CONCLUSIONS

The major point that researchers who use
canonical correlation should keep in mind is that
the larger the sample-to-variable ratio, the more
stable canonical factors and weights become, so
that sample sizes should be as large as possible.
As relationships between variables approach
linearity and data sets approach a multivariate
normal distribution, results should become more
easily interpretable .

By perusing the works that have succrssfully
used canonical correlation to date, it becomes
clear that experimental design and planning are
critical. It is imperative that data be oollected
with canonical correlation in mind. All too often
application of multivariate statistics appears to
be an ad hoc or a posteriori approach rather than
the a priori approach that it should be. The
selection of variables is extremely important and
much of the past troubles with interpretation may
relate to researchers having no idea how the
environmental variables measured actually
influence the organism(s) of interest. The
problem of adequate sample size may make use of
the technique prohibitive, although this has
rarely stopped anyone yet!

With proper planning, canonical correlation
analysis can become a very useful tool for
wildlife habitat studies, and probably will become
more widely used as researchers become familiar
with it. Planning is certainly the key to
successful use and cannot be stressed too heavily.

ACKNOWLEDGMENTS

Frances James kindly read an early draft of
this paper and made many helpful suggestions
concerning terminology and explanations of
examples used in the text. Through her creative
use of multivariate statistics, including
canonical correlation analysis, she has made great
contributions in ecological research and continues
to be a leader in the field of applying
multivariate statistics to ecological problems.

Mary Barkworth pointed out many incon-
sistencies in an earl.er draft. Other comments
were made b.', Doue;1:-iS Anderson, James A. MaoMahon,
Donald Phillips, D^vid Schimpf, Jeffrey J. Short,
and Eric Zuruher. '/hile writing t s p;.per I was
supported by grants DEB 78-05328 (NS. ) to James A.
MacMahon and 03-5-022-84-NOAA to Robert W.
Riseborough .

l:^teraturf citep

Aart, P.J.N' van der, and N. Smeenk-Enserink .
1975. Correlations between distributions of
huntii.g spiders (Lycosidae, Ctenidae) and
environmental characteristics a dune area.
Netherljna. . ddl of Zool i_ ^5:1-45.

Austin, M.P. I95&. An ordinati .1 study of d
chalk jrass.and community . Journal of
Ecology 56:735-757.

89

Barcikowski , R.S.,and J. P. Stevens. 1975. A
Monte Carlo study of the stability of
canonical correl-^tions, canonical weights and
canonical var iate-var iable correlations.
Multivariate Behavorial Research 10:353-364.

Barkham, J. P., and J.M. Norris. 1970. Multi-
variate procedures in an investigation of
vegetation and soil relations of two beech
woodlands, Cotswold Hills, England. Ecology
51 :630-639.

Bartlett, M.S. 1941-42. The statistical

significance of canonical correlations.

Biometrika 32:29-37.
Blackith, R.E., and R.A. Reyment. 1971.

Multivariate morphometries. 412p. Academic

Press, New York, N.Y.
Bock, R.D. 1975. Multivariate statistical

methods in behavioral research. 623p.

McGraw-Hill, New York, N.Y.
Boyce, M.S. 1978. Climatic variability and body

size variation irt the muskrats (Ondatra

zibethicus) of North America. Oecologia

36:1-19.

Bryant, E.H., and W.R. Atchley. 1975.
Multivariate statistical methods, within-
groups covariation. Benchmark papers in
systematic and evolutionary biology 2. 436p.
Dowden, Hutchinson and Ross, Stroudsburg,
Penn .

Calhoon, R.E., and D.L. Jameson. 1970. Canonical
correlation between variation in weather and
variation in size in the Pacific tree frog,
Hyla regilla , in southern California. Copeia
1970: 124-134.

Cassie, R.M. 1969. Multivariate analysis in
ecology. Proceedings of New Zealand
Ecological Society. 16:53-57.

Cassie, R.M., and A.D. Michael. 1968. Fauna and
sediments of an intertidal mudflat: A
multivariate analysis. Journal of

Experimental Marine Biology and Ecology
2:1-23.

Cooley, W.W., and P.R.

Multivariate procedures

sciences. 21 Ip. John

York, N.Y.
Cooley, W.W. , and P.R.

Multivariate data analysis

Wiley and Sons, New York, N.Y.
Dempster, A. P. 1966. Estimation in multivariate

analysis. p. 31 5-334. Iji Krishnaiah,

P. R., editor. Multivariate analysis. 592 p.

Academic Press, New York, N.Y.
DesGrandes, J-L. 1978. Organization of a

tropical nectar feeding bird guild in a

variable environment. Living Bird

17:199-236.

Dueser, R.D., and H.H. Shugart, Jr. 1979. Niche
pattern in a forest floor small-mammal fauna.
Ecology 60: 108-1 18.

Dunn, J.E. 1972. Discussion. p. 128-130. In
Allen R.T., and F.C. James, editors. A
symposium on ecosystematics . University of
Arkansas Museum Occasional Papers No. 4,
Fayetteville , Ark.

Lohnes. 1962.
for the behavioral
Wiley and Sons, New

Lohnes .

364p.

1971 .
John

Dunn, J.E. 1981. Data-based transformations in
multivariate analysis. In Capen, D.E.,
editor. The use of multivariate statistics
in studies of wildlife habitat: Proceedings
of a workshop [Burlington, Vt . , April 23-25,
1980]. USDA Forest Service General Technical
Report. Rocky Mountain Forest and Range
Experiment Station, Fort Collins, Colo. (In
press) .

Gauch, H.G., Jr., and T.R. Wentworth. 1976.

Canonical correlation analysis as an

ordination technique. Vegetatio 33:17-12.
Gittens, R. 1979. Ecological applications of

canonical analysis, p. 309-535. In Orloci,

L., C.R. Rao, and W.M. Stiteler, editors.

Multivariate methods in ecological work.

Statistical ecology series, volume 7. 550p.

International Co-operative Publishing House,

Fairland, Md.
Gnanadesikan , R. 1977. Methods of statistical

data analysis of multivariate observations.

311p. John Wiley and Sons, New York, N.Y.
Green, R.H. 1971. A multivariate statistical

approach to the Hutchinsonian niche: bivalve

molluscs of central Canada. Ecology 52:593-

556.

Harner, E.J., and R.C. Whitmore. 1981. Robust
principal component and discriminant analysis
of two grassland bird species' habitat. l£
Capen, D.E., editor. The use of multivariate
statistics in studies of wildlife habitat:
Proceedings of a workshop [April 23-25, 1980,
Burlington, Vt . ] . USDA Forest Service
General Technical Report. Rocky Mountain
Forest and Range Experiment Station, Fort
'Collins, Colo. (In press).

Harris, R.J. 1975. A primer of multivariate
statistics. 332p. Academic Press, New York,
N.Y.

Harris, R.J. 1976. The invalidity of
partitioned-U tests in canonical correlation
and multivariate analysis of variance.
Multivariate Behavioral Research 11:353-365.

Herrera, CM. 1978. Ecological correlates of
residence and non-residence in a
Mediterranean passerine bird community.
Journal of Animal Ecology 47:871-890.

Hotelling, H. 1935. The most predictable
criterion. Journal of Educational Psychology
26:139-142.

Hotelling, H. 1936. Relations between two sets
of variates. Biometrika 28:321-377.

Hughes, R.E., and D.V. Lindley. 1955.
Application of biometric methods to problems
of classification in ecology. Nature
175:806-807.

James, F.C, and W.P. Porter. 1979. Behavior-
microclimatic relationships in the African
rainbow lizard, Agama agama . Copeia 1979:
585-593.

Jameson, D.L., J. P. Maokey, and M. Anderson.
1973. Weather, climate, and the external
morphology of Pacific tree toads. Evolution
27:285-302.

Karr, J.R., and F.C. James. 1975. Ecomorpho-
logical configurations and convergent
evolution. p. 258-291. In Cody, M.L., and
J.M. Diamond, editors. Ecology and evolution
of communities. 545p. Belknap Press of
Harvard University, Cambridge, Mass.

90

Kettenring, J.R. 1971. Canonical analysis of
several sets of variables. Biometrika
58:H33-451.

Lawley, D.M. 1959. Tests of significance in
canonical analysis. Biometrika 46:59-66.

Lee, P.J. 1959. The theory and application of
canonical trend surfaces. Journal of Geology
77:303-318.

Martindale, S. 1980. On the multivariate
analysis of avian vocalizations. Journal of
Theoretical Biology 83:107-110.

Mclntire, CD. 1 978. The distribution of
estuarine diatoms along environmental
gradients: A canonical correlation.
Estuarine and Coastal Marine Science
6:447-457.

McKeon, J.J. 1965. Canonical analysis: Some
relations between canonical correlation,
factor analysis, discriminant function
analysis, and scaling theory. Psychometric
Monographs I3.

Mendoza, J.L., V.H. Markos, and R. Gonter. 1978.
A new perspective on sequential testing
procedures in canonical analysis: A Monte
Carlo evaluation. Multivariate Behavioral
Research 13:371-382.

Meredith, W. 1954. Canonical correlations with
fallible data. Psychometrika 29:55-65.

Miller, R.G. 1974. The jackknife - a review.
Biometrika 61:1-15.

Morrison, D.F. 1976. Multivariate statistical
methods. Second edition. 415p. McGraw-Hill
Book, New York, N.Y.

Phillips, D.L. 1978. Polynomial ordination:
Field and computer simulation testing of a
new method. Vegetatio 37:129-140.

Pielou, E.C. 1977. Mathematical ecology. 385p.
John Wiley and Sons, New York, N.Y.

Poore, G.C.B., and M.C. Mobley. 1980. Canonical
correlation analysis of marine macrobenthos
survey data. Journal of Experimental Marine
Biology and Ecology 45:37-50.

Rao, C.R. 1955. Estimation and tests of
significance in factor analysis.
Psychometrika 20:93-111.

Rotenberry, J.T. 1980. Dietary relationships
among shrubsteppe passerine birds:
competition or opportunism in a variable
environment? Ecological Monographs

50:93-1 10.

Smith, K.G. 1977. Distribution of summer birds

along a forest moisture gradient in an Ozark

watershed. Ecology 58:810-819.
Sparling, D.W., and J.D. Williams. 1978.

Multivariate analysis of avian vocalizations.

Journal of Theoretical Biology 74:83-107.
Stewart, D. , and W. Love. 1968. A general

canonical correlation index. Psychological

Bulletin 70: I6O-I63.
Thorndike, R.M., 1978. Correlational procedures

for research. 340p. Gardner Press, New

York, N.Y.

Thorndike, R.M., and D.J. Weiss. 1973. A study
of the stability of canonical correlations
and canonical components. Educational and
Psychological Measurement 33:123-134.

Vogt, T., and D.L. Jameson. 1970. Chronological
correlation between change in weather and
change in morphology of the Pacific tree frog
in southern California. Copeia 1970:135-144,

Webb, L.J., J.G. Tracy, W.T. Williams, and G.N.
Lance. 1971. Prediction of agricultural
potential from intact forest vegetation.
Journal of Applied Ecology 8:99-121.

Webb, L.J., J.G. Tracey, J. Kikkawa, and W.T,
Williams. 1973. Techniques for selecting
and allocating land for nature conservation
in Australia, p. 39-52. In Costin A.B., and
R.H, Groves, editors. Nature conservation in
the Pacific. 337 p. Australian National
University Press, Canberra, Australia.

Williams, B.K. 1981. Discriminant analysis in
wildlife research: theory and applications.
In Capen, D.E., editor. The use of
multivariate statistics in studies of
wildlife habitat: Proceedings of a workshop
[April 23-25, 1980, Burlington, Vt.]. USDA
Forest Service General Technical Report.
Rocky Mountain Forest and Range Experiment
Station, Fort Collins, Colo. (In press).

APPENDIX I

Jackknife estimation
(written by James E. Dunn)

§ = usual estimate based on all data

Divide data set at random into g groups of h
observations each, e.g., typically g = n, h = 1.

Compute i.e., usual estimate with ith group

deleted, i=l,2,...,g

Compute pseudo values 0^ = g§ - (g - 1 ) § ^

i = 1,2 g

Jackknifed estimate is
g

0=1 Z 0.

i i=1 ^

p ^---^ g (0.-0)^

s = VAR[0] = I

0 i=l g-1

Test of significance: H : 0 = 0

00

0 - 0^

t = -''t/ 1 N

91

DISCUSSION

JAKE RICE: In your talk you stressed that
canonical correlation was exactly that: a
correlation method, and one does not need to
identify predictor and criterion sets of
variables. Suppose you did have some biological
reason for identifying one set of variables as
predictors and the other set as criteria variables
(as is common in simple regression studies); would
(or could) one do anything different that might
improve the findings? (As in some contexts
regression methods provide results preferable to
correlation methods on the same data set.)

KIMBERLY SMITH: James Dunn has an answer to that
question .

JAMES DUNN: If SAS PROC GLM is available, write a
MODEL statement with the criteria variables on the
left and the predictor variables on the right.
MANOVA mode will work equally well if the
predictors are continuous or categorical. In the
former, it may be necessary to arrange the MODEL
statement properly, specify Type I SS, and
accumulate the derived number of H matrices into a
composite before computing a likelihood ratio

test based on

In ( |E| )

|E + H I
' c '

or a union-intersection test based on max

ch(E~^H ) .
c

JOSEPH FOLSE: In multiple regression analysis,
inclusion of extraneous variables does not bias
other regression estimates whereas deletion of
important variables does. What effects does this
have in canonical correlations?

KIMBERLY SMITH: One way that has been suggested
of attempting to improve canonical correlations is
to eliminate variables that have low loadings (or
weights) and repeat the analysis with the reduced
model. Interestingly, Cassie (1969) has
suggested, however, that if one starts to delete
variables randomly, the relationships obtained
from canonical correlation change rather
capriciously. Thus, the same relationship appears
to hold true for canonical correlation - inclusion
of extraneous variables should not bias other
correlation estimates, whereas deletion of
important variables will.

92

DATA-BASED TRANSFORMATIONS IN MULTIVARIATE
ANALYSIS'
James E. Dunn^

Abstract. — Univariate transformations are considered
initially, because of the common practice of transforming
separately the marginal distribution of each variable of a
multivariate observation. Familiar examples include those
based on a priori assumptions about the underlying sampling
distribution, as well as several general classes of empirical
transformations recommended in a recent text by Hosteller and
Tukey. Multi-normal criteria are considered as a basis for
obtaining and evaluating multivariate transformations,
including the likelihood criterion and various
transformations to uniform statistics. The extension of
power and shi f ted-power transformations to multivariate
analysis is reviewed in detail, including recently published
work involving q-sample problems.

Key words: Multivariate; power transformation; uniform
statistics.

INTRODUCTION

At first exposure, a student without data
typically will be intimidated by the wilderness
which exists in the world of data transformations.
"How," he will ask, "can I interpret a significant
difference between means of logarithms of my
data?" With data he will seem perfectly
comfortable in comparing mean pH values (-log H ).
When life testing with an automatic termination
date, say, after 15 days, he will automatically
compare means of reciprocal response times, where
'starter' values of 1/16 are routinely assigned
for any survivors. An arbitrary error of at most
1/16 for any survivor seems perfectly acceptable
in this context. If asked to compare the means
when sampling from two populations whose variances
clearly are unequal, he will agree that any
comparison is ambiguous until a common metric (o)
is established through use of a variance

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Professor and Statistical Consultant,
Department of Mathematics and Statistics,
University of Arkansas, Fayetteville , AR 72701.

stabilizing transformation.

Even though this manuscript is directed
toward multivariate problems, univariate
transformations are considered initially because
of the common practice of transforming the
marginal distribution of each variable separately.
Even though marginal normality does not guarantee
multivariate normality, as may be demonstrated in
rather contrived examples, numerous cases have
occurred where this approach has seemed
sufficient .

Multivariate transformations are introduced
in the following section in the natural language
of matrix notation. Capital letters are used to
denote matrices (e.g., A, I, i) , lower case
letters underscored for column vectors (e.g., ^,
j^, B.) , and lower case letters in general for
scalars, either random variables or constants,
including the elements of vectors and matrices.

A natural danger in a paper of this type is
to focus on the taxonomy rather than the
systematics (ecology) of transformations. To
avoid this, I have tried to emphasize what I
consider in the process of selecting a data-based
transformation .

93

In the interest of space, two interesting
categorical response situations can only be
mentioned here. The GSK model (Grizzle, Starmer,
and Koch 1969) for analysis of categorical
response using linear models has promise for
studying multivariate relative frequency problems,
such as bird counts by species at different
locations, or stomach contents by taxonomic group.
Dunn and Cappy (1979a, 1979b) have studied the
problem of invertible transformations in this
context. The multivariate logistic model

p = [1 + exp(-6^'2()] ^ has come into focus as a
robust classifier compared to multi-normal
classification functions (cf. Efron 1975, Press
and Wilson 1978). In particular, it lends itself
to use of categorical response variables and
non-linear boundaries.

UNIVARIATE TRANSFORMATIONS

effective for Poisson counts with large rather
than small means. Familiar transformations,
including improvements suggested by Anscombe
(1948) appear in table 1. They are data-based in
the sense that they reflect our prior beliefs
about the parent population being sampled.

A Data-Based Power Transformation

Suppose that y is sampled from a population

with mean p and a^(u) = au*^. i.e the variance is
proportional to some unknown power of the mean.
Using equation (1), one obtains the result

t(p) = /y" ^''^ du

with c = a(1 - B/2)^ if B / 2

lln(y) with c = a if B = 2.

Even though the material in this section is
generally known, it is reviewed here for

2

completeness. If y >/' N(y, a ), then it is evident
from the uniqueness of the cumulant generating

function Ky(t)

yt + o t/2 that the normal

distribution is the only possible distribution in
which the mean and variance are functionally
unrelated. Since most of our familiar hypotheses

If a sample is available from each of q
populations, or if a large sample from a single
population is divided at random into q subsamples,
then one may estimate S from the sample means and

— 2

variances { (y^ , s^ ) : i = 1 , . . . ,q} as the slope of the
2 —

regression of in (s^) on an (y^). Then t = Jin(y)

or t = y^~^^^, depending on the proximity of B to
2.

tests (e.g., t, F, x t etc.) depend on the

normality assumption, finding a transformation of

non-normal data whose variance is unrelated to the

mean seems to be a logical approach to
normalization.

Variance Stabilization From A Priori Assumptions

2

Suppose that y has mean p and variance a (y).
and we require a transformation t = t(y) such that
var[t] is a constant c to some acceptable order of
approximation. Using the familiar stochastic
Taylor's series expansion of t(y) about y leads to
the well-known defining equation (cf. Rao 1952)

t(p) = ;/c/a (y)

(1)

For example, suppose that y has a Poisson

2

distribution with y = x and a (y) =
Then

-1 /2

t(x) = /c ;x dx = 2/-e /X = /X ,

provided we specify the approximate variance to be
c = 1/M. In actual fact if t = /y, then var[t] =

1/4 + 3/(32x) + (O/x^ + . . . = 1/4 + 0(1/x).
Anscombe (1948) produced a still more stable
transformation t = /y + 3/8 with var[t] = 1/4 +

2

0(1/X ). Clearly, variance stabilization is more

Mosteller-Tukey Transformations

Hosteller and Tukey (1977) approach the
problem of selecting a transformation using only
prior knowledge that certain parts of the range of
t(y) should represent a differential shrinkage or
expansion of the domain of y. Selecting a proper
transformation is visualized to be an evolutionary
process, i.e., try, look, and try again, with
interactive access to the data set assumed.
Visual display, to look for linear trend or
parallel trends, is important to see if progress
is being made in the sequence of steps. General
families of transformations suggested by Hosteller
and Tukey (1977) are summarized in terms of the
scale of measurement required. Most readers will
feel at ease with their use of 'starter' values
t(y + c) , though justification of the recommended
c = 1/6 is somewhat mysterious.

Amounts and counts. Assuming that y > 0, one

may use

t(y)

( A(y/A)P/p + (1 - 1/p)A if p /
I A iin(y/A) + A if p = 0

if a match t(y) = y is required when y = A. A
choice of p > 1 will cause expansion of the tails
of the t axis, with the converse holding for
p < 1. If the data contain zeros, then use of
t(y + 1/6) is suggested.

94

Table 1. Familiar univariate, variance-stabilizing transformations.

Distribution/Mean/ Variance

var [t]

Poisson/X/X
Binomial/np/np( 1-p)

Negative binomial/
ra/m(1 + m/k)

s (Normal population)/
a^/(2a^/(n-1 ))

/y + 3/8

sin"V(y + 3/8)/(n + 3/M)

where n must be a constant.

1/4 + 0(1/x )
(4n + 2)"''

-1

sinh"V(y + 3/8)/(k - 3/'i) , k > 2 >p(k)/4
InCy + k/2), k < 2

where k must be known ,
lin(s^)

where n must be a constant.

2/(n-1)

Counted fractions. The typical situation is
"y successes in n trials". If a started fraction
is defined to be

f = (y + 1/6)/[y + 1/6) + (n - y + 1/6)]

= (y + 1/6)/(n + 1/3),

then either the folded square root (froot)

t(y) = /TC/r - /T or the folded

logarithm (flog)

t(y) = 1/2 in f - 1/2 ind-f) = in/T-in/TT

= 1/2 {,n(y + 1/6) - 1/2 in(.n - y + 1/6)

is recommended. Note that f = 1/2 and t = 0 in
either case when y = n/2. Hence, a 50% response
corresponds to a symmetry point for t, with little
differential expansion in the neighborhood of t =
0 and considerable expansion in either tail.

Ranks . If n objects are assigned ranks
ri,...,r^, then the counted fraction trans-
formation may be applied as

t^ = 1/2 iln[(r^ - 1/3)/(n + 1 - r^ - 1/3)]

(i = 1 n) .

Alternatively, 'rankits' variously defined as
[(r. - 1/3)/(n + 1/3)] (Tukey)

t. = [(r. - 3/8)/(n + 1/4)] (Blom)(i=1 q)

* ^ [(r^/(n + 1)] (Vander Waerden)
are also commonly used in this context (cf. SAS

1979), where * is the standard normal cumulative
distribution function (c.d.f.).

Grades . Here, the observations are simply
classifed into one of q mutually exclusive, but
ordered classes a^, •••,3^. Suppose that the

observations are to be scored in terms of an
assumed standard distribution with c.d.f. F(y).
If the relative class frequencies are

q i
p., Po...-.P(, (Z Pi = 1). and s. = Z p., then
^ j = 1 -J j = 1

cut-points are defined by c„ = -<*>, c = » and c. =

0 ' q 1

F Vs^) for i = 1,...,q-1. The score to be

assigned to an observation in the i'th class is
the conditional mean of the standard distribtuion
on the interval (c^ that is

c.

y = f ^ y dF(y)/p (i = 1 q).

°i-1 ^

In the normal case with F(*) = 4i(*), this becomes

u. = [exp(-c^_^/2) - exp(-c^/2)]/p./?S.

(cf. Kendall and Stuart 1967), while for the
logistic as the standard distribution. Hosteller
and Tukey (1977) give

u. = [h(s.) - h(s._^)]/p.,

where h(s) = s iln(s) + (1 - s)iln(1 - s) . Note the
attractive feature that c^,...,c^ need not be

obtained explicitly in the latter case. In any
case, considerable time generally will be required
to complete the data receding in multivariate
applications. Cockrell (1980) will distribute (on
request) a SAS procedure RECODE which seems to fit
this application quite nicely.

95

MULTIVARIATE TRANSFORMATIONS

In a search for a class of generally useful
transformations of a univariate response y, Box
and Cox (1964) proposed the power transformation

scales of measurement are meaningful, only
coordinate dependent approaches will be considered
here [cf. Cox and Small (1978) for coordinate
independent approaches].

.(X)

(y - 1)/X for X i 0,
l!,n(y) for x = 0,

(2)

provided y > 0, or
ation

the shif ted-power transform-

.(X)

[(y + - 1]/X for X ;f 0

lin(y + i)

for X = 0,

(3)

for y + 5 > 0, where ^ either may be a starter
value supplied by the user or else treated as an
additional parameter to be estimated. Since
typical statistics such as t and F are invariant
with respect to changes in location and scale, use

of y^^^ is identical to use of y^ or (y + 5)^,
thus equivalent to the power transformation

derived above. The particular form of y^^^ simply
emphasizes the fact that Jin(y) is a continuity

(X)

point of y

at X = 0, that is

lim y^^^ = lim(y + t.n(y + O - unCy + 5)-

X-0

X-0

If y' = (yi,...y ) is a p-variate response vector,
then Andrews et al. (1971) suggested a

Likelihood Criterion

Let us consider the general q-sample (q >^ 1),
multivariate case [which includes the q-sample,
univariate case discussed by Box and Cox (1954),
and the one-sample, multivariate case treated by
Andrews et al. (1971)]. Suppose that a random
sample Yj^ •] t • • • .y^i ^ °^ size n^ is drawn
i

independently from each of q, p-variate
populations (i=1,...,q). Considering the simpler
case of the transformations defined by equation
(2), we shall want to find an estimate of the
parameter set {X^^} such that

y!^^ -r- N(u.,i;), where
-1 J -1 •

for i = l,...,q; j = 1,...,nj^; and k=l p.

(4)

The

requirement of covariance matrix stabilization,
Z ^ = . . . = = Z ( Z unspecified), as well as

normalization seems realistic in light of the
usual multivariate procedures. Following Dunn and
Tubbs ( 1980), the joint likelihood for the data
set {y.ij^ re-expressed as the concentrated

likelihood function

multivariate

extension

by

defining

(X) r (X,) (X ),, , (X. ) .

y ~ ^'^ y ^ ^p P ■' • "'^^'"^ ^

defined by equations (2) or (3) in terms of
parameter sets {\^] or {Xj^-Ci^} ^o"" ^ - ^ P*

In order to choose 'best' values for either
of the above parameter sets in any particular
application, we must specify our criteria. First,
since almost all multivariate analyses depend on
statistics which are functions of the covariance
matrix, (e.g., principal components, canonical
correlation, discriminant functions, and
multivariate classification), we shall want a
transformation to new variables whose covariance
matrix contains all of the information about
i n t e r - r e 1 a t i o n s h i p s among the variables
(independently of the mean). Again, considering
the uniqueness of the cumulant generating function
^^"^ - li'i. t'Zt/2 for y •p N(£,z), this suggests

that any criteria for the transformation be based
on a characterization of multi-normality. Two of
these possibilities are considered in the
remainder of this section. Second, since the
development of a transformation is likely to be an
iterative process, we shall require an easily
comprehended measure of our progress toward the
optimum and an assessment of our final result.
Finally, since in most applications, the original

^p)

= exp(-np/2)(n/2iT)"P^^*(Xi \ )~'^''^ . (5)

where

P . ^k"'' 2

$(Xt X ) = |G|/(n Xj^yj^ ) .

k= 1

X ^ Xp

-1 "i
^i = "i -l, ^ij '

^ "i
^k = ( .\ ."/ijk)

1/n

i=l j=l

(6)

(k = 1 p).

Dunn and Tubbs (1980) have successfully applied
the Fletcher-Powell conjugate gradient algorithm
to obtain an iterative minimization of (6) with
respect to {Xj^}f equivalent to maximizing (5),

using a FORTRAN IV program locally known as
VARSTB. Elements of the required gradient vector

96

V*(X^ , . . . ,Xp) are given in that paper.

Example. Fisher's classical iris data (1936)
consists of 50 observations from each of three
varieties of iris, virginica, versicolor, and
setosa. Each experimental unit consists of
measurements of sepal length and width and petal
length and width. This data is often used to
illustrate classification and discriminant
functions, implicitly assuming equality of the
covariance matrices. If a check is performed,

2

Bartlett's test gives x = I'^'^.O with 20 degrees
of freedom (P < 0.0001). VARSTB, on an IBM
370/155, converged to estimates = -0.43054,

= 0.51697, = 0.39843, and t. = 0.55468 in three

iterations, requiring 47.02 seconds, starting with
all exponents identically one. At convergence,
the maximum stress between successive iterations

-14

was 3.3 X 10 . Retesting the transformed

2

values, Bartlett's statistic was reduced to x =
65.2. Even though still P < 0.0001, the reduction
of Bartlett's statistic implies that the
transformations have been effective. The
necessity for transformations is substantiated by
X

testing Hq!

'1

X2 = X3 = X^ = 1, using the

generalized likelihood ratio test

X^ = n J!,n[$(1,1,1,1)/*(X^, 1^2. ^3.^4)]

= 150 s,n[1 .5497 x lO^'/l. 30735 x 10'*]

= 25.51 with 4 degrees of freedom

(P<0. 00005). The effectiveness of the transfor-
mation will be examined further in a later
section.

In summary, this procedure seems to satisfy
the three criteria very well. It is based on the
multi-normal likelihood function whose covariance
matrix is independent of the mean (homoscedastic
in the q-sample case). We may observe the
decrease of $ to its minimum in successive
iterations, each step automatically determined by
the last, and test the effectiveness of the
proposed transformations at the end. Finally, the
identities of the original variables are
maintained, since each is simply raised to some
power followed by a shift of location and a change
in scale. Jackknifing is a logical means of
examining the stability of the estimated powers as
well as improving their robustness. Available
computer time may become a limiting factor,'
however, if extensive jackknifing is attempted.

TRANSFORMATIONS TO UNIFORM (U) STATISTICS

Because of their simplistic distributional
properties, transformations to uniform (U)
statistics have universal appeal. Gnanadesikan
(1977) summarized a variety of graphical methods

for assessing marginal and joint normality, based

on approximate transformations to i.i.d. U

statistics. The exactness of his procedures will

always be. suspect, however, in that no

compensation is made for replacing true mean

vectors and covariance matrices by sample

estimates. O'Reilly and Quesenberry (1973)

demonstrated that by conditioning on sufficient

statistics, the conditional distributions of a

properly transformed subset of the original

variables are exactly i.i.d. uniform, U(0,1).

Because their results show the most promise due to

their exactness and because they are scattered

through a relatively recent literature, they are

detailed here so far as they apply to assessing

both marginal and joint normality in single and

q-sample problems. The actual transformations are

considered first, followed by a consideration of

omnibus means of assessing uniformity. Since it

will be required so frequently in the remainder of

this section, let G (•) denote the cumulative
u

distribution function (c.d.f.) of student's t
distribution with u degrees of freedom.

One-sample univariate. Suppose that

y.,...,y are i.i.d. N(y,o ), and let y = z y ./r

in j=1

r

2 — 2

and s = E (y. - y ) /r for r = 3,...,n. Then the

j=1 J *"

n - 2 random variables

1/2,

^-2 = G^.2{(^-2>"^(yr-yr)/

r/ 2 , - ,2-,1/2,

[(r-i)s^- (yj.-y^) ] }

(7)

are i.i.d. U(0,1) for r = 3....,n (O'Reilly and
Quesenberry 1973). In applications, one must
guard against any systematic arrangement of the
data (e.g., ordered from small to large) before
applying the transformation given by (7).

Q-Sample univariate ( het er oscedas t i c ) .
Suppose that a random sample y^i,...,yj^ ^ of size

' ^ 2

n. is drawn independently from each of q N(ii^,a.)
^ q ^

populations (i=1,...,q). Then a set of I (n. - 2)

i=1

i.i.d. U(0,1) random variables will result by
combining the results of applying equation (7)
separately to each sample (Quesenberry et al.
1976). The homoscedastic case will be treated by
means of an analogous result in terms of the
general linear model.

Univariate regression.

Suppose that

y^ ^ N(X^6.ia I) specifies the usual general linear

model of full rank under homoscedastic, normality
assumptions, where is a p x 1 vector of
regression coefficients to be estimated. Let

denote the r'th row and X denote the first r rows

r

97

of the overall design matrix X^, r - p+2,...,n.

Provided that X is full column rank, let
r

= ylEl - X^(X'X^)~^X' ]y^ denote the residual
r — r r r r r — r

sum of squares after fitting the model y^=X^6_+e^^.

Then the n - p - 1 random variables u
^r-p-l^^r^' where

(r-p-1 )^''^[y - x'CX'X )~''x'y ]
r — r r r r— r

r-p-1

(8)

{[1-x'(X'X )~^x ]c -[y -x'CX'X ) ^X'y ]^}^^^
— r r r — r r r — r r r r^r

are i.i.d. U(0,1) for r = p + 2,...,n (O'Reilly
and Quesenberry 1973). In applications, one must
avoid any systematic arrangement of the data while
at the same time insuring that X - is of full
rank.

y =r zy., C = Zyy' -ryy',
— r . i' r . ,—r— r — r— r'

1 = 1 1=1

A'A = C
r r r

(a Cholesky decomposition),

z = A (y -y )/[l - 1/r - (y -y )'C"^S -y ) ] ^
-r r -r -r -r -r r -r -r

= (Zri'^r2 ^rp^' ^'" = P''"2

Then the p(n-p-l) random variables given by

u = G .{z [(r-p+s-2)/

r,s r-p+s-2 rs

(1 + z

rl

z^ (10)
r ,s-l

for r = p+2,...,n and s = l,...,p are i.i.d,

U(0,1) (Rincon-Gallardo et al . 1979).

Q-sample univariate (homoscedastic) . Suppose

that a random sample y. y. of size n. is
^ •'il "i,ni 1

drawn independently from each of q, homoscedastic

2

N(u^io ) populations (i = l,...,q). Let n = i n.

i= 1

represent the total sample size. Noting that the

linear model y. . = p. + e. . for 1-way ANOVA is
ij 1 ij ^

full rank, then the previous result applies
provided that X contains at least one randomly

selected observation from each of the q
populations. If {y^^Qj^} represents a randomized

order across all s^amples of the remaining n-q-1
observations, let y^j^^ represent the sample mean

based on n,., observations from the i'th sample at
(i)

the r'th step of transformation (8) (i = l,...,q).
Let c^ be the corresponding error sum of squares

from 1-way ANOVA at the r'th step. Then the n-q-1
random variables

u^i^li = Vq-i^^-^-^)'''(yr(ij)-y(i))/

r.-, -1 ^ ^ - -,2-,1/2,

[(1-n(.))c^-(y^(. .)-y(.)) ] }

(9)

are i.i.d. U(0,1) for r = q+2 n. It will

usually be informative to reassociate the random
variables in their original sample groups once the
above transformation has been made. In
applications, the most common failure will be to
forget to mix all the samples together in some
random order before applying transformation (9).

One-sample multivariate. Suppose that

. . ,y are i.i.d. fr

-n

N(p z) population. Let

y ■]•••• 'Zn i.i.d. from a full rank, p-variate

Clearly, the multivariate, q-sample
heteroscedastic case can be treated analogously to
the univariate case using (10).

Q-sample multivariate (homoscedastic) .
Suppose that a random sample yj^i.....y^ ^ of size

' i

n^ is drawn independently from each of q,
homoscedastic, p-variate N(£^,z) populations

q

(i = l,...,q). Let n = z n.. Proceeding as in

i=l ^

the q-sample univariate case, let us select p + q
of the observations at random in such a way that
at least one observation is selected from each of
the q samples, and obtain a randomized order
^Zrdj)^ across the samples of the remaining n-p-q

observations. Let y^^ represent the sample mean

vector based on n^^^ observations at the r'th step

of the transformation, and let E be the error

r

matrix of SS and SP obtained from 1-way MANOVA at
the r'th step. If we obtain the Cholesky

decomposition A^A^ = and define

^r = v^yr(ij)-y(i)^/

_ 1 _1 1 /p

[1-n, ,-(y ,..s-y,-s)'E (y ,..^-y,.,)] ,
(i) ^ir(ij) id)"^ r ^r(ij) i(i)

= (Zi,Z-,,...,Z )' ,

r 1 ' r2 ' ' rp '

then by an extension of results given by Rincon-
Gallardo et al. ( 1 979 ), it follows that the
p(n-p-q) random variables

u^^-^^ = G ,{z [ (r-p-q+s-1 )/

r,s r-p-q+s-1 rs

(1 + z^, + ... + z^ J]^^^} (11)
rl r,s-l '

98

for r = p+q+1,...,n and s = 1,...p are i.i.d.
U(0,1). Again, it will usually be informative to
reassociate these random variables in their
original sample groups once the transformations
are completed.

Table 2. Comparison of U statistics for testing
joint normality/homoscedasticity of Fisher's
iris data with and without power transfor-
mations, using 24 random permutations of the
data .

Assessment of Uniformity

Transformation

Suppose that u ^ ^

<

"(2) ^

< u

(m)

represents the ordered values of m uniform
statistics produced by any of the foregoing
transformations. Under normality assumptions, the
expected value of u^.^^ is j/(m+l) for j = 1,...,m,

so that if we plot u^^^ vs. j/(m+1) in Cartesian

coordinates, then we should expect the points to
fall closely about the straight line defined by
g(u) = u. In q-sample problems, it will generally
be informative to assign different symbols to the
points associated with the separate samples, or
produce separate plots, in making a visual
evaluation.

More formally, any of the standard, goodness-

2

of-fit procedures (Pearson's x . Kolmogorov-Smirov
one-sample, etc.) can be applied. Miller and
Quesenberry (1975) have shown that a modified

2

VJatson's U statistic proposed by Stephens (1970)

(12 m)'

j = 1 ^
- m(u - 1/2)'

(j - 1/2)/m]'

(12)

where

'(j)

/m , has attractive power

m

= Z u,
j = 1

properties as an omnibus test of uniformity. It
has the additional advantage of having
approximately constant 10, 5, 2.5, and 1
percentage points of 0.152, 0.187, 0,221, and
0.267 respectively for all m >^ 10 under the null
assumption. The null hypothesis is rejected in

2

this case for large values of U . This test in
association with transformations given by
equations (10) or (11) provides the only known
exact test for multi-normality in either the one
or q-sample cases. The only apparent disadvantage
of these procedures is that the results are
somewhat dependent on the order in which the
observations are transformed, as v;e shall see in
the following example.

Example . In order to study the effects of
data presentation, 24 random permutations of
Fisher's iris data, both with and without power
transformations were subjected to the q-sample
transformation defined by equation (11). The
first p + q = 4 + 3 = 7 observations for each
trial consisted of three randomly chosen
versicolor and two each of virginica and setosa.

Randomization

No

Yes

0.05

2

\ U ;

2

\u )

1

0.

470

0.248

2

0 .

1 1 0

0 . 084

3

0 .

243

0.115

u.

1 OA
\d.Q

0. 031

5

0 .

354

0.130

c
0

U .

0^^ Jl

0. 128

7

0 .

083

0 . 054

Q
0

1 4 1

0.105

9

0.

603

0.217

10

0.

270

0.117

1 1

0.

057

0 . 060

12

0.

260

0.119

13

0.

254

0.095

14

0.

118

0.115

15

0.

199

0.047

16

0.

229

0.060

17

0.

215

0.059

18

0.

218

0.058

19

0.

163

0.030

20

0,

260

0. 174

21

0.

354

0.185

22

0.

072

0.063

23

0.

124

0.069

24

0.

251

0. 153

Mean

0.

231

0. 105

s.d.

0.

130

0.058

= 0.187

u2-025

0

: 0.221

"0.01

= 0.267

Table 2 summarizes the U statistics resulting
from equation (12) for testing the null hypothesis
of joint normality and homoscedasticity . Clearly,
VARSTB has had a beneficial impact, as reflected

2

by the reduction of U in every case. Figures la
-1c show plots of the order statistics from
randomization #21 for each variety before
transformations were made, while figures Id - If
give the analogous results following
transformations. It is evident that the setosa
values are most affected. Applying the one-sample
result of equation (10) to the setosa data alone

yielded U^ = 0.236 (0.01 < P < 0.025) before, and

U^ = 0.208 (0.025 < P < 0.05) after the

2

transformations. Doubtless U could be reduced
still further if VARSTB were applied to the setosa
data in isolation from the versicolor and
virginica data.

99

lICA
ABCIIA

cm

UCIKICIII
• IICU
•IC»
CKD

I 0.3 •

I '

C '

S t.l '

•.« t.l 1.1 «.« ••• •••

t.l l.< I.t I.' •.• t.t

iiraciii Mia

IICIBCb

»c<

•.I •.< >.] t.l t.l t-f

ElPfCUt fM.ll

».2 * U

^ CICt
0.1 • »
V AIKK
\ ■

«.*

o.« o.i *.a a. I *.« 0.3 *.« 0.7 *.■ o.t i.«

[irtcill f«Lll

i I.I '
I

I 0.7 '

I 1

A 1

t t.t I

I '

I \

T 1.9 I

I 1

C 1

I 1.4 <

0.1 •

l.> I.I 1.0 1.3 I.t 1.7 I.I l.l I.I
ilKCIIi HIM

0.2 0.]

0.3 O.i 0.7 o.a o.t
[irtctit !«.■

Figure 1. Plot of order statistics for Fisher's iris data: (a) original sertosa (b) original virginica
(c) original versicular (d) power transformed sertosa (e) power transformed virginica (f) power
transformed versicolor. A = 1 observation, B = 2 observations, C = 3 observations.

100

Table 3. Observed acceptance rates (a = 0.05) for Bonferroni tests of
joint normality/homoscedasticity on Fisher's iris data with and without
power transformations, based on 24 random permutations of the data.

Transformation

No Yes

Summary Trials/test Trials/test

1

2

5

1

2

5

No. of tests

24

23

20

24

23

20

No. of acceptances

9

5

6

22

22

19

Acceptance rate

0.375

0.217

0.300

0.917

0.957

0.950

A disconcerting finding in table 2 was the

2

extreme variability of U due simply to

rearrangements in the order of the data. For the

2

untransformed cases, U was observed to range from
a nonsignificant 0.057 to a highly significant
0.603. As summarized in table 3. the null
hypothesis acceptance rates (a = 0.05) were 0.375
and 0.917 respectively for the original and the
transformed data. In order to produce a test of
greater overall power in the presence of this lack
of symmetry, the logical recommendation is to
subject s random permutations of the data to the
foregoing procedures. If a signficance level of
a/s is used in each trial, and we reject the
overall hypothesis if a rejection occurs for any
trial, then the familiar Bonferroni result
guarantees an overall significance level of at
most a. Table 3 summarizes empirical acceptance
rates for the Bonferroni procedure with s = 2 and

2

s = 5 applied to the U statistics of table 2, in
the order shown. When s = 2, each successive pair

2 2

was compared to ^^^^ = 025 " 0-221 for an

overall 0.05 level test. The analogous procedure

for s = 5 employed ^^-q^/^ = "o 01 ^ 0-267. (In

actual practice, either exactly s = 2 or s = 5
permutations would be tested.) Clearly, some
increase in power is suggested by the decrease in
acceptance rates, particularly when changing from
s = 1 to s = 2 in terms of the untransformed data.

ACKNOWLEDGMENTS

The author wishes to thank Mr. Mike Meredith
for sharing his SAS macro MULFIT for the
computations involving Fisher's iris data. This
work was partially supported by the Department of
Energy Contract No. EY-76-S-05-5147.

LITERATURE CITED

Andrews, D.F., R. Gnanadesikan , and J.L. Warner.

1971. Transformations of multivariate data.

Biometrics 27:825-840.
Andrews, D.F., R. Gnanadesikan, and J.L, Warner.

1973. Methods of assessing multivariate

normality. p. 95-116 l£ Krishnaiah, P.R.,

editor. Multivariate Analysis III. Academic

Press, New York, N.Y.
Anscombe, F.J. 1948. The transformation of

Poisson, binomial, and negative binomial

data. Biometrika 35:246-252.
Box, G.E.P., and D.R. Cox. 1964. An analysis of

transformations. Journal of the Royal

Statistical Society (B) 26:211-252.
Cockrell, D. 1980. SAS PROCEDURE RECODE.

Computer Services Division, University of

South Carolina. (Private communication.)
Cox, D.R., and N.J.H. Small. 1978. Testing

multivariate normality. Biometrika

65:263-272.

Dunn, J.E., and G. Cappy. 1979a. A class of
invertible functions for analysis of
categorical response using linear models,
p. 182-187. In Fourth Annual SAS Users' Group
International Conference. SAS Institute,
Raleigh, N.C.

Dunn, J.E., and G. Cappy. 1979b. A class of
invertible functions for analysis of
categorical response using linear models.
University of Arkansas Statistical Laboratory
Technical Report No. 11.

Dunn, J.E., and J. Tubbs. 1980. VARSTB: A
procedure for determining homoscedastic
transformations of multivariate normal
populations. Communications in Statistics
Simulation and Computation B9(6) :589-598.

Efron, B. 1975. The efficiency of logistic
regression compared to normal discriminant
analysis. Journal of American Statistical
Association 70:892-898.

Gnanadesikan, R. 1977. Methods for statistical
data analysis of multivariate observations.
311 p. John V/iley and Sons, New York, N.Y.

Grizzle, J.E., C.F. Starmer, and G.G. Koch. 1969.
Analysis of categorical data by linear
models. Biometrics 25:489-504.

101

Kendall, M.G., and A. Stuart. 1967. The advanced
theory of statistics II. 437 p. Hafner
Publishing Co., New York, N.Y.

Miller, F.L., and C. Quesenberry. 1975.
Statistics for testing uniformity on the unit
interval. Technical Report UNCCNC-CSD-12,
Union Carbide Corporation, Nuclear Division.

Mosteller, F., and J. Tukey. 1977. Data analysis
and regression. 588 p. Addison-Weley
Publishing Company, Reading, Mass.

O'Reilly, F, , and CP. Quesenberry. 1973. The
conditional probability integral

transformation and applications to obtain
composite chi-square goodness-of-f it tests.
Annals of Statistics 1:74-83.

Press, S.J., and S. Wilson. 1978. Choosing
between logistic regression and discriminant
analysis. Journal of American Statistical
Association 73:699-705.

Quesenberry, CP., T.B. Whitaker, and J.W.
Dickens. 1976. On testing normality using
several samples: an analysis of peanut
aflatoxin data. Biometrics 32:753-759.

Rao, CR. 1952. Advanced statistical methods in
biometric research. 390 p. John Wiley and
Sons, New York, N.Y.

Rincon-Gallardo, S., CP. Quesenberry, and F.J.
O'Reilly. 1979. Conditional probability
integral transformations and goodness-of-f it
tests -for multivariate normal distributions.
Annals of Statistics 7:1052-1057.

SAS Institute. 1979. SAS users' guide. 494 p.
Raleigh, N. Car.

Stephens, M.A. 1979. Use of the Kolmogorov-
Smirov, Cramer-von Mises and related
statistics without extensive tables. Journal
of Royal Statistical Society (B) 32:115-122.

102

Applications: Ecological Theory, Habitat Management,
Inventory

103

MULTIVARIATE ANALYSIS OF NICHE, HABITAT, AND ECOTOPE'
Andrew B. Carey^

Abstract. — Comprehension of the components of a species'
response to an environmental complex can be achieved best by
partitioning the full range of environmental factors
potentially affecting the species (the ecotope hyperspace)
into intercommunity factors (the habitat hyperspace) and
intracommunity factors (the niche hyperspace). Hyperspaces,
and the parts of hyperspaces occupied by a species (hyper-
volumes), are defined by multidimensional coordinates.
Reduction in the number of dimensions of these hyperspaces
and hypervolumes to increase comprehension of a species'
response to them can be accomplished through multivariate
analyses. The analysis of an ecosystem on a south-facing
slope of the montane zone in Rocky Mountain National Park,
Colorado is presented as an example. Principal component
analysis was used to determine the major gradients in a
habitat hyperspace defined by 21 environmental variables.
Five principal components were interpreted as gradients of
soil depth, soil moisture, ground cover, mammal distribution,
and shrub abundance. Responses of five species of rodents to
these gradients were determined by examining their relative
positions on the principal components. Stepwise discriminant
analysis (DA) was used to mathematically describe habitat and
ecotope hypervolumes of the mammals. Comparison of the major
determinants of each hypervolume of each species clarified
the niche of the species. Furthermore, the distribution of a
major ectoparasite of the mammals was analyzed by DA of
environmental variables. Three levels of abundance of the
wood tick Dermacentor andersoni could be predicted using only
five environmental variables. The presence of a virus
infecting both the ticks and mammals could be determined
using seven environmental variables.

Key words: Discriminant analysis, ecotope, Eutamias ,
habitat, niche, principal component analysis, Spermophilus .

INTRODUCTION

Understanding the relationships between a
species and its environment is the basic premise
for wildlife management; to further this

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Research Wildlife Biologist, Northeastern
Forest Experiment Station, USDA Forest Service,
Morgantown, WV 26505.

understanding is the foremost goal of wildlife
research (Sanderson et al. 1979). Wildlife
management and wildlife research have progressed
to the point where precise and mathematical
definition of basic terminology is necessary for
further rapid progress. The response of a species
(or ecotype) to its environment is usually
examined in one of two contexts: the habitat (its
response to extensive features) or the niche (its
response to local features). Unfortunately, the
communication of research results depends on the
assumption of mutual understanding of terminology
rather than on a basis of precise definition or an

104

organized system of mathematical descriptions.
Haskell (1940) pointed out that this assumption is
characteristic of a poorly advanced science.

Development of multidimensional treatments of
habitat and niche (Hutchinson 1958) was concurrent
with the adaptation of multivariate statistical
methods to ecology (Hughes and Lindley 1955).
These methods made practical the application of
multidimensional thinking to field studies of
species-environment relationships (Cody 1968,
Green 1971, Hespenheide 1971, James 1971, Shugart
and Patten 1972).

Despite these developments, there is still no
unanimity on precise definitions of habitat and
niche. Most ecology texts refer to habitat as the
"address" and niche as the "profession" of an
animal in its environment and later, in a
discoursive context, refer to Hutchinson's
n-dimensional hypervolume (Odum 1971, 1975;
Kendeigh 1974; Smith 1974, 1977; Richardson 1977;
Brewer 1979; McNaughton and Wolf 1979).
Vandermeer (1972) pointed out that such
definitions are excessively vague and inadequate.
Whittaker et al. (1973) stated that the two terms,
habitat and niche — perhaps the two most important
in ecology — are among the most confused in usuage,
and that their unsystematic usage has led to
further confusion of other terms and concepts.

A Landscape. . . Made Up 01 Blotopes. . .Containing Communities
BIOTOPE

DOUGLAS- FIR

HABITAT
VARIABLES

-extensive

-across communities

-often scenopoetic

-may Include :
elevation
slope exposure
soli fertility
communities

PONDEROSA PINE-
ANTELOPE

BITTERBRUSH

ECOTOPE
VARIABLES

the full range
of

external factors
affecting
tfie species

N ICHE
VARIABLES

- intensive
-witiiin a community

- often bionomic
-may include^

food

other species

foliage height diversity

den sites

BiOTOPES; Homogeneously diverse spaces
that contain recognizable communities...
defined relative to the range of the
motile organism under consideration.

Nudds (1979) explained the need for applied
wildlife research in a theoretical context, and
Sanderson et al . ( 1979) reminded us that one of
our major reasons for reporting wildlife research
is to transfer that information from the research
community to the management community. The need
for clarity in interpretation and reports of
research results and for some degree of unanimity
on terminology is apparent. Whittaker et al .
(1973, 1975) proposed a precise terminology (fig.
1) and found support in Hutchinson (1978).
Despite objections to this terminology (Kulesza
1975, Rejmanek and Jenik 1975), I believe it
offers the precise definition required to maintain
a desirable level of clarity in the presentation
of species-environment relationships. In this
paper I will first discuss the terminology of
Whittaker et al. (1973) with some additions from
Hutchinson (1978), and then illustrate how it can
be applied to a field study of a complex
ecosystem. Finally, I will discuss the utility of
the terminology in wildlife research and in
management .

NICHE, HABITAT, AND ECOTOPE

Landscapes are characterized by spatial
gradients of structural (stage-setting or sceno-
poetic) variables (e.g., elevation, slope, soil
fertility), and are composed of biotopes —
locations (physical spaces) that have convenient
arbitrary upper and lower boundaries, and that are
horizontally homogeneously diverse (their
structural elements are small compared with the
range of an individual) relative to the larger
motile organisms within them. These biotopes in

Figure 1. A suggested terminology.

the landscape pattern contain recognizable
communities (fig. 1). Biological communities,
however defined, are vaguely bound in space and
time (Inger and Colwell 1977). The biotic context
of a population is its community — an association
of coexisting populations bound functionally by
their interactions and spatially by their
co-occurrence in a biotope (Colwell and Fuentes
1975).

The environmental variables with extensive
spatial components are intercommunity or habitat
variables ; axes derived from these variables
describe a multidimensional habitat hyperspace.
Each species in the landscape occurs over some
range of the habitat variables; the limits of
these ranges define a habitat hypervolume — a
fraction of the habitat hyperspace where a species
occurs. Multivariate analyses can be used to
reduce the number of dimensions defining
hyperspaces and hypervolumes , and this allows
comparison of the habitat relationships of the
many species populations found in the habitat
hyperspace. The response of a species population
to habitat variables within its hypervolume
describes its habitat; Maguire (1973) provided a
method for examining such responses in terms of
isopleths of population parameters projected on
axes representing the reduced dimensions.

The intracommunity or niche variables
(intensive or local environmental variables, e.g.,
height above ground, seasonal time, prey size,
"microhabitat" variables) likewise define a niche

105

hyperspace that interrelates the species of a
community. The niche hyperspace is analagous to
the fundamental or preinteractive Hutchinsonian
niche. Each species in the community

differentially utilizes, occurs in, or is affected
by some range of these variables; limits of these
ranges define the niche hypervolume — a fraction of
the niche hyperspace where the species occurs.
The niche hypervolume is analagous to the realized
or postinteractive Hutchinsonian niche including
the "included niche" of Miller (1964). Niche
variables are generally bionomic variables
(resources for which there may be competition) and
may define axes representing other member species
of the community. The dimensions of niche
hyperspaces and hypervolumes can be reduced
through multivariate analyses, and isopleths of
the population response to the niche variables
(the niche) can be projected on them (Maguire
1973).

The landscape of communities (the compound
hyperspace that represents the full range of
external circumstances to which species in the
landscape are adapted) is the ecotope hyperspace
and contains an ecotope hypervolume defined by a
species' limits on the ranges of the ecotope
variables. The ecotope represents the full range
of a species' adaptation to external factors and
is the ultimate arena for consideration of its
relations to its environment (especially in a
broad evolutionary context) , whereas the niche
focuses on the role of the species within its
community (especially competitive interactions),
and the habitat relates to the distributional
response of the species to the intercommunity
environmental factors (those with extensive
spatial components) . Niche differences are
intracommunity differences and involve genetic
characteristics evolved in relation to other
species; habitat differences reflect the
evolutionary response to a gradient of
environmental factors external to, although often
modified by, the community. Niche breadth
measures intrapopulation genetic characteristics;
habitat breadth measures interpopulation
differentiation based on mechanisms different from
those responsible for niche breadth, for example,
ecotypic and subspecific differentiation.

It can be seen readily that most wildlife
management efforts do deal with habitat variables;
however, the strength of the concepts is that they
provide a conceptual scheme for different kinds of
diversity (Hutchinson 1978) and, with increased
interest in managing diversity (Pimlott 1969i
Siderits and Radtke 1977), the wildlife biologist
may want to deal with niche differences (within
habitat or alpha-diversity) , habitat differences
(between habitats or beta-diversity), or with
broad geographical differences (gamma-diversity).

The basic principle underlying multivariate
analyses of species-environment relationships is
the determination of points (limits of ranges of
niche, habitat, or ecotope variables) in
multidimensional space followed by the
mathematical reduction of the number of dimensions

so that those remaining are all orthogonal,
independent, and significant (Hutchinson 1978).

AN ILLUSTRATION: A MONTANE ECOSYSTEM

The landscape chosen for study (Carey et al.
1980) included two south-facing slopes of the
upper montane forest climax region in the Rocky
Mountain National Park, near Estes Park, Colorado.
The areas encompassed about 20 ha of varied
terrain, ranging from steep 030°) slopes with
massive rock outcrops to gentle (2-8° ) slopes with
deep soil. Elevation was between 2,487 and 2,585
m, and the areas were representative of the upper
montane, and contained all major plant communities
found there: aspen stand complex, ponderosa pine
complex, big sagebrush complex, and dry montane
grassland complex (Marr 1967). Five species of
rodents were abundant on the areas; in order of
decreasing abundance they were deer mice
(Peromyscus maniculatus) , Richardson's ground
squirrels (Spermophilus richardsonii ) , golden-
mantled squirrels (S. lateralis) , least chipmunks
( Eutamias minimus ) , and Uinta chipmunks (E.
umbrinus ) .

Methods

Sampling grids were surveyed on the two areas
and consisted of 269 intersections (marked by
stakes) 30 m apart (13x13 stakes on one area,
10x10 stakes on the other). A number of
structural variables (table 1) were measured on
the 225 30m x 30m quadrats covering the 20 ha.
Seven plant-frequency counts were taken from 20cm
X 20cm quadrats randomly placed in the vicinity of
each stake. Four traps (for small mammals) were
placed in the vicinity of each stake; these were
operated for 46,000 trap nights. Fecal samples
were collected from droppings beneath traps
containing a chipmunk or ground squirrel. Fecal
samples were analyzed by the microhistological
method (Sparks and Malechek 1968) to obtain the
dietaries of the species populations. Blood
samples for Colorado tick fever virus isolation
attempts were collected from a subsample of the
captured rodents. Traps for adult, free-ranging
ticks were placed in the center of 128 of the 225
quadrats and were operated for 1,200 trap nights.
For detailed procedures see Carey et al . ( 1980).

Statistical Analyses

Structural variables that were highly
correlated (>0.90) with simpler (not
transgenerated or more easily measured) variables
or that were invariant (e.g., DIRNSL) were deleted
from the data set. The remaining variables were
averaged over the sets of four contiguous quadrats
surrounding each point location (stake) in the
landscape. These variables were taken to
represent habitat variables, and mean values for
each point location were paired with values of the
rodent-capture variables for that point location
(total captures of the five most abundant

106

Table 1. Structural variables measured on 30m x
30m quadrats or transgenerated from measured
variables .

Mammal-capture variables were included in the
analysis to obtain a measure of the habitat, or
response, of each species to the hyperspace
gradients .

Acronym

Structural variable

SOILD:

XS0IL2:

XEXPRK:

RKRK:

DSLOPE:
DIRNSL:
GRASSL:
PINEL:

WDDEBR:

LNFTLG:

DECOMP:

SMPINE:
LGPINE:
NASPEN:
%SHRUB:
PUTR-P:

RICE-P:
ARTR-P:

NJUSC:
DJUCO:
ARLU-RK:

ARFR-RK:

ROAC-RK:
OPPO-RK:

PESI-RK:

CHVI-P:

EXPRKRK:

SHPUTR:

SHRICE:

SHARTR:

SLPEXR:

TOPINE:

LNFTDC:

15 cm dbh
15 cm dbh

mean depth of soil exclusive of areas
with < 5 cm of soil

percent of quadrat with soil < 5 cm in
depth

percent of quadrat covered by exposed
rock

rank (0-4)' of exposed rock for

interstices

degree of slope

aspect (compass degrees)

ranked abundance (0-4) of grass litter

ranked abundance (0-4) of pine needles

and cones

ranked abundance (0-4) of wood debris

less than 15 cm in diameter

length of logs > 15 cm diameter

multiplied by diameter

rank (0-4) of decomposition of log

litter

number of coniferous trees <
number of coniferous trees >
number of quaking aspen
percent of quadrat covered by shrubs
presence-absence (1,0) of antelope
bitterbr ush

presence-absence (1,0) of wax currant
presence-absence (1,0) of basin big
sagebrush

number of Rocky Mountain junipers
diameter (cm) of common junipers
ranked abundance (0-4) of Louisiana
sagewort

ranked abundance (0-4) of fringed
sagewort

ranked abundance (0-4) of prickly rose
ranked abundance (0-4) of plains
pr icklypear

ranked abundance (0-4) of mountain
ball cactus

presence-absence (1,0) of Douglas
rabbitbr ush
KXPRK X RKRK

XSHRUB
?SHRUB
XSHRUB
DSLOPE
SMPINE
LNFTLG

PUTR-P
RICE-P
ARTR-P
XEXPRK
LGPINE
DECOMP

Point locations were grouped into species'
ecotope and habitat hypervolumes and remaining
hyperspaces according to the presence of each
rodent species except for Richardson's ground
squirrels and deer mice. The criterion for
grouping stations in terms of Richardson's ground
squirrels was that two or more ground squirrels
had been captured near the point location. This
criterion was used because single captures were
scattered, and multiple-capture locations were
clumped in distribution. The scattered captures
presumably resulted from breeding season movements
and dispersal behavior (Hansen 1962, Yeaton 1972,
Michener and Michener 1977) and therefore, did not
reflect habitat. Deer mice were trapped at
virtually all locations.

The habitat hypervolumes of chipmunks and
ground squirrels were mathematically described by
stepwise discriminant analysis (DA) (Dixon 1976)
of 21 of the habitat ( scenopoetic) variables. The
ecotope hypervolumes of the sciurids were
similarly described by stepwise discriminant
analysis of 9 habitat variables, 11 plant-
frequency variables representing the distribution
of the major food items of sciurids, and 4
mammal-capture variables (the four abundant
species of rodents other than the species under
consideration); these 24 variables collectively
constitute ecotope variables. Results of the
stepwise procedures (the best discriminating
variables are chosen first) for the habitat and
ecotope hypervolumes of a single species were
compared to determine if the species was
responding to habitat (scenopoetic) variables or
to niche (bionomic) variables.

Distributions of adult ticks and virus in the
landscape were described mathematically by
discriminant analysis of structural variables
(Carey 1979). The purpose of this analysis was to
determine if enough information was contained in
the structural variables to allow DA to generate
discriminant functions (DF) that would be useful
in classifying other parts of the landscape into
categories of relative abundance of ticks and
virus. A jackknife procedure (Brown 1977) was
used to obtain less biased estimates of the error
of classification of these discriminant functions.

Results

'Rank (0-4): Absent, 0; sparse, 1; scattered, 2;
common, 3; abundant, 4.

species). Principal component analysis (PCA)
(Dixon 1976) was used to generate a smaller number
of new variables (to reduce dimensions of the data
set) that were interpreted as spatial gradients or
major axes of the habitat hyperspace.

Principal Component Analysis

Five principal components (PC) were
interpreted (table 2). These accounted for 64% of
total variance in the data.

PCI is a soil depth gradient in the habitat
hyperspace. Measures of shallow soil, exposed
rock, and slope were heavily weighted at one end
with deep soil measures at the other end. CHVI-P,

107

ROAC-RK, and SRICH were at the deep soil end.
Douglas rabbitbrush (Chrysothamnus vlscidiflorus)
was most abundant in areas of deep dry soil, and
prickly rose (Rosa acicularls) was most abundant
in areas of deep moist soil. Richardson's ground
squirrels were abundant in areas of deep soil.

PC2 is a soil moisture gradient. Quaking
aspen ( Populus tremuloides) , lush grass, and
prickly rose were found on moist soil. Big
sagebrush (Artemisia tridentata) grew only on deep
dry soil, as did Douglas rabbitbrush. Fringed
sagewort (Artemisia f r ig id a ) and shrub cover
(antelope bitterbrush, Purshia tridentata , etc.)
were well distributed over areas with dry soil.

PC3 is ground cover and abundant vegetation
measures oppose litter measures. The order is
vertical vegetative diversity, ranging from trees
to shrubs to litter. PINEL was not highly
correlated with TOPINE (r=0.l6); pine litter
accumulated in relatively closed stands and around

scattered over-mature trees.

PC4 is mammal distribution in the habitat
hyperspace. Richardson's ground squirrels
(SRICH)are at one extreme of the component and the
golden-mantled squirrels (SLAT) and chipmunks
(TOTCH) are at the other extreme; the deer mice
(PMAN) are in the middle.

^PC5 is a shrub abundance variable. Shrub
measures are at one end and exposed rock measures
are at the other end. The presence of CHVI-P
seems to contradict the interpretation of PC5.
However, CHVI-P was a presence-absence variable,
not a measure of abundance. Douglas rabbitbrush
was found in the absence of other shrubs in some
deep dry soils and the stands of big sagebrush and
antelope bitterbrush in other areas; it was rarely
found in stands where antelope bitterbrush grew
together with wax current ( Ribes cereum) and
boulder raspberry (Rubus deliciosus) . TOPINE is
associated with low shrub abundance in PC5; shrub
cover was very low in pine stands.

Table 2. The principal components and their variable coefficients (from Carey et. al . 1980).

PCI— Soil
depth

PC2~Soil
moisture

PC 3 — Ground
cover

PC 4— Mammal
distribution

PC 5— Shrub
abundance

Variable

Coeff .

Variable

Coeff.

Variable

Coeff.

Variable

Coeff.

Variable

Coeff.

%S0IL2

0,

.36

NASPEN

-0. 35

PINEL

-0. 36

SLAT

0.37

SHARTR

-0.39

%EXPRK

0. 31

ROAC-RK

-0. 29

WDDEBR

-0. 27

TOTCH

0.34

%SHRUB

-0.37

TOPINE

0.28

GRASSL

-0. 24

LNFTLG

-0.23

WDDEBR

0.29

ARLU-RK

-0.36

EXPRKRK

0.26

NJUSC

-0.23

SLAT

-0. 13

EXPRKRK

0.23

DJUCO

-0.31

ARLU-RK

0,

.26

SOILD

-0.22

CHVI-P

-0. 07

JSHRUB

0.23

LNFTLG

-0.29

DSLOPE

0. 2U

DJUCO

-0. 17

%S0IL2

-0.07

SHARTR

0.21

NJUSC

-0.21

NJUSC

0,23

PINEL

-0. 15

DJUCO

-0. 01

SOILD

0. 17

ROAC-RK

-0. 14

PMAN

0.

. 17

TOTCH

-0.08

ARLU-RK

0.03

%EXPRK

0. 17

WDDEBR

-0. 15

LNFTLG

0. 15

SRICH

-0. 06

NJUSC

0.04

DJUCO

0. 17

NASPEN

-0. 09

SLAT

0

. 15

LNFTLG

-0.05

SOILD

0.08

CHVI-P

0. 15

ARFR-RK

-0. 09

WDDEBR

0

. 1 1

WDDEBR

-0. 00

ARFR-RK

0. 11

NASPEN

0. 12

%S0IL2

-0.06

TOTCH

0,

.07

%S0IL2

0.00

%SHRUB

0. 15

ROAC-RK

0. 1 1

GRASSL

-0.05

ARFR-RK

0,

. 04

%EXPPRK

0. 01

SHARTR

0. 16

PINEL

0. 10

DSLOPE

0.00

PINEL

0.

.02

ARLU-RK

0.01

%EXPRK

0. 19

PMAN

0. 06

TOTCH

0.02

DJUCO

-0. 05

PMAN

0. 03

SRICH

0. 18

LNFTLG

0. 02

SRICH

0.05

SHARTR

-0,

.09

SLAT

0.05

TOTCH

0. 19

GRASSL

0.02

SOILD

0. 07

NASPEN

-0,

. 10

EXPRKRK

0. 07

EXPRKRK

0. 20

TOPINE

0. 02

PMAN

0. 09

JSHRUB

-0,

. 12

TOPINE

0.09

PMAN

0.22

NJUSC

0.01

PINEL

0. 13

GRASSL

-0,

. 14

DSLOPE

0. 15

GRASSL

0. 27

JS0IL2

-0.03

TOPINE

0. 15

ROAC-RK

-0,

. 17

CHVI-P

0.26

NASPEN

0.28

DSLOPE

-0. 10

JEXPRK

0. 19

SRICH

-0,

.22

ARFR-RK

0.33

DSLOPE

0. 30

ARLU-RK

-0. 16

SLAT

0.20

CHVI-P

-0,

.28

SHARTR

0. 34

TOPINE

0.31

ARFR-RK

-0.35

CHVI-P

0.21

SOILD

-0,

.28

JSHRUB

0. 34

ROAC-RK

0. 32

SRICH

-0. 35

. EXPRKRK

0.27

Eigenvalue

6,

.3

3.6

2.2

2. 1

1.7

Cumulative

0,

.25

0. 40

0. 49

0.57

0. 64

percent '

'Cumulative proportion

of total

variance

explained by

the principal

components.

108

Thus, the major gradients in the habitat
hyperspace were soil depth and soil moisture, and
they accounted for two-thirds of the explained
variance. Smaller portions of the variance were
explained by ground cover, mammal distribution,
and shrub abundance. Barkham and Norris
(1970) pointed out that it is not uncommon for
minor components to increase in complexity and to
represent interaction effects. The last three
components probably do, in part, represent the
interaction of soil depth and soil moisture. PCA
illustrated the plants' and mammals' responses
(habitats) to the habitat hyperspace.
Richardson's ground squirrels were strongly
associated with deep soils (PCI). Golden-mantled
ground squirrels were moderately associated with
shallow soils (PCI) and exposed rock (PCI, PC5).
Chipmunks were associated with golden-mantled
squirrels (PC4). Deer mice were associated with
grasses (PC3) and were intermediate on the other
PC's. All of the mammals occupied intermediate
positions on the soil moisture gradient. This
indicates that soil moisture was not a major
determinant of any species' habitat hypervolume
and suggests that mammal distributions were a
function of soil depth. Richardson's ground
squirrels almost exclusively used ground burrows
for escape and denning. They assumed a
characteristic "picket pin" posture for
observation. They did not use rocks for escape or
observation in the study areas. Golden-mantled
ground squirrels used large rocks as observation,
feeding, and basking posts. They commonly used
rock interstices for escape cover and den sites.
Chipmunks also used rocks as observation posts,
escape cover, and den sites.

Discriminant Analyses on Mammals

Results of DA on data sets of the habitat and
ecotope hyperspaces are illustrated in table 3.
The first five steps and the last step of the DA
are shown. Group means of the habitat hypervolume
and the remaining hyperspace were significantly
different (P<0.001) at each step for all species
except the least chipmunk. For the least
chipmunk, P=0.012 at the first step and P=0.019 at
the last step. Group means of the ecotope
hypervolume and the remaining hyperspace were
significantly different (P<0.001) at each step for
all species. Rates of change in the U statistics
and the percentages of stations correctly
classified showed that most information contained
in the DF's was contributed by the first few
variables in each case. More stations were
classified correctly with the ecotope hyperspace
variables than with the habitat hyperspace
variables. U statistics were lower and station
classifications were more successful for
golden-mantled ground squirrels and Richardson's
ground squirrels than for Uinta chipmunks and
least chipmunks.

Comparison of discriminating variables
between the habitat hypervolumes and the ecotope
hypervolumes showed that the ecotope hypervolume
of the Richardson's ground squirrel differed
little from its habitat hypervolume, and that its

Table 3. Results of stepwise discriminant
analysis of hyperspaces for Spermophi lus
lateralis (modified from Carey et al. 1980).

Variable
set

ouep

n
U

*

Habitat

1

%S0IL2

0.87

66

2

WDDEBR

0. 81

72

3

CHVI

0.79

75

4

PINEL

0.78

75

5

EXPRKRK

0.77

76

21

GRASSL

0.72

77

Ecotope

1

SRICH

0.67

86

2

PMAN

0. 65

86

3

ARLU-F

0.64

87

4

EM IN

0.63

87

5

%SHRUB

0.62

86

24

EXPRKRK

0.59

88

'Step in stepwise procedure, variables included,
Wilks X, cumulative percent of stations properly
assigned. All group means signif incantly
different at all steps (P<0.01).

distribution was a response primarily to a
structural variable, soil depth, and incidentally
to a bionomic variable, captures of golden-mantled
ground squirrels. The golden-mantled ground
squirrel's distribution was a function of bionomic
variables — captures of other mammals (primarily
Richardson's ground squirrels) — rather than a
response to habitat (structural) variables.
Habitat variables (e.g., EXPRKRK) were equal in
importance to bionomic variables (e.g., captures
of golden-mantled ground squirrels) in describing
the distribution of least chipmunks. The Uinta
chipmunk responded to habitat variables.

Summary of Species-Environment Relationships

Richardson's Ground Squirrel. — Food habits of
the Richardson's ground squirrels were similar to
those of the golden-mantled ground squirrels.
Spatial overlap between the two species in the
habitat hyperspace was moderate, and ratios of the
mean captures of the two in each other's habitat
hypervolume were inversely related. Diet overlap
with the least chipmunk was small; spatial overlap
moderate. Richardson's ground squirrels were at
one end of the mammal distribution component, and
golden-mantled ground squirrels and chipmunks at

109

the other end. PCA and means of the

discriminating variables showed that the habitat
hypervolume of Richardson's ground squirrel was
characterized by deep soil, vegetative ground
cover, and slight slopes. It was not

characterized by either extreme of soil moisture,
or by high values of exposed rock or shrub cover.
Discriminating variables for the ecotope
hypervolume demonstrated the overwhelming
influence of the habitat (structural) variables,
especially soil depth measures, but also indicate
a degree of negative response to the
golden-mantled ground squirrels.

Golden-mantled Ground Squirrels. --Trophic
overlap was great with Richardson's ground
squirrel, and moderately low with the least
chipmunk. Spatial overlap was small with
Richardson's ground squirrel, and great with the
least chipmunk. Capture^ ratios were inversely
related with the ground squirrel, and one-sided
with the chipmunk (in favor of the golden-mantled
ground squirrel). Their habitat hypervolume was
characterized by shallow, somewhat dry soil on
steep slopes with sparse grass cover, moderate
tree (ponderosa pine, Pinus ponderosa ) cover,
abundant litter, and abundant exposed rocks with
interstices. The best discriminating variable for
the ecotope hypervolume was the captures of
Richardson's ground squirrel, a bionomic variable.
Other discriminating variables contributed little
to the description of the ecotope hypervolume.

Least Chipmunk. — Overlap between the diet of
least chipmunks and those of ground squirrels was
small; the least chipmunk made greater use of
arthropods for food than did ground squirrels.
Spatial overlap with the golden-mantled ground
squirrel was high. The ground squirrel was near
92% of the point locations in the chipmunk habitat
hypervolume and outnumbered chipmunks throughout
the habitat hyperspace. The habitat hypervolume
of the least chipmunk was characterized by
intermediate soil depth and moisture, moderate
vegetative ground cover, and higher than average
values of exposed rock, rock interstices, prickly
rose, aspen, pine trees (P_. ponderosa and P^.
contor ta ) , and common juniper ( Juniperus
communis) . Exposed rock with numerous interstices
was the single best descriptor of the habitat
hypervolume, regardless of whether rocks were in
conjunction with deep or shallow soils, pine trees
or aspens, or shrubs or grass. The ecotope
hypervolume was characterized by the presence of
golden-mantled ground squirrels. Deep soil
indicators (NASPEN, POPR-F, ARTR-F) were less
important discriminating variables, and described
the part of the hypervolume not characterized by
golden-mantled ground squirrels, for example,
small rock outcrops in deep soil areas.

Uinta Chi pmunk . — Uinta chipmunks were
relatively few in number and were caught in only
15% of the trap stations. Their habitat and
ecotope hypervolumes were characterized by the
highest mean values for shallow soil, exposed
rock, rock interstices, steep slopes, log litter,
and pine trees, and by the lowest mean values for

soil depth, shrub cover, grass litter, and deep
soil indicators, such as prickly rose and Douglas
rabbitbrush .

Discriminant Analyses on Ticks and Virus

Ticks .• — The relationship between habitat
variables and adult wood tick abundance is
illustrated in table 4. High tick abundance was
assocjiated with shallow soil, moderate shrub
cover, abundant exposed rock and rock interstices,
steep slopes, relatively abundant pine, and
abundant log litter. Moderate tick abundance was
associated with intermediate values of the same
variables, except for shrub cover, w^iich was
abundant. The distribution of ticks in the
habitat hyperspace reflects their physiological
tolerance for soil temperatures and moisture
(Wilkinson 1967).

The objectives of the DA of tick distribution
were to obtain DF's of variables that were easily
and quickly measured and that provided a high
degree of discriminance in classifying areas on
the basis of tick abundance. The discriminant
functions in table 4 met those objectives. The
two best discriminating variables were DSLOPE and
%S0IL2. Less biased estimates of errors of
misclassif ication were obtained with jackknifing
procedures; they differed little from the
cumulative percentages in table (Carey 1979).

CTF Virus. — The relationship between virus
activity and the ecotope hyperspace variables was
illustrated by Carey (1979). Virus was isolated
from mammals captured in areas characterized by
shallow soil, abundant exposed rocks and rock
interstices, relatively abundant pine, moderate
shrub cover, moderately steep slopes, relatively
high numbers of golden-mantled squirrels, least
chipmunks, Uinta chipmunks, deer mice, and
immature ticks, and low numbers of Richardson's
ground squirrels. The objective of the DA of
virus distribution was to obtain OF of easily
measured variables that would permit
identification of areas maintaining virus
circulation with a low probability
of misclassifying areas with virus. The DF met
this objective (Carey 1979).

DISCUSSION

I believe the foregoing is a demonstration of
precise definition combined with methods of
multivariate analysis. After the terminology is
grasped, the results should be ordered and clear
to the reader. The distinction between structural
(scenopoetic) variables and bionomic (potentially
measuring competition) variables should be clear;
also the differences in scope between habitat
(intercommunity), niche ( intracommunity ) , and
ecotope variables should be clear. The enforced
distinction between niche, habitat, and ecotope
results in heuristic data analysis and
interpretation of results that is clear. The
compatibility between the terminology and the
analysis is obvious.

110

Table 4. Discriminating variables and discriminant functions for the
distribution of adult, free-ranging wood ticks. All group means
significantly different at P<0.01 using approximate F tests.

Variable U Cumulative' DF Coefficients'

% None-Few Few-Many

ticks ticks

n c^ 1 Q

ftp R

0.21 450

0. 25643

DSLOPE

0. 360

68.8

0. 19518

0. 44237

%SHRUB

0. 324

72.7

0.24368

0.21 132

ARLU-RK

0.302

72.7

1. 16317

2. 19884

SOILD

0.291

73.4

1.23637

1.21552

GRASSL

0.281

74.2

-0. 17259

-0.78330

NASPEN

0.263

76.6

0.33857

2.63709

Constant

-14.55902

-21.57874

'Cumulative percentages of stations correctly assigned by the DF.
'Coefficients of the variables for the discriminant function separating the
two groups.

These concepts (the ecological and the
statistical) have been used to good advantage in
the past, but, I believe, without precise
definition. James (1971) clearly distinguished
between intercommunity and intracommunity
relationships in addition to formulating the
niche-gestalt. Anderson and Shugart (1974) used
the same (Whittaker et al. 1973) definition of
habitat hyperspace. Dueser and Shugart (1978,
1979) also recognized the distinction between
inter- and intracommunity relationships, but used
a plethora of terms (habitat, microhabi tat ,
habitat type, habitat association, habitat patch,
patch type, forest stand type, forest type, desert
community, rodent community, forest biome,
fine-grained environment, etc.) in the 1978 paper
(I am not criticizing the quality of their
scientific contribution, I am pointing out the
lack of precise terminology). Similarly M'Closkey
(1975, M'Closkey and Fieldwick 1975) recognized
the inter- and intracommunity distinction and the
applicability of multivariate analysis (M'Closkey
1976) to the study of animal-environment relation-
ships but not the use of precise terminology.

The use of the suggested terminology and
multivariate analysis in the study of animal-
environment relationships is clear, but what about
the applicability to wildlife management? Aside
from the basic understanding of how a species (or
ecotope) responds to its environment, I think the
inter- and intracommunity distinction is one that
will prove to be of great value to the wildlife
biologist, especially, if the trend towards
management for diversity continues. The
distinction between structural and bionomic
variables is useful at the management level;

extensive management must focus on scenopoetic
variables, but intensive management can consider
bionomic variables. Understanding the concepts of
spatial gradients and species' responses to them
is fundamental to predicting results (or
"environmental impact") of a management decision.
The concept of homogeneously diverse biotopes
coupled with inter- and intracommunity
distinctions is of great value in determining
scale or size for both sampling units and managing
units.

The use of the terms "hyperspace" and
"hypervolume" may be confusing and, therefore, of
less benefit. Simpler terms, for example
"preinteractive" (or fundamental) and
"postinteractive" (or realized) could be used to
modify "niche space" and "habitat space" without
changing the concepts. Applying theoretical
concepts to field studies can be rewarding. In
the example given, discriminant functions for the
ticks and virus could be used to locate
recreational facilities and activities for minimum
human exposure to the virus. The tick
discriminant function could be used in the
application of acaricides. Richardson's ground
squirrel is a reservoir of plague (Yersinia
pestis) , and the golden-mantled ground squirrel is
a reservoir of Colorado tick fever virus. Results
of the analysis suggest that if Richardson's
ground squirrels were controlled (killed) to
minimize human exposure to plague (such a control
effort was instituted in 1976 and 1977), they
would be replaced by golden-mantled ground
squirrels. This would not increase the risk of
human exposure to Colorado tick virus, because the
golden-mantled ground squirrels would have moved

m

out of the area that was climatically suited for
ticks .

Many authors, especially Green, (1971, 1974)
have discussed the applicability of multivariate
analyses in ecology; most of the comments apply
equally to wildlife research. However, I have
seen only two reports in the Journal of Wildlife
Management (Klebenow 1969, Martinka 1972)
describing the application of multivariate
analyses to animal-environment relationships. I
believe a functional organization and a
standardization of ecological terminology can do
much to promote the application of multivariate
statistics to the more applied areas of wildlife
reasearch .

LITERATURE CITED

Anderson, S.H., and H.H. Shugart, Jr. 197^+.
Habitat selection of breeding birds in an
east Tennessee deciduous forest. Ecology
55:828-837.

Barkham, J. P., and J.M. Norris. 1970.
Multivariate procedures in an investigation
of vegetation and soil relations of two beech
woodlands, Cotswald Hills, England. Ecology
51 :630-639.

Brewer, R. 1979. Principles of ecology. 299 p.

W.B. Saunders Co., Philadelphia, Penn.
Brown, M.B., editor. 1977. BMDP-77. Biomedical

computer programs. P-series. 880 p.

University of California Press, Berkeley.
Carey, A.B. 1979. Discriminant analysis: a

method of identifying foci of vector-borne

diseases. American Journal Tropical Medicine

and Hygiene 28:750-755.
Carey, A.B., R.G. McLean, and G.O. Maupin. 1980.

The structure of a Colorado tick fever

ecosystem. Ecological Monograph 50:131-151.
Cody, M.L. 1968. On the methods of resource

division in grassland bird communities.

American Naturalist 102:107-147.
Colwell, R.K., and E.R. Fuentes. 1975.

Experimental studies of the niche. Annual

Review of Ecology and Systematics 6:281-310.
Dixon, W.J., editor. 1976. B.M.D. Biomedical

computer programs. 773 p. University of

California Press, Berkeley.
Dueser, R.D., and H.H. Shugart, Jr. 1978.

Microhabitats in a forest-floor > small-mammal

fauna. Ecology 59:89-98.
Dueser, R.D. , and H.H. Shugart, Jr. 1979. Niche

pattern in a forest-floor small-mammal fauna.

Ecology 60: 108-1 18.
Green, R.H. 1971. A multivariate statistical

approach to the Hutchinsonian niche: bivalve

molluscs of central Canada. Ecology

52:543-556.

Green, R.H. 1974. Multivariate niche analysis

with temporally varying environment factors.

Ecology 55:73-83.
Hansen, R.M. 1962. Dispersal of Richardson's

ground squirrel in Colorado. American

Midland Naturalist 68:58-66.
Haskell, E.J. 1940. Mathematical systematization

of "environment," "organism" and "habitat."

Ecology 21:1-16.

Hespenheide, H.A. 1971. Flycatcher habitat
selection in the eastern deciduous forest.
Auk 88:61-74.

Hughes, R.E., and D.V. Lindley. 1955.
Application of biometric methods to problems
of .classification in ecology. Nature
175:806-807.

Hutchinson, G.E. 1958. Concluding remarks. Cold

Spring Harbor Symposium on Quantitative

Biology 22:415-427.
Hutchinson, G.E. 1978. An introduction to

population ecology. 260 p. Yale University

Press, New Haven, Conn.
Inger, R.F., and R.K. Colwell. 1977.

Organization of contiguous communities of

amphibians and reptiles in Thailand.

Ecological Monographs 47:229-253.
James, F.C. 1971. Ordinations of habitat

relationships among breeding birds. Wilson

Bulletin 83:215-236.
Kendeigh, S.C. 1974. Ecology with special

reference to animals and man. 474 p.

Prentice-Hall, Inc., Englewood Cliffs, N.J.
Klebenow, D.A. 1969. Sage grouse nesting and

brood habitat in Idaho. Journal of Wildlife

Management 33:649-667.
Kulesza. G. 1975. Comment on "Niche, habitat,

and ecotope." American Naturalist 109:

476-479.

M'Closkey, R.T. 1975. Habitat succession and
rodent distribution. Journal of Mammalogy
56:950-955.

M'Closkey, R.T. 1976. Community structure in

sympatric rodents. Ecology 57:728-739.
M'Closkey, R.T., and B. Fieldwick. 1975.

Ecological separation of sympatraic rodents

( Per omyscus and Microtus ) . Journal of

Mammalogy 56:119-129.
McNaughton, S.J., and L.L. Wolf. 1979. General

ecology. 702 p. Hold, Rinehart and Winston,

New York, N.Y.
Maguire, B. , Jr. 1973. Niche response structure

and the anlaytical potentials of its

relationship to the habitat. American

Naturalist 107:213-246.
Marr, J.W. 1967. Ecosystems of the east slope of

the front range in Colorado. 134 p.

University of Colorado Studies Series in

Biology No. 8, Boulder, Colo.
Martinka, R.R. 1972. Structural characteristics

of blue grouse territories in southwestern

Montana. Journal of Wildlife Management

36:498-51 0.

Michener, G.R., and D.R. Michener. 1977.
Population structure and dispersal in
Richardson's ground squirrels. Ecology
58:359-368.

Miller, R.S. 1964. Ecology and distribution of
pocket gophers (Geomyidae) in Colorado.
Ecology 45:256-272.

Nudds, T.D. 1979. Theory in wildlife conser-
vation and management. Transactions North
American Wildlife and Natural Resources
Conference 44:277-288.

Odum, E.P. 1971. Fundamentals of ecology, third
edition. 574 p. W.B. Saunders Co.,
Philadelphia, Penn.

Odum, E.P. 1975. Ecology: the link between
the natural and social sciences. 244 p.
Holt, Rinehart and Winston, New York, N.Y.

112

Pimlott, D.H, 1969. The value of diversity.
Transactions North American Wildlife and
Natural Resources Conference 3^*: 265-280.

Rejmanek, M. , and J. Jenik. 1975. Niche,
habitat, and related ecological concepts.
Acta Biotheoreologica 24:100-107.

Richardson, J.L. 1977. Dimensions of ecology.
112 p. The Williams and Wilkins Co.,
Baltimore, Md.

Sanderson, G.C., E.D. Abies, R.D, Sparrowe, J.R.
Grieb, L.D. Harris, and A.N. Moen. 1979.
Research needs in wildlife. Transactions
North American Wildlife and Natural Resources
Conference 44:166-175.

Shugart, H.H., Jr., and B.C. Patten. 1972. Niche
quantif iation and the concept of niche
pattern. p. 283-325. In B.C. Patten,
editor. Systems analysis and simulation in
ecology. Volume 2. Academic Press, New
York, N.Y.

Siderits, K. , and R.E. Radtke. 1977. Enhancing
forest wildlife habitat through diversity.
Transactions North American Wildlife and
Natural Resources Conference 42:425-434.
Smith, R.L. 1974. Ecology and field biology.
850 p. Second edition. Harper and Row
Publishers, New York, N.Y.
Smith, R.L. 1977. Elements of ecology and field
biology. 497 p. Harper and Row Publishers,
New York, N.Y,
Sparks, D.R., and J.C. Malechek. 1968.
Estimating percentage dry weight in diets
using a microscope technique. Journal of
Range Management 21:261-265.
Vandermeer, J.H, 1972, Niche theory. Annual

Review of Ecology and Systematics 3:107-132,
Whittaker, R.H., S,A. Levin, and R.B. Root, 1973,
Niche, habitat, and ecotope. American
Naturalist 107-321-338.
Whittaker, R.H., S,A, Levin, and R,B, Root, 1975,
On the reasons for distinguishing "niche,
habitat, and ecotope," American Naturalist
109:479-482,
Wilkinson, P,R. 1967.
Dermacentor ticks in
zones ,

The distribution of
Canada in relation to
Canadian Journal of

bioclimatic

Zoology 45:517-537,
Yeaton, R,I. 1972, Social behavior and social
organization in Richardson's ground squirrel
(Spermophilus richardsonii) in Saskatchewan,
Journal of Mammalogy 53:139-147,

BOB CLARK: By removing biases due to dispersal,
would you improve the discrimination of the
habitats?

A,B, CAREY: Yes. However, the sample size was
large and dispersal movements probably accounted
for a small proportion of the captures used as
captures seemed to be very clumped. In regards to
other biases, it should be emphasized that I was
interested only in generating a relatively crude
measure of population response with the objective
of determining hypervolumes.

KEN MORRISON: Given that there are behavioral
biases inherent in trapping data, would this bias
any results and/or interpretations of such an
analysis?

A,B, CAREY: Yes, but see previous response to Bob
Clark.

R. DUESER: How important is the notion of the
"homogeneously diverse biotope" to your
distinction between "niche" and "habitat"? What
would be the implications of the non-existence of
homogeneously diverse biotopes?

A.B, CAREY: The distinction between niche and
habitat is that between intra- and inter-
community, thus one must be able to assign a point
location in the landscape to a recognizable
community which occupies the homogeneously diverse
space called the biotope, thus the notion is
fundamental, "Homogeneously diverse" is relative
to the range of a motile organism; if communities
exists, so do biotopes; if they do not, niches
don't either.

JAMES DUNN: (1) If the principal components are
properly named, can variation in soil really be
independent of moisture? (2) Can variation in
mammal distribution really be independent of
variation in ground cover? (3) Why not confess
that the fundamental variables need not be
independent and proceed with an oblique factor
solution. The additional benefit might be a
measure of the association between say mammal
factor and cover factor if essentially the same
named factors result?

mm

' "i

rlllir

'IIPI
)

DISCUSSION

BOB CLARK: Were there age and sex biases in the
captures?

A.B. CAREY: Possibly. I attempted to minimize
biases due to age and sex by disregarding single
captures of Richardson's ground squirrels, to
trap-happiness by only using the first capture of
an individual in a sampling period, and to
competition for traps by placing four traps at
each point location in the landscape and by
checking the traps up to four times daily.

A.B. CAREY: (1) Soil depth and soil moisture can
be quite independent as is reflected by the
community pattern in the landscape e.g., basin-
big sagebrush on deep dry soils, and aspen stand
complexes on deep wet soils. (2) Variation in
mammal distribution can be independent of "ground
cover" especially in a homogeneously diverse
space. (3) Principal components beyond the 1st
and 2nd components may show interaction effects
(Barkham & Norris 1970) as mine do. However, I
think my principal components nicely describe the
spatial gradients in the landscape. I would say
den sites (ground dens, rock dens) were more
important than vegetative cover; there is also
good evidence that food is not limiting and that
the behavior of the species causes them to avoid
the densest cover.

113

FORHAB: A FOREST SIMULATION MODEL TO PREDICT
HABITAT STRUCTURE FOR NONGAME BIRD SPECIES'
T.M. Smithy H.H. Shugart', and D.C. Wesf

Abstract. — FORHAB (a deciduous forest stand stimulation
model) was used to predict changes in available breeding
habitat for the avian community inhabiting the Walker Branch
Watershed in east Tennessee. A census was conducted to
locate all breeding territories of the various bird species
on the watershed. Data on vegetational structure of these
territories were used to calculate linear decision scales, a
classification procedure based on discriminant function
analysis, which could be used to classify forest stands as
potential breeding habitat for the various bird species.
FORHAB was used to simulate changes in forest structure of
the watershed due to both natural succession and certain
introduced forest management practices (diameter-limit cut).
Variables describing the vegetational^ structure of the forest
stands generated by FORHAB were used to determine
availability of potential breeding habitat for each bird
species through time using a subroutine based on the
above-mentioned classification procedure. Predictions of
available habitat for the ovenbird (Seiurus aurocapillus) on
the Walker Branch Watershed are presented as an example of
model output.

Key words: Avian community; deciduous forest;
discriminant function analysis; forest management; linear
decision scales; nongame birds; simulation model; vegetation
structure .

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25,1980, Burlington Vt.

^Graduate Research Fellow, Environmental
Sciences Division, Oak Ridge National Laboratory,
Oak Ridge TN 37830.

^Senior Research Staff Member, Environmental
Sciences Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37830.

■•Research Staff Member, Envr ionmental
Sciences Division, Oak Ridge National Laboratory,
Oak Ridge TN 37830.

INTRODUCTION

The effects of modern forest management
practices on wildlife has been a subject of
growing concern in recent years (Slusher and
Hinckley 1974, Thomas 1979). A major portion of
this interest has been directed toward the nongame
bird component of an animal community (Smith 1975,
DeGraaf 1978) particularly the effect that
management-related disturbances have on the
population dynamics of either a single species
(e.g., endangered species) or the avian community
as a whole. The majority of work in this area has
been of a descriptive nature. Researchers have
dealt primarily with general habitat preferences
of selected species (fig. 1) or the role of
structural heterogenity on the overall avian
diversity of the forest (MacArthur and MacArthur

114

1961). This information, coupled with a knowledge
of effects of various silvicultural practices on
structure of the forest, has been the basis of
timber-wildlife management decisions to date.
Problems of time and expense have limited the
number of actual field studies which have
monitored directly the effects of timber
management on bird populations (Hagar 1950,
Franzreb and Ohmart 1978). However, even these
studies are of limited value. Site-specific
conditions and past history of the forest greatly
limit the generalization of results from a
specific study to other forests.

Recently, techniques of multivariate
statistics have allowed analyses beyond
qualitative methods of describing habitat
preferences of avian species. With the use of
multivariate statistical procedures, one can
classify forest stands quantitatively with respect
to habitat potential. Multivariate data analysis
has been used to quantify the microhabitat
selection patterns at both the species and
community level (James 1971, Shugart and Patten
1972, Anderson and Shugart 1974, Whitmore 1975).
Conner and Adkisson (1976) proposed a method of
classifying forest stands as suitable woodpecker
habitat using a discriminant function analysis
procedure based on variables describing structure
of forest vegetation and stressed the potential of
such techniques as management tools. However,
applicability of such statistical procedures is
hindered by lack of habitat data necessary for
dynamic habitat analyses. Management is by
definition a dynamic process. To assess potential
effects of various management strategies on
availability of habitat, quantitative data
describing changes in structure of the forest
through time is necessary. To date, quantitative
information on structural changes of the forest
resulting from various timber management practices
is not available. A forest stand simulation model
may very well remedy problems inherent in
assessing habitat modifications associated with
various management techniques.

Conofita.l *^ White tailed ■ ^\ Red

rabbit ;Ker 'i>* ^» saon.ei''*. Fo»

f ^ . Rutted giouse

Grass Low shrub High shrub Sitult-t.i'e Opening L-nv 'lee

Figure 1. General habitat preference of various
animal species inhabiting a northeastern conifer
forest. [Reprinted with permission from Smith
(1980)].

Forest stand simulators (Shugart and West
1980), can be used to assess effects of alternate
forest management strategies on selected bird
species or for entire avian communities. Models
vary in their mathematical structure, but
typically function by considering tree-by-tree
changes over time for an area that corresponds to
that of a canopy tree or some sample unit. The
spatial scale of these models corresponds to what
has been termed the microhabitat scale for birds.
Because such simulators have the ability to
predict structural changes in the forest through
time, one can couple this quantitative data with
statistical classification procedures previously
mentioned to project long-term consequences of
different management practices on available
habitat .

Objectives of this research have been 1) to
integrate techniques of multivariate
classification with the predictive ability of a
forest stand simulation model, and 2) to use the
resulting model to determine effects of forest
management practices on availability of nongame
bird habitat. This synthesis requires: a) a
structural classification of forest stands in
terms of suitability for specific bird species,
and b) ability of the forest stand simulator to
generate specific variables on which the
classification is based. By introducing
disturbances (e.g., fire, timber harvest) to the
model, we can evaluate effects of natural and
man-induced perturbations on availability of
habitat for a specific species of bird.

The remainder of this paper will be devoted
to presenting an example of this method, FORHAB, a
forest stand simulation model designed to predict
impacts of certain forest management decisions on
availability of habitat for the avian community
inhabiting the Walker Branch Watershed.

The Walker Branch Watershed is a 97.5 ha site
on the D.O.E. reservation in Anderson County,
Tennessee (fig. 2). The watershed ranges in
elevation from 285 m to 375 m and occupies an area
of steeply sloping ridges and narrow valleys.
Girgal and Goldstein (1971), in an analysis of the
structure of the watershed, found dominant forest
types to be pine (predominately Pinus echinata) ,
yellow popular (dominated by Li r iodendron
tulipif era) , oak-hickory (mixed Quercus spp.,
Carya spp.) and chestnut oak (typified by Quercus
prinus) .

MODEL

The following model, FORHAB, is a modified
version of FORET (Shugart and West 1977), an
Appalachian deciduous forest stand simulator. A
detailed description of the model (FORET) can be
found in Shugart and West (1977). For the purpose
of this paper we will review briefly the general
form of the model, but will discuss only
modifications in detail.

115

ORHL-DWG 67-7601

Figure 2. Location of Walker Branch Watershed,
Oak Ridge, Tennessee.

FORHAB, like its parent model, simulates the
annual change of a forest stand (0.085 ha circular
plot) by calculating the growth increment of each
tree growing on the stand (subroutine GROW), by
tabulating the addition of new saplings to the
stand (subroutines BIRTH and SPROUT), and by
tabulating the death of trees present on the stand
(subroutine KILL). These processes are modeled as
stochastic functions. Growth parameters for tree
species and climatic conditions included in the
model are based on sites of lower slope positions
located in eastern Tennessee. A flow chart for
FORHAB is provided in appendix I. The main
equations for the above subroutines are summarized
in appendix II. CUT, HABIT and DISCRM subroutines
will be dealt with separately.

Subroutine CUT

The CUT subroutine simulates various forest
management practices which are applicable to the
southeastern deciduous forest type. The version
of this subroutine used for the following analysis
was a diameter-limit cut. In this subroutine all
commercially valuable species for sawtimber
greater than 23 cm dbh (diameter at breast height)
were removed from the plot on a 60-year rotation.
This form of timber management was pra'cticed on
the watershed prior to 19^0. The rotation period
of 60 years was determined by analysis of stem and
basal area curves generated by FORHAB after
initial simulations of logging on the watershed.

Subroutine HABIT

The process of classifying stands by their
potential to provide habitat for a given bird
species is carried out in subroutine DISCRM. The
classification is based the following biomass
variables which describe vegetational structure of
the forest stand:

Foliage biomass of trees 1.2-8.M cm dbh
Foliage biomass of trees 8.5-22.8 cm dbh
Foliage biomass of trees >22.8 cm dbh
Branch biomass of trees 1.2-8.U cm dbh
Branch biomass of trees 8.5-22.8 cm dbh
Branch biomass of trees >22.8 cm dbh
, Bole biomass of trees 1.2-8.4 cm dbh
Bole biomass of trees 8.5-22.8 cm dbh
Bole biomass of trees >22.8 cm dbh
Number of trees 1.2-8,4 cm dbh
Number of trees 8.5-22.8 cm dbh
Number of trees >22.8 cm dbh

Foliage biomass of average tree 1.2-8.4 cm
dbh

Foliage biomass of average tree 8.5-22.8 cm
dbh

Foliage biomass of average tree >22.8 cm dbh
Branch biomass of average tree 1.2-8.4 cm dbh
Branch biomass of average tree 8.5-22.8 cm
dbh

Branch biomass of average tree >22.8 cm dbh
Bole biomass of average tree 1.2-8.4 cm dbh
Bole biomass of average tree 8.5-22.8 cm dbh
Bole biomass of average tree >22.8 cm dbh

Model output in the form of species and tree
diameters must be used to calculate these biomass
variables .

The HABIT subroutine divides all trees on the
simulated plot into two groups, conifer and
deciduous, and then into three size classes within
each of these two groups. The foliage, branch and
bole biomass for each tree is then calculated
using regression equations in table 1 , which are
site specific to the Walker Branch Watershed
(Harris et al. 1973). These values are then
summed for all trees on a plot for each size class
to provide the variables listed above.

Subroutine DISCRM

The classification of simulated forest stands
as potential habitat for a given bird species is
carried out in subroutine DISCRM. The
classification is based on the statistical
procedure of two-group discriminant function
analysis (Morrison 1967). Classification criteria
were constructed using vegetation data collected
on 298 0,085-ha permanent census plots. Breeding
territories of various bird species were located
and mapped if they either contained or overlapped
any of the 298 plots. If a plot was located
within the territory of an individual bird
(breeding pair), that plot was considered as
potential habitat for that species. Conversely,
if a given plot was not within the boundary of a
territory of an individual of that species, that
plot was classified as not providing habitat for
that species. Thus data were obtained on areas of
both suitable habitat and areas considered
inadequate for the needs of the various species.
Data on vegetation of these census plots, in the
form of species and diameter for each tree on the
plot, then were used to generate biomass variables
for classification using the same regression
equations as those in subroutine HABIT.

116

Table 1. Uncorrected regressions (In-ln) of tree component weight (kg) on tree
diameter breast height (cm) used in subroutine HABIT,

Dependent

Intercept

Slope

variable

(a)

(b)

Leal

All spp .

-J . 'lyo

1 . oy 5

U , 00

1 '311

Hardwoods

-3.862

1.740

0.88

178

Conifers

-2.907

1.674

0,91

65

Branch

All spp.

—J, I 00

n 01
u , y 1

£1 yo

1 9^

Hardwoods

-3.173

2.224

0.89

231

Conifers

-3.461

2.292

0.95

51

Bole

All spp.

-2.437

2.418

0.97

298

1 .08

Hardwoods

-2.270

2.385

0.98

231

Conifers

-3.787

2.767

0.96

51

Branch-bole

All spp.

-2,126

2.393

0,96

371

'N = Number of samples in each regression

is the correction factor for bias on logarithmic transformation which is
multiplied by exp [In T]

Figure 3 is a hypothetical example of the
actual classification procedure in subroutine
DISCRM. For each bird species, census plots were
placed into one of two groups, suitable habitat or
unsuitable habitat, based on the process described
above. The two elipses in figure 3 represent two
populations of census plots plotted in two-
dimensional space (with XI and X2 being two
biomass variables from table 1 ) . These two
populations were then subjected to two group
discriminant function analysis.

Discriminant function analysis is a
statistical procedure for finding a linear
combination of the original predictor variables
(XI and X2) which results in the largest
difference between the two mean vectors (i.e.,
maximizes the ratio of among-group to within-group
sums of squares) . Let us examine the case of two
populations, suitable habitat and unsuitable
habitat plots, with samples of Nl and N2
independent observations. The populations of P
(number of variables) responses are multivariate
normal with a common variance-covariance matrix I,
but different mean vectors u and u^- 2^1 ^^'^ ^2

are the sample mean vectors for the two groups and
S is the pooled estimate of z, our intention is to
find a coefficient vector a_ of the linear compound
a^X of the responses which will maximize the
distance between the two groups in P-dimensional
space. It can be shown that

a = (XI - X2);

and the general form of the resulting discriminant
function is

Y = (XI - X2)'S~^X

with

+ a X
P P

(where p is the number of variables in the
classification) .

GROl P
SUITABLE HABITAT

m

mi

m

Figure 3. Linear decision scale from subroutine
DISCRIM.

117

In the case of two populations with
multivariate normal distributions and respective
mean vectors and X^, common covariance matrix S

and prior probabilities of group membership h and
(1-h), the Bayes, or minimum expected loss,
classification rule states that the observation
vector should be assigned to population 1 if

X'S'^U^-X^) - l/SCX^+X^) 'S"^ (X^-X^) > In (1-h)

and assigned to population 2 if this relationship
does not hold.

It should be noted that the first term in
this equation is the linear discriminant score for
the observation vector X and the second term is
the point midway between the means of the
discriminant function as computed for each group
or population. For the purpose of classifying
simulated observation vectors (subroutine HABIT)
the prior probabilities of group membership can be
assigned to be equal since there is no a priori
reason for assuming group membership. As a result
of this assumption, the term to the right of the
inequality becomes zero and the equation can be
expressed as

X'S~\x^-X2) - 1/2(X^+X2)'S"^(X^-X2)

with the term to the left of the inequality being
the discriminant score for the observation vector
to be classified and term to the right being the
midpoint between the discriminant scores as
computed for the group mean vectors.

By calculating the scores corresponding to

the mean vectors XI and X2 (where XI = suitable
habitat and X2 = unsuitable habitat) and then
taking the midpoint of those values;

^mid = . Y2)/ 2

one can compute the linear decision scale as shown
in figure 3.

To classify a newly sampled response (such as
a plot output by the model) into either group 1 or
2, the Y. score must be calculated for that given
response vector. If the Y. value is to the group
1 direction of Y^^^ then the plot would be

classified as belonging to group 1 , with the
converse holding true if the Y^ value was to the

opposite direction of Y^^^.

Subroutine DISCRM consists of a series of
these linear decision scales, one corresponding to
each bird species comprising the avian community
on the watershed. Each simulated plot is input to
subroutine HABIT where the biomass variables
necessary for classification are generated. These
variables are then input to subroutine DISCRM
where the Y. value is calculated for that plot.
This value is then compared to the Y^^^ value for

each bird species and the decision as to whether
that plot provides potential breeding habitat for
each of the species of interest is output from the
model .

RESULTS

A test of homogeneity of within covariance
matrices for the two group discriminant function
analysis of ovenbird (Seiurus aurocapillus)
habitat found no significant difference between

covariance matrices (as approximated by x i P 2.
0.05) and they were pooled for the analysis
(Kendall and Stuart 1961). The two groups
(suitable and unsuitable habitat) were found to be
significantly different (P >^ 0.05) with respect to
those structural variables measured, using the

2

Hotelling T test statistic. Bayesian
classification of initial observations (298 plots)
based on the two group discriminant function
analysis accurately classified group membership
(suitable and unsuitable habitat) for 96% of the
forested plots, with prior probabilities of group
membership being set proportional to the number of
observations in each group.

Figure ^ shows results of a 500-year
simulation of available habitat for the ovenbird
(Seiurus aurocapillus) on Walker Branch Watershed.
Results are presented as percentage of total land
area which provides potential breeding habitat for
the ovenbird over a 500-year period. Year zero
represents the present structural configuration of
the forest on the watershed. This was
accomplished by initializing the model with 25
randomly chosen census plots from the watershed
using vegetation data collected on these plots in
1977. Results are given for both simulations
including timber management (diameter-limit cut)
and undisturbed conditions. Simulations of
undisturbed forest dynamics show an initial
increase in available habitat for the ovenbird
from 20 percent of the land area to approximately
65 percent over the next 60 years. This decline
to year 90 is followed by a general oscillation of
available habitat from 10 to 20 percent for the
remainder of the simulation with only one period
of increase above 30 percent, that being at
approximately year 250.

In comparison, results of the simulation
which included timber management showed
considerable divergence from simulations of the
undisturbed forest. Dynamics previous to the
first cut at year 60 were identical for both
managed and undisturbed simulations. Following
the first cut at year 60, however, we see a
divergence with the managed stands showing an
increase in available habitat to 85 percent by
year 90 as compared to the 5 percent available
habitat for the undisturbed simulations. This
increase is followed by a continuous decline over
the next 30 years until the second cut at year
120. At this time once again we see an increase
in available habitat for the ovenbird to 50

118

OVENBIRD

OVENBIRD

10 -

— UNDISTURBED

- ■ TIMBER HARVEST

200 300

YEAR

500

Figure 4. Results from 500-year simulation of
ovenbird habitat on the Walker Branch Watershed
in east Tennessee. Available habitat expressed
as percentage of total land area of the
watershed .

CD
<
X

CO

<

]

<

>
<

u

0 10 20 30 -40 50 60 70 80 90 100 110 120

YEAR

Figure 5. Results from 120 year simulation of
available ovenbird habitat on the Walker Branch
Watershed under conditions of timber management.
Available habitat expressed as percentages of
the total land area of the watershed.

percent of the total forested area of the
watershed. Following the second timber harvest at
year 120 there is a decline in available habitat
continuing to the next cut at year 180. Following
the third cut there is a continuing decline to
year 210, when the model predicts there will be no
potential breeding habitat for the ovenbird on the
watershed. In all remaining cuts there is an
initial increase in available habitat following
harvest. This is followed by a decline, which
generally continues until the next cut, the last
being at year 480.

Figure 5 presents results of the same
simulation under conditions of timber management
that were presented in figure 4, only over a
shorter time scale showing the short-term dynamics
of available habitat following timber harvest
(year 60). Once again there is an initial
increase in availability of suitable habitat
followed by a decline at year 30. This decline
continues until the forest is cut at year 60, when
those trees greater than 23 cm dbh and of
commercial value as sawtimber are removed from the
forest. This thinning of the forest results in an
increase of potential habitat for the ovenbird
over the next 35 years, at which time a downward
trend begins and continues through the remainder
of the simulation.

DISCUSSION

Results presented in Figures 4 and 5
represent predicted dynamics of available habitat

for the ovenbird on the Walker Branch Watershed.
The 500-year simulation in figure 5 is meant to
show that extrapolations cannot necessarily be
made from results of a single timber cut to cuts
pending in the future. Resulting availability of
habitat for the ovenbird following an initial
diameter-limit cut on a 60-year rotation may not
represent availability of potential habitat the
forest will provide following future
diameter-limit cuts. Availability of habitat
following the first cut increased dramatically,
but subsequent cuts on a 60-year rotation yielded
quite different results. Secondly, failure to
look at potential effects of repeated harvests may
mislead the manager with respect to long term
habitat dynamics. As can be seen in results of
the 500-year simulation, the first cut was
followed by a four-fold increase in available
habitat for the ovenbird. Subsequent cuts,
however , resulted in less dramatic increases and
in some cases led to elimination of potential
habitat .

These results show the importance of historic
considerations in determining effects of
particular timber management practice on a given
forest. Structural configuration of the forest
prior to cutting is of utmost importance in the
case of repeated long-term management plans. To
date this type of information has been lacking.
FORHAB and models of its type can be used to
provide information on long-term management plans
and combinations of management schemes before
their actual implementation.

Results (figs. 4 and 5) show an initial

119

increase in potential habitat after the initial
cut at 60 years. This increase contrasts with
decline of habitat availability for undisturbed
simulations. The increase after cutting is a
result of thinning or general decrease in density
of stands on the watershed. At present the forest
has gone approximately 70 years without a timber
harvest, so by year 60 the stand is approximately
130 years in age. Thinning of larger trees, which
leads to an initial increase in available habitat,
is quickly followed by an increased density of the
understory, reducing available habitat. This is
the reason for the pattern of initial increase
after cutting followed by a subsequent decline in
availability of habitat for the ovenbird. These
results are in general agreement with the
structure of vegetation chosen as breeding habitat
by the ovenbird on the watershed. Optimal habitat
for the ovenbird on the Walker Branch Watershed
(as determined by the relationship between
vegetational structure and prey abundance) is a
deciduous stand with a sparse understory and brush
and little ground cover.'

It should be noted that the model simulates
availability of potential breeding habitat
expressed as a percentage of the total area under
consideration. The model does not simulate
population dynamics of a given bird species per
se. The ability of the ovenbird population to
track changes in availability of habitat such as
the initial increase in habitat following the
first timber harvest, or to reinvade after the
disappearance of potential breeding habitat in an
area are not considered explicitly in the model.
These considerations would depend on immigration
into the area or the existence of a "floater"
population of nonbreeding individuals unable to
establish territories due to lack of suitable
habitat. Likewise, the model does not consider
quality of habitat provided by a given stand.
Some marginal areas may become potential habitat
depending on size of the ovenbird population.
These points must be kept in mind when
interpreting results of simulation from the model.

The potential of predicting effects of
proposed management schemes on habitat
availability need not be limited to the breeding
season during spring and summer months. For many
resident bird species, availability of suitable
habitat during fall and winter seasons may be the
major limiting factor. Figure 6 shows seasonal
variation in habitat selection for several species
of birds on the watershed as expressed by the
first two discriminant functions describing the
vegetational structure of the forest stands
(Shugart et al . 1975). It can be seen that in
many cases the habitat needs of certain species
change seasonally. By including information on
seasonal habitat requirements into the
classification procedure, FORHAB and models of its
type could be used to determine habitat
availability not only on a yearly basis, but

*T.M. Smith, unpublished data on file at Oak
Ridge National Laboratory.

ORNL-OWG 75-3478

1 \ \ \ \ \ \ \ I \ \ — \ — r

F W

-0.5 0 0.5

HABITAT DISCRIMINANT FUNCTION II

Figure 6. Seasonal variation in habitat selection
for several species of birds inhabiting the
Walker Branch Watershed.

seasonal variations in available habitat.

Two-group discriminant function analysis (as
described in subroutine DISCRM) can be used to
construct habitat maps which reflect the potential
of a site to provide habitat for a given species
of bird. A map of potential habitat for five
woodpecker species on the Haw Ridge Watershed on
the D.O.E. reservation in east Tennessee is shown
in figure 7 (Shugart et al. 1978). The map was
constructed using a classification scheme
identical to that presented for subroutine DISCRM.
By initializing the model for a given forested
region such as the Haw Ridge Watershed (initial
conditions based on sample plots from the Haw
Ridge Watershed as was done for the Walker Branch
Watershed) it would be possible to construct a
series of maps representing changes in available
habitat for the area through time. This would aid
in more site specific, small scale management
problems .

The model FORHAB which has been presented as
an example of the process and methodology of
habitat simulation has dealt solely with the
Appalachian deciduous forest of the Southeast.
However, the general form of the forest simulation
model underlying this method (FORET) has been
adapted for a wide variety of forested areas,
including mixed conifer-hardwood forests of the
Northeast (Botkin et al. 1972), loblolly pine
forests of Arkansas (Milke 1978) and rain forests
of Puerto Rico (Doyle 1980). Another version of

120

ORNL OWG, 77-20962

Figure 7. Map of potential habitat for five species of woodpecker on Haw Ridge as defined by discriminant
function classification procedure.

the model FORMIS' simulates flood plain forests of
the Mississippi River and could be modified to
simulate riparian habitats in other areas.

Potential applications of the model to
predict effects of proposed and untried management
schemes on specific forested areas for both
individual species and the avian community as a
whole, as well as its versatility and adaptability
to a diverse array of forested regions, make
models like FORHAB potentially important tools for
forest and wildlife managers in the future.

ACKNOWLEDGMENTS

Research supported by National Science
Foundation under Interagency Agreement 40-700-78
with U.S. Department of Energy under contract
W-7405-eng-26 with Union Carbide Corporation.

LITERATURE CITED

Anderson, S.H., and H.H. Shugart. 1974, Habitat
selection of breeding birds in an East
Tennessee deciduous forest. Ecology
55:828-837.

Botkin, D.B., Jr., J.F. Janek, and J.R. Wallis.
1972. Some consequences of a computer model
of forest growth. Journal of Ecology
60:849-872.

Conner, R.N., and C.S. Adkisson. 1976.
Discriminant function analysis: A possible
aid in determining the impact of forest
management on woodpecker nesting habitat.
Forest Science 22:122-127.

DeGraaf, R.M., technical coordinator. 1978.
Workshop on management of southern forests
for nongarae birds. USDA Forest Service
General Technical Report SE-14, 175 p.
Southeastern Forest Experiment Station,
Asheville, N. C.

'M.L. Tharp, unpublished model available at
Oak Ridge National Laboratory.

Doyle, T.W. 1980. FORICO: Gap dynamics model of
the lower montane rain forest in Puerto Rico.
94 p. M.S. Thesis, University of Tennessee,
Knoxville.

Franzreb, K.E., and R.D. Ohmart. 1978. The
effects of timber harvest on breeding birds
in a mixed coniferous forest. Condor
80:431-441 .

Grigal, D.F., and R.A. Goldstein. 1971. An
integrated ordination-classification analysis
of an intensively sampled oak-hickory forest.
Journal of Ecology 59:481-492.

Hagar, D.C. I960. The interrelationships of
logging, birds, and timber regeneration in
the douglas-fir region of northwest
California. Ecology 41 (1 ): 1 16-125 .

Harris, W.F., R.A. Goldstein, and P. Sollins.
1973. Net above ground production and
estimates of standing biomass on VJalker
Branch Watershed, p. 41-64. In Young, H.E.,
editor. lUFRO Biomass Studies. University
of Maine Press, Orono, Me.

James, F.C. 1971. Ordinations of habitat
relationships among breeding birds. Wilson
Bulletin 83:215-236.

Kasanaga, H. , and M. Mousi . 1954. On the light
transmission of leaves and its meaning for
the production of matter in plant
communities. Japanese Journal of Botany
14:302-324.

Kendall, M.G., and A. Stuart. 1961. The advance
theory of statistics. p. 266-282. Volume 3.
Charles Griffin and Co., London.

Ker, J.W., and J.H.G. Smith. 1955. Advantages of
the parabolic expression of height-diameter
relationships. Forest Chronology 31:235-246.

Loomis, R.S., W.A. Williams, and W.G. Duncan.
1967. Community architecture and the
productivity of terrestial plant communities,
p. 291-308. In San Pietro, A., F.A. Greer,
and T.J. Army, editors. Harvesting the sun.
Academic Press, New York, N.Y.

MacArthur, R.H., and J.W. MacArthur. 1961. On
bird species diversity. Ecology 42:594-598.

121

Milke, D.L.. H.H. Shugart, and D.C. West. 1978.

Users manual for FORAR, a stand model for
composition and t^rowth of upland forests of
southern Arkansas. ORNL/TM-5767 , ESD
Publication No. 1021. Oak Ridge National
Laboratory, Oak Ridge, Tenn.

Morrison, D.F. 1967. Multivariate statistical
methods. 415 p. McGraw Hill, New York, N.Y.

Perry, T.O., H.E. Sellers, and CO. Blanchard.
1969. Estimation of photo-synthetically
active radiation under a forest canopy with
chlorophyll extracts and from basal area
measurements. Ecology 50:39-'^4.

Shugart, H.H., S.H. Anderson, and R.H. Strand.
1975. Dominant patterns in bird populations
of the eastern deciduous forest biome. p.
90-95. In Smith, D.R., techincal

coordinator. Management of forest and range
habitats for nongame birds: Proceedings of a
symposium [Tucson,' Ariz., May 6-7, 1975].
USDA Forest Service General Technical Report
WO-1, p. Washington, D.C.

Shugart, H.H., and B.C. Patten. 1972. Niche
quantification and the concept of niche
pattern. p. 284-326. In Patten, B.C.,
editor. Systems analysis and simulation in
ecology II. Academic Press, New York, N.Y.

Shugart, H.H., T.M. Smith, J.T. Kitchings, and
R.L. Kroodsma. 1978. The relationship of
nongame birds to southern forest types and
successional stages. p. 5-16. In DeGraaf,
R.M., techincal coordinator. Management of
southern forests for nongame birds:
Proceedings of a workshop [Atlanta, Ga . ,
January 24-26, 1978] USDA Forest Service
General Technical Report SE-14, 175 p.
Southeastern Forest Experiment Station,
Asheville, N. C.

Shugart, H.H., and D.C. West. 1977. Development
of an Appalachian deciduous forest succession
model and its application to assessment of
the impact of the chestnut blight. Journal
of Environmental Management 5:161-179.

Shugart, H.H., and D.C. West. 1980. Forest
succession models. Bioscience 30 (5 ): 308-31 3 .

Slusher, J. P., and T.M. Hinckley. 1974.
Timber-wildlife management: Proceedings of a
symposium. [Columbia, Mo., January 22-24,
1974] 131 p. Missouri Academy of Science.
Occasional Paper No. 3, Columbia, Mo.

Smith, D.R., technical coordinator. 1975.
Management of forest and range habitats for
nongame birds. Proceedings of a smyposium
[Tucson, Ariz., May 6-9, 19751. USDA Forest
Service General Technical Report WO-1, 343 p.
Washington, D.C.

Smith, R.L. 1980. ■ Ecology and field biology.
Third edition. 835 p. Harper and Row, New
York, N.Y.

Thomas, J.W., editor. 1979. Wildlife habitats in
managed forests the Blue mountains of Oregon
and Washington. 512 p. USDA Forest Service,
Agriculture Handbook 553, Washington, D.C.

Whitmore, R.C. 1975. Habitat ordination of
passerine birds of the Virgin River Valley,
southwestern Utah. Wilson Bulletin 87:65-74.

Appendix II. Model equations in FORHAB.

Process

Equations

Growth of each tree
under optimal con-
ditions^

Height/diameter
relation^

2

d[D H]
dt

R LA (1 -

DH

-)

'^ax^ax

R = growth rate parameter, LA = leaf area of tree,

D = diameter at breast height, H = height of tree,

D = maximum diameter for a particular species,
max

t^ax " maximum height for a particular species,
d2h = index of tree volume.

H - 137 + b2D - b3D2,

b, = 2(H - 137)/D
2 max max'

ax >

and b^ determined by setting
H = Umax t^^^/dt = 0 when D = U^^y^.

Extinction of 1 ight
as a function of
leaf area in forest
canopies'^

Reduction of
photosynthesis
due to shading^

Crowding effects
related to
stand biomass
(competition)^

Intrinsic tree
mortal ity '

Mortal ity of trees
with suppressed

Q(h)

Q exp(-k / LA(h')dh')
° h

growth

LA(h') = distribution of leaf area as a function of height

= incident radiation,
Q(h) = radiation at height (h),
K = constant^.

Various empirical equations fitted to light-photosynthesis
curves for shade-tolerant or shade-intolerant species found in
each forest. These equations are used to reduce the magnitude
of the growth equation (above) for shaded trees.

S(BAR) = 1 - BAR/SOILQ,

BAR = total biomass (basal area) of simulated stand,
SOILQ = maximum biomass (basal area) recorded.

p = 1 - (1 - £)",

p = probability of mortality at year n is chosen such that
p = 0.99 when n = AGEi^jx 't^^ maximum age for the species)

If growth is less than a critical value for the species,
p = 0.368.

iBotkin et al . 1972.
2Ker and Smith 1 955.

^Kasanaga and Monsi 1954, Loomis et al . 1967, Perry et al. 1969.
'^Kramer and Kozlowski 1960.

123

A WINDSHIELD AND MULTIVARIATE APPROACH TO THE
CLASSIFICATION, INVENTORY, AND EVALUATION
OF WILDLIFE HABITAT: AN EXPLORATORY STUDY'
C.E. Grue^ R.R. Reid», and N.J. Silvy^

Abstract. — Techniques are described for evaluating the
habitats of breeding mourning dove ( Zenaida macroura) , and
bobwhite (Colinus virginianus) and scaled quail (Callipepla
squamata) from within a vehicle. Audio counts of the three
species and habitat surveys were conducted on 133 (24 km)
transects in Texas in 1976. The linear distance of each
habitat type intersecting a transect and the number of
structural features within ca . 0.8 km were recorded. Habitat
variables, including indices of habitat interspersion and
diversity, were analyzed for correlations with audio counts
of the three species using stepwise multiple regression. We
used discriminant analysis to determine the accuracy of using
habitat variables to identify transects supporting below or
above average .densities of the three species.

Habitat variables accounted for 28 to 87% of the
variation in mourning dove call counts, and 60 to 95% and 32
to 93% of the variation in bobwhite and scaled quail whistle
counts, respectively. Discriminant analyses correctly
classified 78 to 89% of the mourning dove call-count surveys.
Comparable values for whistle counts of bobwhite and scaled
quail were 71 to 96% and 39 to 98%, respectively. Results
suggest techniques developed may be applicable to analyzing
the habitat of wildlife species for which transects are used
to obtain population estimates.

Key words: Bobwhite quail; discriminant analysis;
habitat classification; habitat evaluation; habitat
inventory; mourning dove; multiple regression; scaled quail;
Texas; windshield approach.

'Texas Agricultural Experiment Station
Technical Article 16235. Paper presented at The
use of multivariate statistics in studies of
wildlife habitat: a workshop, April 23-25, 1980,
Burlington, Vt.

^Graduate Research Assistant, Department of
Wildlife and Fisheries Sciences, Texas A&M
University, College Station, TX 77843. Present
position: Research Biologist, U.S. Fish and
Wildlife Service, Patuxent Wildlife Research
Center, Laurel, MD 20811.

'Graduate Research Assistant, Department of
Wildlife and Fisheries Sciences, Texas A&M
University, College Station, TX 77843. Present
position: Staff Biologist II, Espey-Huston and
Associates, Inc., P.O. Box 519, Austin, TX 78767.

"Associate Professor, Department of Wildlife
and Fisheries Sciences, Texas AiM University,
College Station, TX 77843. Present position:
Assistant Leader, Florida Cooperative Fish and
Wildlife Research Unit, 117 Newins-Ziegler Hall,
University of Florida, Gainesville, FL 32611.

124

INTRODUCTION

Classification

The abundance of wildlife is usually
determined by habitat conditions which alter the
carrying capacity of the land. These conditions
do not remain static. Our understanding of trends
in habitat quality and quantity is limited, even
for those wildlife species for which habitat needs
are generally known. It is, therefore, essential
to determine the habitat requirements of wildlife,
develop the capability to assess the abundance of
critical habitats, and monitor changes in their
quantity and quality.

The mourning dove is a species which has not
received management commensurate with its
popularity as a game bird (Amend 1969, Sandfort
1977). This species has the widest range of any
game bird in the United States and is the most
important in North America in terms of numbers and
hunter harvest (Keeler 1977). Management of
mourning doves in the United States, however, has
been almost entirely restricted to control of
harvest based on fluctuations in breeding
populations monitored nationwide by call-count
surveys (Dolton 1977). While some breeding
populations have decreased (Dolton 1977), hunter
harvest has increased to more than 49 million
(Keeler 1977). If demand for utilization of the
dove resource continues to increase, habitat
management may become essential. First steps in
initiating management plans will be habitat
inventory, analysis, and evaluation. The
objective of our study was to develop techniques
for evaluating the habitat of breeding m.ourning
dove from a vehicle ("windshield approach").
Texas was well suited for such a study because of
its size and habitat diversity (Gould 1975). In
addition, we concurrently evaluated habitats of
breeding bobwhite and scaled quail using these
same techniques to determine if they were
applicable to other wildlife species.

METHODS

Our study of habitats of breeding mourning
dove and \bobwhite and scaled quail in Texas
consisted of four steps: classification,
inventory, analysis, and evaluation. Classi-
fication was defined as identification and
placement of habitats into specific habitat types
according to established criteria. Inventory was
defined as the process by which abundance of a
particular wildlife species or habitat parameter
was determined along an audio-count transect. We
defined analysis as examination of habitat, its
types and structural features, and their
relationships to audio counts and one another.
Evaluation was defined as the process by which
habitats were ranked according to estimated
densities of the three species they supported.
From these data, use of habitat variables to
predict audio counts of the three species was
examined.

We developed a method of classifying habitats
from within a vehicle. Habitat type was defined
as a description of the vegetation of an area
consisting of a unique combination of canopy
composition and spatial distribution and ground
cover height and composition. Our hierarchial
habitat classification (fig. 1) was divided
horizontally into three major strata. (Figures
and tables follow literature cited). The first,
"physiognomic class", was used to describe
vertical and horizontal distribution of canopy
vegetation within a given area (for detailed
descriptions, see Grue 1977). The second level
subdivided habitats containing trees on the basis
of canopy composition (deciduous, coniferous, or
mixed) and presence or absence of understory. In
the present study, we also separated mesquite
( Prosopis spp . ) from other deciduous species. In
the third level, cropland, pasture, savannah,
parkland, desert scrub, woodland, and forest were
further divided based on height and/or composition
of ground cover. Classification of canopy and
ground cover composition and ground cover height
was based on relative abundance within each
composition and height category. If at least 75%
of canopy or ground cover was similar in
composition and height, it was considered
homogeneous. Canopy or ground cover in which less
than 75% was similar in composition was considered
mixed .

We also considered structural features within
habitat types; structures or characteristics other
than height and composition of ground cover, and
composition and spatial distribution of canopy,
which others (for review, see Grue 1977, Reid
1977) have suggested may be important to breeding
dove and quail. Included within this category
were the number of fences, shrubrows, windbreaks,
powerlines, roads, and railroad rights-of-way, and
whether or not these structures paralleled or
intersected the call-count transects. The number
of edges (an abrupt change in the physiognomy of
the vegetation excluding ecotones), permanent
water sources, buildings with associated
vegetation, washes, livestock feeders and
feedlots, gravel pits, irrigation and oil pumps,
and presence of snags (dead, defoliated woody
shrubs or trees) within 0.8 km of each transect
were included. Type of road surface on the survey
route (asphalt, gravel, sand, or dirt) and the
width of the road shoulder were recorded at each
stop.

Inventory

Call-count and whistle-count data for 1976
were obtained for the first 15 (3-min) listening
points (stops) located at 1 . 6 km intervals along
each of the 133 Federal and State mourning dove
call-count transects in Texas. Call counts were
conducted on each transect four times between 20
May and 10 June 1976 by personnel of the Texas
Parks and Wildlife Department (Dunks 1976). Quail

125

whistle counts were conducted concurrently on the
last three of the four surveys of each transect.
Audio-counts are believed to be reliable indices
of the relative abundance of breeding dove (Dolton
1977) and quail (Bennitt 1951, Elder 1956, Rosene
1957, Norton et al . 1961, Campbell et al . 1973,
Brown et al . 1978) within relatively large areas.

The habitat intersecting the transects was
surveyed during this 20-day time period using two
vehicles, each with a two-man team (Grue et al.
1976). To reduce error due to differences in
observers, trial surveys were conducted as a group
along several transects throughout most of Texas;
each team worked within different ecological areas
(Gould 1975) and one member of each team was
designated driver and odometer reader, while the
other person classified habitat throughout the
study. A team traveled and recorded habitat data
on both sides of each transect starting 0.8 km
before and ending 0.8 km after each stop. Each of
these 1.6 km units was defined as a transect
interval. The linear distance of each observation
of a habitat type intersecting the survey route,
measured to the nearest 0.02 km, and the number of
structural features present within 0.8 km (maximum
radius of audibility of each species; Davey 1955,
Baxter and Wolfe 1973) were also recorded within
each transect interval.

Analysis and Evaluation

Habitat variables were analyzed for
correlation with audio counts of the three species
by transect, statewide, and within ecological
areas, using stepwise multiple regression (Barr
and Goodnight 1972). Ecological areas of Gould
(1975) were selected because call counts were more
homogeneous within their boundaries than those of
other physiographic divisions of the State (Grue
1977) and have been used by the Texas Parks and
Wildlife Department to analyze annual call-count
data (Dunks 1976). Call-count and whistle-count
data for all surveys considered valid by the Texas
Parks and Wildlife Department were included in
analyses because variation in audio counts of the
three species between surveys was significant
(Grue 1977, Reid 1977). Independent variables
(habitat variables) entered and remained in models
if values for their partial F-statistics were
significant (P<0.05).

Habitat inter spersion and diversity indices
were included in all analyses. Interspersion was
calculated by summing frequencies of occurrence
for each habitat type within the 15 transect
intervals. Habitat diversity was calculated for
each transect using the Shannon-Weiner Index
(Shannon 19^*8). Transect call counts and whistle
counts were equal to the sum of the number of dove
and quail heard calling on the 15 stops,
respectively.

To determine importance of spatial
distribution of the canopy, canopy composition,
and ground cover height and composition in
predicting audio counts of the three species, we

evaluated seven simplifications of our habitat
classification. Simplifications corresponded to
horizontal strata within the hierarchy of the
initial classification from the most complex
(habitat type = physiognomic class + canopy
composition + ground cover height and composition)
to the least complex (habitat type = physiognomic
class). Simplifications were evaluated for each
of the three species by ecological area using
transect audio counts and habitat variables.
Structural features were excluded from stepwise
multiple regression analyses so that multiple
correlation coefficients represented differences
between habitat classifications. Indices for
habitat interspersion and diversity were also not
included because of the number of simplifications
examined. The simplified habitat classification
with the fewest habitat variables which also
resulted in high multiple correlation coefficients
statewide was judged the "best" classification
system for each species.

We developed an index to minimum habitat
interspersion applicable to the simplified habitat
classifications selected. Habitat interspersion
was more difficult to calculate than habitat
diversity (we continued to use the Shannon-Wiener
Index) because the number of times the habitat
types actually changed on both sides of a
call-count transect was not known. The new
interspersion index was based on the number of
habitat types present within a transect as well as
the presence or absence of each habitat type
within adjacent transect intervals. If a
particular habitat type was present within a
transect interval but was absent within an
adjacent interval, the value of the interspersion
index increased by 1. Conversely, if a particular
habitat type was present or absent within two
adjacent transect intervals, interspersion was
equal to 0 and the value of the index remained
unchanged. This process was continued until all
habitat types were examined within the 15 transect
intervals of each of the 133 call-count transects.
The interspersion index on a particular transect
was equal to this value plus the number of habitat
types present within the transect. The latter was
used as an indirect measure of minimum
interspersion within the transect intervals.

We used discriminant analysis (Barr and
Goodnight 1972) to determine the accuracy of
predicting audio-count classes of the three
species using the habitat variables within the
simplified habitat classifications selected.
Analyses were conducted by ecological area and
only those habitat parameters within the multiple
linear regression models for a given ecological
area were included. A mean transect audio-count
was determined for each species and ecological
area using data from all call-count and
whistle-count surveys conducted in 1976 (table 1).
Transect audio counts for each species and survey
were then classified into one of two classes:
average or above, or below average. The use of
habitat variables to predict audio-count classes
was evaluated in terms of percent of call-count
and whistle-count surveys correctly classified

126

within each class by the discriminant functions.
Multiple analysis of variance (Barr and Goodnight
1972) was used to test for statistically
significant (P<0.05) differences in values for the
habitat variables between the two audio-count
classes .

RESULTS

Habitat variables within the initial habitat
classification accounted for up to 87% of
variation in mourning dove call counts, and up to
94 and 93% of variation in bobwhite and scaled
quail whistle counts, respectively (table 2).
Analyses within ecological areas resulted in an
increase in multiple correlation coefficients
and/or a decrease in the number of independent
variables in the models. Both habitat types and
structural features accounted for a significant
portion of variation in audio counts of the three
species (table 3).

Simplification of the initial habitat
classification resulted in a significant reduction
in the number of habitat types, but a relatively
slight decrease in multiple correlation
coefficients (table 4). The number of habitat
types within the simplified habitat classification
selected for each of the three species was reduced
90 to 96%, while multiple correlation coefficients
decreased by only 4 to 14%. Of seven simplified
habitat classifications evaluated, physiognomic
class with canopy composition and cropland
divisions of grain, nongrain, forage, and plowed
ground accounted for the most variation in
mourning dove call counts. Physiognomic class
with canopy composition with mesquite was the
simplified habitat classification selected for
both species of quail.

Models for predicting audio counts of the
three species incorporating both structural
features and habitat types within the simplified
habitat classifications selected are presented in
table 5. Multiple correlation coefficients for
these models were similar to those associated with
the initial habitat classification (table 2).
Models accounted for 28 to 88% of variation in
mourning dove call counts, and 60 to 95? and 32 to
93% of variation in bobwhite and scaled quail
whistle counts, respectively, within the 10
ecological areas.

Discriminant functions incorporating the
habitat variables within these models correctly
classified 75 to 89% of mourning dove call-count
surveys within each ecological area into the two
audio-count classes (table 6). Comparable values
for whistle-count surveys of bobwhite and scaled
quail were 71 to 96% and 39 to 98%, respectively
(table 6). Multiple analyses of variance
indicated that there were significant (P<0.05)
differences between values for the habitat
parameters within the discriminant functions for
the two audio-count classes for each species.

DISCUSSION

Results suggested that the habitat
classification and techniques we employed may
identify habitat variables significantly
correlated with audio counts of mourning dove and
bobwhite and scaled quail. Both habitat types and
structural features appeared to be important
components of habitats selected by the three
species during the breeding season. Analyses by
transect within ecological areas usually resulted
in an increase in multiple correlation
coefficients or a decrease in the number of
independent variables in models for the three
species. These trends may be explained by
increased homogeneity in audio counts (Foote et
al. 1958) or habitat within ecological areas
(Blankenship et al . 1971). Analysis within
ecological areas may also more accurately describe
the relationships between audio counts of the
three species and habitat variables. Simple
correlation analyses (Grue 1977; Reid et al. 1978,
1979) suggested audio counts of the three species
were correlated with habitat variables that may
have provided requisites necessary for survival
and reproduction. Habitat parameters which
provided these requisites differed between
ecological areas and appeared to depend on the
abundance and distribution of the habitat types
and structural features present. For example,
mourning dove call counts were positively
correlated with cropland within the Trans-Pecos,
but were negatively correlated with this habitat
type on the High Plains. In the Trans-Pecos,
nesting substrate was abundant, whereas sources of
food and water were generally restricted to
cultivated areas. The opposite was true on the
High Plains, where food and water were more
abundant, but nest sites within woody vegetation
were limited. Results suggest future analyses and
evaluations should be conducted within physio-
graphic divisions.

With a few exceptions, the amount of
variation in audio counts of the three species
accounted for by habitat types within the initial
and simplified habitat classifications selected
was similar. Exclusion of mesquite as a separate
canopy type from the simplified habitat
classification selected for mourning dove may
account for the low multiple correlation
coefficient associated with models for the South
Texas Plains and Rolling Plains. Selection of
mesquite habitats by nesting doves on the Rolling
Plains ha,s been reported (Jackson 1940).
Inclusion of mesquite as a separate canopy type
appears to be important in evaluating the habitat
of nesting mourning doves within these ecological
areas, as well as habitats of both species of
quail throughout most of Texas during the breeding
season. Similarly, absence of ground cover
characteristics within the simplified habitat
classification selected for bobwhite and scaled
quail may have resulted in low multiple
correlation coefficients associated with models
for the Edwards Plateau, Rolling Plains, and High
Plains, ecological areas in which overgrazing or
cultivation was extensive. Ground cover height

'mm

:ii!ir
m

m

iiiiiiiiili '
""?[

It

127

and composition, however, appear not to be as
important as the spatial distribution and
composition of the canopy in evaluating habitats
of the three species during the breeding season in
Texas.

Results of multiple regression and
discriminant analyses suggest that habitat
variables might be used to predict audio counts of
the three species and identify areas supporting
above or below average densities within most of
the ecological areas of Texas. Reasons for the
low multiple correlation coefficients associated
with models for mourning dove call counts within
the Gulf Prairies and Marshes and scaled quail
whistle counts within the Trans-Pecos are not
known. High habitat heterogeneity in conjunction
with low and uniform mourning dove call counts
within the Gulf Prairies and Marshes (Grue 1977)
may account for the low multiple correlation
coefficient for this model. Conversely, habitat
on transects within the Trans-Pecos was relatively
homogeneous while whistle counts of scaled quail
varied (Reid 1977), suggesting our habitat
classification may not have included an important
component of the habitat which this species
selected for nesting.

Several improvements may be useful in testing
and applying the techniques presented. Two
improvements could be made in methods used to
inventory habitat parameters. Habitat data could
be collected using "mark-sense" computer data
forms which would eliminate manual transfer of
data to computer cards. Collection of habitat
data in the sequence observed would permit direct
calculation of habitat inter spersion at any level
within the habitat classification. In addition,
aerial photographs and satelite imagery may
facilitate inventory of habitat data, particularly
within remote areas.

Data analysis could be improved by increasing
the number of transects within physiographic units
so that the number of observations exceeds the
number of habitat types within the habitat
classification selected. In the present study, we
included audio-count data for all surveys
conducted on the 133 call-count transects because
differences in audio counts between surveys were
significant. By including variation in call
counts between surveys, high multiple correlation
coefficients were more difficult to obtain (Grue
1977). However, inclusion of multiple surveys of
individual transects artificially increased sample
sizes within ecological areas, as surveys of the
same transect were not statistically independent.

Limitations of stepwise multiple regression
analyses should also be considered. Habitat
parameters within models may not be the only ones
significantly correlated with density. In
addition, individual regression coefficients may
depend on other variables within a model.
Examination of correlation matrices may,
therefore, prove useful in 1) identifying all
habitat variables significantly correlated with
density of the three species, 2) identifying

habitat variables which may be more easily
inventoried than those within the models and
substituted without a significant reduction in
multiple correlation coefficients, and 3)
establishing guidelines for improving habitat for
nesting mourning dove and bobwhite and scaled
quail .

Use of the techniques presented to predict
fluctuations in wildlife density and evaluate and
manage wildlife habitat are outlined in figure 2.
Procedures may be divided into three stages: 1)
development of multiple linear regression models
and discriminant functions which account for the
greatest amount of variation in density indices
and classes, respectively, and identification of
habitat parameters significantly correlated with
density of the wildlife species of interest; 2)
testing of multiple linear regression models and
discriminant functions; and 3) use of models to
predict fluctuations in wildlife density, or use
of discriminant functions and simple correlation
analyses to evaluate and manage wildlife habitat.
Procedures within the first stage have already
been discussed here and elsewhere (Grue 1977, Reid
1977, Reid et al. 1978, 1979).

Multiple linear regression models may be
tested by inventorying habitat parameters and the
wildlife species of interest simultaneously on
transects within the physiographic unit the models
represent and comparing predicted densities and
their confidence limits with observed values
through time. If predicted and observed densities
remain similar over time (i.e., habitat condition
is the major factor governing fluctuations in
density), the model(s) may be used to predict
fluctuations in density. However, if predicted
and observed densities differ significantly over
time, other environmental factors (e.g., weather,
disease, hunting) may be governing fluctuations in
the density of species.

Discriminant functions should be tested by
inventorying both habitat parameters and wildlife
species of interest on additional transects within
the physiographic unit for which each function was
developed. If the percentage of the test
transects correctly classified into each density
class is high, the functions may be used to
evaluate the habitat intersecting other transects
within the physiographic unit for which they were
developed. In areas for which the predicted
density class of the wildlife species of interest
is low, management recommendations could be made
to increase those habitat parameters positively
correlated with density of the species.

CONCLUSIONS

We believe the habitat classification and
techniques described may be applicable to
evaluating habitats of breeding mourning dove and
bobwhite and scaled quail throughout their ranges,
and habitats of other wildlife species for which
line transects are used to collect population
data. Methods presented may prove useful in

128

predicting annual fluctuations in wildlife
density, in determining effects of habitat
modification on wildlife density, and in
evaluating and managing wildlife habitat. Further
research into development and testing of a
windshield approach to the evaluation of wildlife
habitat is needed and appears justified.

ACKNOWLEDGEMENTS

This study was funded by the U.S. Fish and
Wildlife Service, the Caesar Kleberg Research
Program in Wildlife Ecology, and The Agricultural
Experiment Station, Texas A&M University, in
cooperation with the Texas Parks and Wildlife
Department, We gratefully acknowledge the
assistance of F.W, Martin, Director, Migratory
Bird and Habitat Research Laboratory, U.S. Fish
and Wildlife Service, J.H. Dunks and J,T.
Robertson, Texas Parks and Wildlife Department,
and personnel of the Texas Parks and Wildlife
Department who conducted the audio counts. We are
also indebted to J,L, Folse and W,E. Grant for
review of the manuscript. This paper constitutes
part of a dissertation and master's thesis by the
senior and second authors, respectively.

LITERATURE CITED

Amend, S.R, 1969. Progress report on Carolina
Sandhills mourning dove studies. Proceedings
Annual Conference Southeastern Association
Game and Fish Commissioners 23:191-201.

Barr, A.J,, and J,H, Goodnight, 1972, A user's
guide to the statistical analysis system,
260 p. SAS Institute, Inc, Raleigh, N, Car.

Baxter, W.L., and C.W. Wolfe. 1973. The
intersper sion index as a technique for
evaluation of bobwhite quail habitat. p.
158-165. In Morrison, J. A. and J.C. Lewis,
editors. Proceedings First National Bobwhite
Quail Symposium [Stillwater, Okla . , April
23-26, 1972]. Oklahoma State University.

Bennitt, R. 1951. Some aspects of Missouri quail
and quail hunting, 1938-19'48. 51 p.
Missouri Conservation Commission Technical
Bulletin 2.

Blankenship, L.H., A.B. Humphrey, and D.
MacDonald. 1971. A new stratification of
mourning dove call-count routes. Journal of
Wildlife Management 35:319-326.

Brown, D.E., C.L, Cochran, and T.E. Waddell.
1978. Using call-counts to predict hunting
success for scaled quail. Journal of
Wildlife Management 42:281-287.

Campbell, H. , D,K. Martin, P.E. Ferkovich, and
O.K. Harris. 1973. Effects of hunting and
some other environmental factors on scaled
quail in New Mexico. 49 p. Wildlife
Monograph 34.

Davey, P. 1953. The mourning dove in southern
Michigan. M.S. Thesis. 47 p. University of
Michigan, Ann Arbor, Mich.

Dolton, D.D. 1977. Mourning dove status report,
1976. 27 p. U.S. Fish and Wildlife Service
Special Scientific Report Wildlife 208.

Dunks, J.H. 1976. Statewide mourning dove
research; Job No. 1: Density, distribution,
and movement; Texas. Report on Federal Aid
Project W-95-R-10. 9 p. Texas Parks and
Wildlife Department.

Elder, J.B. 1956. Analysis of whistling patterns
in the eastern bobwhite, Colinus _v.
virginianus L. Proceedings Iowa Academy
Science 63:639-651.

Foote, L.E., H.S. Peters, and A.L. Finkler. 1958.
Design tests for mourning dove call count
sampling in seven southeastern states.
Journal of Wildlife Management 22:402-408.

Gould, F.W. 1975. Texas plants - a checklist and
ecological summary. 121 p. Texas A&M
University Agricultural Experiment Station.
MP-585/Rev.

Grue, C.E. 1977. Classification, inventory,
analysis, and evaluation of the breeding
habitat of the mourning dove ( Zenaida
macroura) in Texas. Ph.D. Thesis. 187 p.
Texas A&M University, College Station, Tex,

Grue, C.E., R.R. Reid, and N.J. Silvy. 1976. A
technique for evaluating the breeding habitat
of mourning doves using call-count transects.
Proceedings Annual Conference Southeastern
Association Game and Fish Commissioners,
30:667-673.

Jackson, A.S. 1940. The mourning dove in
Throckmorton County, Texas. M.S. Thesis.
122 p. North Texas State University,
Denton, Tex.

Keeler, J.E., chairman. 1977. Mourning dove
(Zenaida macroura) . p. 275-298. 1t\
Sanderson, G.C., editor. Management of
migratory shore and upland game birds in
North America. 358 p. International
Association Fish and Wildlife Agencies,
Washington, DC.

Norton, H.W., T.G. Scott, W.R. Hanson, and D.W.
Klimstra. 1961. Whistling-cock indices and
bobwhite populations in autumn. Journal
of Wildlife Management 25:398-403.

Reid, R.R. 1977. Correlation of habitat
parameters with whistle-count densities of
bobwhite ( Colinus virginianus) and scaled
quail (Callipepla squamata) in Texas. M.S.
Thesis. 101 p. Texas A&M University, College
Station, Tex.

Reid, R.R., C.E. Grue, and N.J. Silvy. 1978.
Breeding habitat of the bobwhite in Texas.
Proceedings Annual Conference Southeastern
Fish and Wildlife Agencies 32:62-71.

Reid, R.R., C.E. Grue, and N.J. Silvy. 1979.
Competition between bobwhite and scaled quail
in Texas. Proceedings Annual Conference
Southeastern Association Fish and Wildlife
Agencies 33: 146-153.

Rosene, W. , Jr. 1957. A summer whistling cock
count of bobwhite quail as an index to
wintering populations. Journal Wildlife
Management 21:153-158.

Sandfort, W.W, 1977. Introduction. p. 1-3. In
Sanderson, G,C,, editor. Management of
migratory shore and upland game birds in
North America. 358 p. International
Association Fish and Wildlife Agencies,
Washington, D.C.

Shannon, C.E. 1948. A mathematical theory of
communication. Bell System Technical Journal
27:379-423: 623-656.

HABITAT
I I

Nol Inundated

O -I

> S:
u O

O IS

PLOWED FIBER ROOT GRAIN VEGETABLE FORAGE
LAND CROPS CROPS CROPS CROPS CROPS

(W/0 GROUND COVER]

TALL
> 0.50m

Figure 1. Schematic diagram of habitat classification (after Grue et al . 1976). Habitats were keyed to
habitat types from top to bottom. Names were assigned habitat types from bottom to top using words in
bold capital letters describing the height and compostion of the ground cover, composition of the canopy,
and the physiognomic class, in that order (e.g., tall grass deciduous savannah). Words in brackets
specifying the presence or absence of ground cover or understory comprised the last portion of the name
for habitat types where appropriate (e.g., mixed woodland with understory).

130

Table 1. Mean transect audio counts for mourning dove (MD), bobwhite quail (BW),
and scaled quail (SQ) within the 10 ecological areas of Texas in 1977. Means are
rounded to nearest whole bird. N represents the number of valid surveys
conducted .

MD BW SQ

Ecological area

N

X

SD

N

X

SD

N

X

SD

Pineywoods

32

10

7.'»

23

13

12. 8

23

0

Gulf Prairies and Marshes

18

7

3.8

13

43

16. 4

13

0

Post Oak Savannah

33

17

13.2

24

30

19.5

24

0

Blackland Prairies

HO

17

9.5

28

29

13.2

28

0

Cross Timbers and Prairies

62

27

21.3

44

46

27.7

44

0

South Texas Plains

70

28

17.6

52

27

18.9

52

2

3.8

Edwards Plateau

72

18

12.7

54

12

15.6

54

5

8.3

Rolling Plains

88

45

26.3

64

38

20.4

64

4

7.1

High Plains

52

7

7.9

38

6

8.2

38

3

4.7

Trans-Pecos

35

16

18.9

26

0

26

10

5.0

Table 2. Correlations between audio counts of mourning dove (MD) , bobwhite quail
(BW), and scaled quail (SQ) and habitat variables including habitat types within
the initial habitat classification and structural features. Multiple correlation
coefficients are expressed as a percent. The number of variables remaining in
the models is given in parentheses.

No. habitat
variables

Ecological area present MD BW SQ^

Pineywoods

98

83.3

(3)

91.1

(3)

Gulf Prairies and Marshes

96

28.3

(1)

82.2

(2)

Post Oak Savannah

126

85.9

(3)

94.3

(4)

Blackland Prairies

105

68. 1

(3)

58.9

(3)

Cross Timbers and Prairies

145

79.3

(6)

83.6

(5)

South Texas Plains

117

78.2

(4)

93.9

(8)

92.6

(4)

Edwards Plateau

66

64.1

(6)

85.7

(6)

92.9

(5)

Rolling Plains

92

73.5

(10)

89.2

(10)

78. 1

(4)

High Plains

52

62.9

(2)

93.8

(5)

81.7

(4)

Trans-Pecos

36

86.5

(3)

31.9

(1)

Texas

194

79.8

(54)

89.5

(51)

65.8

(16)

131

Table 3. Correlations between transect audio counts of mourning dove (MD), bobwhite (BW), and scaled quail
(SQ) and habitat variables considering structural features and habitat types within the initial habitat
classification, separately. Multiple correlation coefficients are expressed as percents. The number of
variables remaining in the models is given in parentheses.

Structural features Habitat types

Number

jJi c 0 1 1 u

MD

BW

SO

Number

MD

BW

so

Pi fio vu/^r^H ^

1 Q

80.2(3)

7R

1 J

83.

.2(3)

ftiil'T Pr'a'ir'Tfic:

and Marshes

19

28.

,3(1 )

80. 0(2)

73

22.

.8(1 )

82.

,2(2)

Post Oak Savannah

20

87.

,0(U)

80.7(2)

102

85.

.9(3)

94.

,9(5)

Blackland Prairies

20

47.

.5(1 )

46.2(2)

81

70.8(4)

58.

,9(3)

Cross Timbers
and Prairies

20

78.

.4(10)

65.5(5)

121

79.

,3(6)

83.

,5(6)

South Texas Plains

22

70.

,2(8)

76. 1(6)

92.6(3)

91

78.

,2(4)

94.

,3(9)

92.7(5)

Edwards Plateau

16

32.

,7(4)

54.7(6)

44.8(3)

47

64,

,3(6)

83.

,6(5)

93.2(10)

Rolling Plains

19

47.

,7(5)

40.9(5)

44.8(3)

70

72.

1(10)

86.

5(10)

78.0(4)

High Plains

17

51.

3(4)

75.2(4)

74.7(3)

32

62.

9(2)

93.

5(5)

81.7(4)

Trans-Pecos

16

84.

1(3)

31.9(1)

17

81.

5(2)

30.0(1)

Mean

19

60.

7(4)

67.8(4)

57.8(3)

71

70.

1(4)

85.

4(5)

75. 1(5)

Table 4. Correlations between transect audio counts of mourning dove, bobwhite quail, and scaled quail and
habitat types within simplified habitat classifications. Multiple correlation coefficients are expressed
as percents; those underlined represent the habitat classification scheme selected as 'best.' The number
of potential habitat types within each classification scheme is given in parentheses.

Species

Ecological
area

Initial

(501)

(19)

Physiognomic class

w/mesquite
(33)

(29)

w/canopy composition

w/crops w/mesquite

(34) (43) (44)

w/under story

w/ crops

(49)

Mourning
Dove

Pineywoods 83.2

Gulf
Prairies &
Marshes

22. 8

83.3

0.0

83.3

0. 0

82.9

0.0

82.9

22. 8

82.9

0.0

82.2

0.0

82.2

22. 8

Post Oak
Savannah

85.9

86.7

85.8

87.6

85.9

84.5

85.5

85.9

Blackland
Prairies

70.8

70.0 71.0 70.2 70.2

65.2

69.3 56.2

(continued)

132

(table 4 continued)

Cross
Timbers &
Prairies

79.3

61 . 9

68.9

71.7

78.it

79. 1

73. 1

79.0

South Texas
Plains

Edwards
Plateau

Rolling
Plains

78.2

64.3

72. 1

High Plains 62.9

Trans-Pecos 81.5

Mean 70. 1

Bobwhite Pineywoods 91.1
Quail

Gulf
Prairies &
Marshes

Scaled
Quail

Post Oak
Savannah

Blackland
Prairies

Cross
Timbers &
Prairies

South Texas
Plains

Edwards
Plateau

Rolling
Plains

82.2
94.9
58.9

83.5
94.4
83.6

86.5

High Plains 93.5
Mean

South Texas
Plains

Edwards
Plateau

Rolling
Plains

High

Plains 81.7

Trans-Pecos 30.0

Mean 75.1

85.4
92.7
93.2
78.0

35.0

42.6

35.8
64.0
80.7
56.0
90.6

79.3
92.2
0.0

0.0
68.8
75.3

46.4
75.5
58.7

19.5

44.6

58.7

66.2
0.0
37.8

74.3

42.6

40. 1
64.0
24.3
55.4
90.6

81 .9
87.9
63.7

58.7

69.4

79.7

72.5
75.4
75.5

70. 1

36.9

67.3

81.7
30.0
57.2

28.2
55. 1

36.4

64.0
80.7
57.6
88.7

80.5
94.2
59.4

0.0

68. 1

75.6

50.2
75.5
65.8

19.5

44.6

58.7

66.2
0.0

37.8

60.5
55. 1

36.4
63.9
81.5
63.8

69.3
94.4
16.6

60.2

79.8

75.6

57. 1
80.0
69. 1

19.5

44.6

58.7

74.6
15.5
42.5

75.6
57.8

50.2
64.0
24. 3
58.4
88.7

81.9
89.0
64.7

79. 1

94.7

81.5

70.3
72.4

80.3

85.3

42. 0

67.3

81.7
30.0
61.3

29.3

55. 1

36.4
64.0
80.7
57.6
90.2

80.4
94.2
52.9

0.0

75.6

52.8
88.6
67.0

19.5

44.6

58.7

70.3
0.0
38.6

60.4
55. 1

36.4
64.0
81.5
62.3
90.2

69.3

38.5

60.2

79.8

75.6

52.8
88.6
71.5

19.5

44.6

58.7

74.4
15.5
42.5

1*11

•I'll

lllll'

n

133

Table 5. Multiple linear regression models for transect audio counts of mourning dove (MD), bobwhite quail
(BW), and scaled quail (SQ) and habitat variables including habitat types within the simplified habitat
classifications selected and structural features. Multiple correlation coefficients are expressed as
percents .

Ecological Area

Model

Pineywoods

Gulf Prairies & Marshes

Post Oak Savannah

Blackland Prairies

MD = -0.4 + 0.8EPARALLEL SHRUBROWS]
+ 0.4[PARALLEL POWERLINES]

4.0[DECIDU0US FOREST]

BW = -15.7 + 12.4[DECIDU0US SAVANNAH] + 9.5CDECIDUOUS FOREST]
+0.3[PARALLEL FENCES]

MD r 1,3 + 0. 1 [INTERSECTING FENCES]

BW = 22.5 + 41.5[SAND ROAD SURFACE] + 18.2[SHRUB SAVANNAH]

MD = 28.0 + 18.8[HAY] - 7.0[GRAVEL PITS] + 10.3CSHRUB PARKUND]
- 1.4[PASTURE OR FIELDS]

BW = 11.6 + 54.5EMESQUITE WOODLAND] + 33.9[MIXED MESQUITE SHRUB
PARKLAND] + 101 . 6[C0NIFER PARKLAND] + 307. ^[ORCHARDS]

MD = 55.6 - 1. 1 [INTERSECTING ROADS] + 86. 7 [ORCHARDS ] -0. 1 [BUILDINGS
AND ASSOCIATED VEGETATION]

BW = 19.6 - 99. 1 [MESQUITE SHRUB SAVANNAH] + 0. 5 [BUILDINGS AND
ASSOCIATED VEGETATION] + 1.1 [PARALLEL POWERLINES]

81.3

90.9
28. 3
82.0

87.5

93.8

68. 1

60. 1

Cross Timbers & Prairies

South Texas Plains

Edwards Plateau

MD = 98.8 + 6.1 [GRAVEL PITS] - 0. 3 [ INTERSECTING FENCES]

+ 6.8[DECIDUOUS WOODLAND] - 5.0[WASHES] + 108. 4[C0NIFER PARKLAND]

- 37.0[HABITAT DIVERSITY] + 3.4[PLOWED LAND] 71.5

BW = 611.8 + 22.5[PARALLEL FENCES] - 181.7[BARREN LAND]

- 59.3[SHRUB PARKLAND] + 0.8[ROAD SHOULDER WIDTH] + 5.8
[GRAVEL PITS] - 259.2[MIXED MESQUITE SHRUBLAND] + 34.3[MIXED

MESQUITE SHRUB SAVANNAH] 77.9

MD = 7.6 + 19.5[HAY] - 4.9[PASTURE OR FIELDS] + 2.2[SNAGS] -
0.6[INTERSECTING SHRUBROWS] + 14.8[HABITAT DIVERSITY]

- 2. 1 [DECIDUOUS PARKLAND] 74.7

BW = 12.0 + 5.1 [MESQUITE PARKLAND] + 12.0[HABITAT DIVERSITY]

- 2.0[GRAVEL ROAD SURFACE] + 3. 0[LI VESTOCK FEEDERS]

+ 5.0[BRUSHLAND] + 5. 9 [ INTERSECTING RAILROADS] + 0.4[BRUSH
W/MESQUITE] + 5.7[DECIDUOUS WOODLAND] + 1.0[MIXED MESQUITE
SHRUBUND] + 0.4[PARALLEL POWERLINES]

SQ = 8.9 - 0.2[INTERSECTING POWERLINES] + 0.3[PARALLEL WINDBREAKS]
+ 4.1 [URBAN DEVELOPMENT] - 1 . 8 [ INTERSECTING RAILROADS]

- 2.9[IRRIGATION PUMPS] + 2.6 [BRUSHLAND] - 0.2[ROAD SHOULDER
WIDTH] - 1.5[DECIDU0US SAVANNAH] - 0.3 [MESQUITE PARKLAND]
+ 0.2[ASPHALT ROAD SURFACE] + 0. 3[SHRUBLAND]

MD = 2.9 + 14.5[URBAN DEVELOPMENT] + 4. 3[BRUSHLAND] + 4.9[MIXED

PARKLAND] +2.9 [PASTURE OR FIELDS] + 0. 3[INTERSECTING FENCES]

- 1. 1 [MIXED WOODUND] + 5.2 [INTERSECTING RAILROADS]

BW = 1.9 + 7. 6[BRUSHLAND] + 3- 8[DECIDU0US SAVANNAH]
WOODLAND] - 1.9[MESQUITE SHRUBLAND]

2.2[MESQUITE

SQ = 7.2 - 3.9[HABITAT DIVERSITY] + 6.5[SHRUB SAVANNAH]

+ 17.3[INTERSECTING RAILROADS] + 0. 4 [SHRUBLAND] - 1.0[WASHES]
- 0.7[INTERSECTING ROADS] +0.4 [DECIDUOUS SAVANNAH]
+ 0.1 [PARALLEL POWERLINES]
(continued)

95.0

93. 1

63.9

81.5

93.3

134

(table 5 continued)
Rolling Plains

MD = 2.1 + 3.i|[DECIDU0US PARKLAND] + 2,0[SNAGS] + 26.0[HABITAT
DIVERSITY] - 3.3 [PARALLEL WINDBREAKS]

55.7

High Plains

Trans-Pecos

BW = - 18.6 + 66.8CHABITAT DIVERSITY] + 33 . 1 [MESQUITE SHRUB

PARKLAND] + H.1 [INTERSECTING WINDBREAKS] - 1 . 7 [INTERSECTING

ROADS] - 2.3[MESQUITE SAVANNAH] + 98.3[CONIFER SAVANNAH]

+ 3. 4 [BUILDINGS AND ASSOCIATED VEGETATION] - 2.7 [INTERSECTING

POWERLINES] - 371 . 1 [ORCHARDS] + 198. ^[CONIFER PARKLAND]

+3. 1 [MESQUITE PARKLAND] - 5. 9 [MIXED MESQUITE PARKLAND]

+ 56.2[DECIDU0US WOODLAND] - 1.3[HABITAT INTERSPERSION ] 87.4

SQ = - 4.5 + 2.UWASHES] + 4.2[SHRUB PARKLAND] - 1 1 . 3 [MESQUITE

SHRUBLAND] +0.4 [PARALLEL POWERLINES] + 3.8[MESQUITE SHRUB

PARKLAND] + 1 . 7 [INTERSECTING RAILROADS] 75.4

MD = 0.3 + 71.9[DECIDUOUS SAVANNAH] + 0.7[PASTURE OR FIELDS] 63.9

BW = 3.0 + 62.8[SHRUBLAND] + 0.9[GRAVEL ROAD SURFACE] - 32.0[SHRUB
PARKLAND] + 0. 3[BUILDINGS AND ASSOCIATED VEGETATION]

- 0.6[ROAD SHOULDER WIDTH] - 3.3 [MESQUITE SHRUB SAVANNAH] 92.6

SQ = 0.7 + 25.7[MESQUITE SHRUBLAND] + 74.8[SHRUB SAVANNAH]

+ 33.2[SHRUB PARKLAND] + 4.7[URBAN DEVELOPMENT] 81.7

MD = 16.1 + 23.5[PLOWED LAND] - 4 . 7 [BUILDINGS AND ASSOCIATED

VEGETATION] - 0.5 [HABITAT INTERSPERSION] 86.6

SQ = 8.9 + 0.6[IRRIGATION AND OIL PUMPS] 31.9

Table 6, Percent audio-count surveys of mourning dove (MD), bobwhite quail (BW) and scaled quail (SQ)
correctly classified as below average or average and above by discriminant analyses. Discriminant
functions incorporated the habitat variables within multiple linear regression models for each ecological
area (see table 5); differences in habitat variables between audio-count classes were significant (MANOVA:
P<0.05). N represents the number of audio count surveys.

Ecological area

MD

BW

SQ

Pineywoods

32

81.3

23

95.7

Gulf Prairies and Marshes

18

77.8

13

84.6

Post Oak Savannah

33

84.8

24

70.8

Blackland Prairies

40

82.5

28

82. 1

Cross Timbers and Prairies

62

85.5

44

90.9

South Texas Plains

70

84.3

52

94.2

96.2

Edwards Plateau

72

77.8

54

74. 1

98. 1

Rolling Plains

88

79.5

64

95.3

93.8

High Plains

52

75.0

38

89.5

89.5

Trans-Pecos

35

88.6

26

38.5

Mean

81.7

86.4

83.2

'1

iiir

135

Figure 2. Flow diagram for the development, testing, and use of a windshield approach to the evaluation of
wildlife habitat.

SELECT
WILDLIFE

SPECFES

SPECIES RANGE

MSTINC
lYSIOGRAPHIC
VISIONS

SELECT

PHYSIOGRAPHIC
DIVISIONS

HOMOGENEITY OF
DENSITY INDICES FOR
WILDLIFE SPECIES

LSTABLISH RANDOM
LINE TRANSECTS
FOR INVENTORY OF
HABITAT PARAMETERS
AND WILDLIFE SPECIF.S

RANDOM LINE
TRANSECTS PREVIOUSL'
ESTABLISHED

ESTABLISHED HABITAT

CLASSIFICATION

SCHEMES

POTENTIALLY
IMPORTANT HABITAT
PARAMETERS

SELECT HABITAT
CLASSIFICATION
SCHEME

SELECT APPROPRIATE
DENSITY INDEX FOR
WILDLIFE SPECIES
OBTAINABLE FROM

DENSITY INDEX

PREVIOUSLY

ESTABLISHED

ESTABLISH TECHNIOUES
FOR INVENTORY OF
HABITAT PARAMETERS
AND WILDLIFE SPECIES

TECHNIOUES
PREVIOUSLY
ESTABLISHED

LITERATURE REVIEW

ALUATC DATA

CONDUCT 7KIM.
INVENTORY OF
HABITAT PARAMETERS
AND WILDLIFE SPECIES
SIMULTANEOUSLI

PARAMETERS AND
WILDLIFE SPECIES

SIMULTANEOUSLY

136

(figure 2 continued)

GENERATE
INITIAL
PREDICTIVE
MODELS

REVISE

PHYSIOGRAPHIC
DIVISIONS

COMBINATION OF
PHYSIOGRAPHIC
DIVISIONS

15

PERFORM SIMPLE
CORRELATION ANALYSES
■ BETWEEN WILDLIFE
DENSITY AND HABITAT
PARAMETERS

HABITAT AS AN
INDEX TO DENSITY
OF WILDLIFE SPECIES

I

EVALUATION AND
MANAGEMENT OF
HABITAT FOR
WILDLIFE SPECIES

DEVELOP MODEL
FOR ENTIRE
STUDY AREA

I

WEIGHT COMPONENT
MODELS BY PERCENT
LAND AREA

DETERMINE DENSITY
LEVELS TO
DISCRIMINATE

DEVELOP

DISCRIMINANT

FUNCTION{S}

I

TEST MODEL(S)

19 INVENTORY HABITAT
PARAMETERS AND WILD
LIFE SPECIES ON SAME
TRANSECTS AT ESTA
6LISHED TIME INTERVAL

COMPARE PREDICTED
DENSITIES AND CON-
FIDENCE LIMITS TO
OBSERVED DENSITIES

HABITAT PARAMETERS
WITHIN DISCRIMINANT
FUNCTION

I

19 INVENTORY HABITAT
PARAMETERS AND WILD-
LIFE SPECIES ON NEW
TRANSECTS

I

20

PREDICT DENSITY
LEVEL(S) AND
COMPARE TO OBSERVED
LEVEL(S)

® ©

21

DEVELOP -MARK SENSE-
DATA COLLECTION
FORMS AND GUIDELINES
FOR CONDUCTING
SURVEYS

WILDLIFE DENSITY
INDEX AND/OR
HABITAT PARAMETERS

PREDICT ANNUAL
FLUCTUATIONS IN
WILDLIFE DENSITY

DETERMINE EFFECT OF
HABITAT MODIFICATION
ON WILDLIFE DENSITY
OVER TIME

22

HABITAT EVALUATION
AND MANAGEMENT
USING DISCRIMINANT
FUNCTION AND SIMPLE
CORRELATION ANALYSES

23

INVENTORY HABITAT
PARAMETERS ON
SAME TRANSECTS
DURING ESTABLISHED
TIME INTERVAL

HABITAT PARAMETERS
WITHIN DISCRIMINANT
FUNCTION

INVENTORY HABITAT
PARAMETERS ON
TRANSECT WITHIN AREA
TO BE EVALUATED

CALCULATE PREDICTED
DENSITIES AND
CONFIDENCE LIMITS

PREDICT DENSITY
LEVEL OF WILDLIFE
SPECIES WITHIN

COMPARE PREDICTED
DENSITIES OVER TIME
AND ESTABLISH TREND

COMPARE TRENDS IN
WILDLIFE DENSITY
AND HABITAT OVER
TIME

2S

MANIPULATE HABITAT
TO INCREASE HABITAT
PARAMETERS POSITIVE-
LY CORRELATED WITH
WILDLIFE DENSITY

137

DISCUSSION

PAUL GEISSLER: I have two comments. First, you
have an average of about eight times as many
variables as transects in your initial analysis.
Even counting the artificially increased sample
sizes which resulted from using multiple surveys'
of transects as independent observations, you have
an average of over twice as many variables as
observations. In addition to the variables
examined in your initial analysis, you also
examined additional variables that were
combinations of the original variables in the
process of evaluating seven simplifications of
habitat classifications. It does not matter that
simplified habitat classifications were evaluated
in separate computer runs, because these
additional variables were still included in the
search for a simple model with high predictive
power .

Very high R^'s can be obtained from random
numbers which have no predictive power whatsoever
(figure ID). This is due essentially to the
repeated analyses on the same set of data by the
stepwise procedure. When there are few
observations relative to the number of variables,
there is a tendency for the procedure to fit the
random variations in the data as well as the
underlying biological process. This results in
prediction bias, the over-estimation of the

0 10 20 30 40 50 60 70 80 90 100
Sample Size

Figure ID. Effect of sample size on in
stepwise regression. One hundred sets of data
were generated for each of the sample sizes of
3, 5, 10, 20, 30, no, 50, 75, and 100
observations. Each observation consisted of a
dependent variable and 10 independent variables
generated as normally distributed random
numbers. Each set of data was analyzed using
the SAS Stepwise Procedure with default options
(Barr et al . , op. cit.) and the mean and the
95% confidence limits for the mean R' plotted as
a function of sample size.

model's predictive ability (R^) based on the data
used to construct the model as compared to the
model's true predictive ability on other data
(Neter, J., and W. Wasserman. ^9^^. Applied
linear statistical models. Richard D. Irwin,
Inc., Homewood, 111. p. 388). In other words, a
very high indicating good predictive power may
be obtained from the original set of data, but
poor predictions and low R^ result when the model
is applied to another set of data.

Using the Pineywoods as an example, you
obtained a R^ of 83.3% using a model with three
variables selected from 98 variables present.
There were nine transects in this area. Using
independent normally distributed (0,1) random
numbers in place in these 98 prediction variables
and the response variable for nine independent
transects, I calculated an R^ of 100% 24 out of 25
times using the SAS Stepwise Procedure with
default options (Barr, A.J. et al. 1979. SAS
User's Guide. SAS Institute, Gary, N. C.). There
are similar problems with the discriminant
analysis unless there are substantially more
independent observations than there are variables.

My second comment concerns the correlation
among your call-count surveys. You indicate that
audio count data for all of the surveys conducted
on the 133 call-count transects were included
instead of using transect means. This was done
because differences in audio counts between
surveys were significant. However, you note that
"...inclusion of multiple surveys of individual
transects artificially increased sample sizes
within ecological areas, as surveys of the same
transect were not statistically independent." A
basic assumption in any regression analysis is
that the error terms are uncorrelated (Neter and
Wasserman, op. cit., p. 31). The test for a
difference among surveys is not relevant to the
determinations of the unit that constitutes an
independent observation. That decision must be
based on the lack of correlation among units. It
is possible for there to be a perfect correlation
(r=1) between the call-counts on multiple surveys
of individual transects, and at the same time be
significant differences among the surveys. What
you need to demonstrate is that the survey results
are uncorrelated, not that there is a significant
difference among surveys. Analyzing the mean of
the multiple surveys of a transect will result in
uncorrelated observations and meet the assumptions
required for regression analysis.

CHRIS GRUE: We were aware of the possibility of
prediction bias throughout our study. The number
of potential habitat types within our habitat
classification was, unfortunately, nearly four
times the number of call-count transects within
Texas. We, however, considered the habitat
classification to be suitable for classifying
habitats from within a vehicle and an unbiased
method of selecting the habitat variables to be
included in the regression analyses; the
classification scheme was developed prior to the
collection of any field data. We believed the
elimination of levels within the classification

138

hierarchy was an objective means of simplifying
the habitat classification and reducing the number
of habitat variables therein. We also realized
that, depending on the level within the habitat
classification, only a fraction of the potential
habitat variables might actually be observed along
the call-count transects. With respect to the
discriminant analyses, we decided to include only
those habitat variables which entered the
regression models.

Because the number of call-count transects
was lower than the potential number of habitat
variables, the decision of whether to conduct the
regression analyses statewide or within ecological
areas was difficult. Both approaches had apparent
problems, statistical or biological. By
conducting regression analyses statewide, the
potential for prediction bias would be reduced.
However, this approach appeared to suffer
biologically. Habitat variables important to
doves may not be expected to be the same within
any two ecological areas due to differences in the
abundance and distribution of habitat types.
Analyses within ecological areas, though probably
more valid biologically, increased the potential
for prediction bias; we were restricted to
sampling the existing call-count transects due to
budget constraints. Since the two approaches
appeared to have merit, we decided, apriori, to

utilize both in the study.

To counter the possible effects of prediction
bias in the regression analyses conducted within
ecological areas, we included call-counts for all
surveys as the dependent variable instead of
transect means. Variability in call-counts
between surveys was great and we believed its
inclusion in the regression analyses would make it
more difficult to obtain high multiple correlation
coefficients. Because surveys of the same
transect were not independent observations and
their inclusion artificially increased sample
size, we increased the significance level for
entry of habitat variables into the models from
the default value of P<0.10 (Barr et al., op.
cit.) to P<0.05.

Multiple correlation coefficients for models
from statewide and within-ecological-area analyses
using call-counts for all surveys and transect
means are presented in table ID. Inclusion of
call-counts for all surveys did make it more
difficult to obtain high multiple correlation
coefficients. Some prediction bias is undoubtedly
present in the data, particularly models for
ecological areas based on transect means. The
amount of prediction bias present, however, cannot
be assessed until the models are tested. We
believe the amount of prediction bias present is

Table ID. Correlations between mourning dove call-counts (all surveys
conducted and transect means) and habitat variables (structural features
and habitat types within the simplified habitat classification selected) .
Multiple correlation coefficients are expressed as percents. The number
of habitat variables remaining in the models is given in parentheses.

No. habitat All surveys Mean of

variables surveys

Ecological area

present

N

N

R

2

Pineywoods

32

81.

3

(3)

9

100

.0

(6)

Gulf Prairies and Marshes

to

18

28.

3

(1 )

6

100

.0

(3)

Post Oak Savannah

49

33

87.

5

(4)

9

100

.0

(6)

Blackland Prairies

44

40

68.

1

(3)

10

93

.6

(3)

Cross Timbers and Prairies

46

62

71.

5

(7)

17

27

.0

(1)

South Texas Plains

43

70

74.

7

(6)

18

86

. 1

(4)

Edwards Plateau

35

72

63.

9

(7)

18

81

.7

(4)

Rolling Plains

37

88

55.

7

(4)

23

56

.6

(2)

High Plains

34

52

63.

9

(2)

14

89

.5

(2)

Trans-Pecos

31

35

86.

6

(3)

9

99

.7

(4)

Texas

54

502

44.

6(16)

133

38

.2

(4)

139

substantially less than that suggested by your
example. One would not expect to account for only
27? of the variation in call-counts given 17
transects and 46 habitat variables if prediction
bias was severe. If one is willing to accept the
results of our regression analyses in which the
number of independent observations substantially
exceeded the number of independent variables
(e.g., 3X) , then habitat variables still accounted
for a significant portion of the variation in
mourning dove call counts (ca. 40%) throughout
Texas (table ID). That regression models for
ecological areas accounted for a greater
proportion of the variation in call counts than
the model for the State, however, appears to be
reasonable biologically. Prediction bias in our
discriminant analyses should not be great; the
number of transects was two to seven times the
number of habitat variables included.

BERNARD MORZUCH: What are the implications of
using stepwise regression on the properties of the
resulting parameter estimates in the regression
analyses?

CHRIS GRUE: Compared to "all possible regression"
procedures, predictive models generated by
stepwise may not provide the best fit to a data
set because a limited amount of stepping back is
done. Models generated by either procedure may
not include all independent variables
significantly correlated with a dependent variable
and individual regression coefficients may depend
on the other parameters within a model. Models
generated by either procedure are primarily
restricted to predictive uses. Examination of
correlation matrices may, therefore, prove useful
in identifying all habitat parameters
significantly correlated with wildlife density and
in establishing guidelines for improving wildlife
habitat.

140

Examples: Multivariate Analyses of Wildlife Habitats

141

INTERSPECIFIC DIFFERENCES IN NESTING HABITAT OF
SYMPATRIC WOODPECKERS AND NUTHATCHES'
Martin G. RaphaeP

Abstract. — To test for nest site differences among nine
sympatric species of woodpeckers and nuthatches (hole
excavators) in a Sierra Nevada mixed conifer forest, I
located 306 active nests, measured forest stand
characteristics on a 0.04 ha plot centered at each nest,
measured characteristics of the nest tree, and used
discriminant analysis to compare these nest site
characteristics among bird species.

Three discriminant functions were considered. The first
was associated most strongly with live tree basal area and
canopy height. The second function was associated with nest
tree species and nest tree height; the third was identified
by nest tree diameter and top condition. Mean discriminant
scores differed significantly among all but 3 of 36 possible
pairs of species along at least one of these discriminant
axes, indicating that nearly all bird species chose distinct
nest sites. The distributions of discriminant scores along
each axis, however, showed considerable overlap with nest
sites of two sapsucker species being the most similar.
Euclidian distance between mean scores was the most useful
measure to characterize the similarity of species' nest
sites .

Nest stand and nest tree variables contributed nearly
equally to the discrimination between bird species. This
analysis suggested that both of these sets of variables
should be included in management prescriptions to provide
habitat for cavity nesting birds.

Key words: Cavity nesting birds; cluster analysis;
discriminant analysis; Euclidian distance; nuthatches; Sierra
Nevada; woodpeckers.

'Paper presented at The use of multivariate
statistics in the studies of wildlife habitat: a
workshop, April 23-25 1980, Burlington, Vt.

^Staff Research Associate, Department of
Forestry and Resource Management, University of
California, Berkeley, CA 9'<720.

INTRODUCTION

Cavity nesting birds comprise about 30? of
the breeding bird species in western forests.
Habitat management for these species requires,
among other factors, the provision of adequate
numbers of suitable nest sites, usually standing
dead trees (Raphael and White 1978, Thomas et al .
1979). The question arises whether each species
selects distinct nest sites or whether groups of

142

species nest in trees with similar character-
istics. Habitat management is simplified in the
latter case since species with similar nest sites
can be grouped into a reduced set of functional
management units.

The purpose of this study was to determine
whether the nest sites of each species in a group
of sympatric primary cavity nesters were distinct
or whether one can identify subsets of species
using nest sites with similar characteristics. To
do so, I performed a one-way multivariate analysis
of variance using discriminant analysis , .followed
by an examination of the relative separation of
all possible species-pairs. The basic advantages
of using discriminant analysis instead of a series
of single-variable comparisons among all species
ar« that it: a) accounts for correlations among
the variables used in^ the analysis, and b) allows
more rigorous control over the experiment-wise
(type I) error rate.

Primary cavity nesters excavate their own
cavities and to do so they must choose the
appropriate substrate for the nest. Secondary
cavity nesters, on the other hand, choose
appropriate cavities, usually abandoned woodpecker
holes (Raphael 1980), I limited the analysis to
primary cavity nesters because these species
select trees; the secondary cavity nesters choose
cavities and this confounds an analysis of tree
characteristics.

METHODS
Study Area

Field studies were conducted at the
University of California Sagehen Creek Field
Station, located on the east side of the Sierra
Nevada, 13 km north and 6 km west of Truckee,
California. Elevations in the 39 km^ Sagehen
Creek drainage vary from 1800 m to 2300 m. The
drainage is dominated by a mix of Jeffrey pine
(Plnus jef freyi) and white fir (Abies concolor )
and by brushfields or conifer plantations on the
site of the I960 Donner Ridge fire which burned
the eastern quarter of the basin. Meadows,
lodgepole pine (Pinus murryana) , and aspen
(Populus tremuloides) occur in mesic sites, and
red fir (Abies magnif ica) and mountain hemlock
(Tsuga mertensiana) dominate at higher elevations.

Searches for active nests were conducted
throughout the Sagehen Creek basin from 1976 to
1979. Active nests were confirmed by observing
adults entering a cavity to incubate eggs or feed
young and by the sounds of young calling from a
nest.

Measurement of Nest Site Characteristics

I measured seven variables describing the
characteristics of the stand within a 0.04 ha
circular plot centered at each nest. These were
1) habitat (classified as burned or unburned); 2)

canopy height (maximum height, measured using a
relaskop); 3) basal area of live trees (computed
from diameters of all live trees > 8 cm DBH
excluding nest tree); 4) shrubcover (estimated
percent canopy cover of all woody perennials
including trees £ 8 cm DBH); and 5-7) density of
small (< 23 cm DBH), medium (23-38 cm), and large
(> 38 cm) snags.

Characteristics recorded for each nest tree
included: 1) tree condition (live or dead); 2)
diameter (DBH, measured with a diameter tape); 3)
height (measured with a relaskop); 4) bark cover
(estimated percent of stem covered by bark); 5)
top condition (broken or intact); 6) twig
condition (most foliage-bearing twigs present or
most broken); 7) tree species (Jeffrey pine,
lodgepole pine, red fir, white fir, or other).

All binary variables were coded as 0 or 1 for
subsequent analyses. Tree species was converted
to four dummy binary variables. If the tree was a
given species it was assigned a value of 1;
otherwise, it was assigned a 0. The fifth group,
other, was left out of the analysis to avoid
redundancy. In total, then, 17 variables were
included in the analysis.

Statistical Analyses

A linear discriminant analysis was performed
on all 17 variables (percentages analyzed using an
arcsine transformation) using the SPSS (version
8.0) package (Nie et al . 1975). The null
hypothesis was that nest sites of all species are
equal, that is, the mean discriminant scores do
not differ between members of any possible pair of
species .

The maximum number of functions derived in a
multi-group discriminant analysis is either one
less than the number of groups (in this case, bird
species) or equal to the number of variables
entered into the analysis, whichever is smaller.
The first function derived explains the greatest
proportion of the total variance, and each
additional function explains successively less.
There are two approaches in deciding how many of
the possible functions to consider. First, one
can test the statistical significance of each
function by comparing the additional variance
explained by that function to an expected value
(Klecka 1975, Morrison 1976). Alternatively, one
can arbitrarily define a minimum proportion of
explained variance and accept only those functions
that explain more than that minimum. In this
study, I chose the latter approach and considered
only those functions explaining 5% or more of the
total variance. This alternative allows more
powerful planned comparisons of mean discriminant
scores among the groups as opposed to post-hoc
comparisons which would have been necessary had I
used the former approach.

To interpret the biological meaning of each
discriminant function, I computed the correlation
of each variable with the discriminant score

143

derived for each function (structure matrix).
Variables with the highest correlations were used
to interpret functions. Some researchers (e.g.,
Klecka 1975:443) prefer to use the standardized
discriminant function coefficients (pattern
matrix) to interpret functions, but these
coefficients can be highly unstable.^

I used a t-test to compare mean discriminant
scores of each pair of species along each of the
discriminant axes accepted for analysis. Given S
species, there are [S(S-l)]/2 possible pairwise
comparisons on each axis. To control the total
type I error rate at <^ 0.05 for all comparisons on
each axis, I used Dunn's (1961) procedure for
multiple planned comparisons. Assuming equal
variance in discriminant scores, the formula used
for each pairwise test comparing species i and j
on the kth discriminant axis is simply:

— — 1 /2 —

t, = (d., - d., ) / (1/n. + 1/n.) where d is the

mean discriminant score and n is the sample size
for each species. Values of t were compared to
values tabled by Dunn (1961) to accept or reject
the null hypothesis of no difference between mean
scores at the 0.05 significance level.

The value of t is dependent on sample size as
well as the magnitude of the difference between
the discriminant scores. Given the same value of
(d^l^ - d ji^) , a comparison involving two species

with large sample sizes may be statistically
significant while a comparison of species with
smaller sample sizes may not be. Values of t may
also be affected by unequal variance in scores
among species. To reduce this dependency on
sample size and to account for the possibility of
unequal variance, I divided each discriminant axis
into equal segments, computed the frequency of
discriminant scores for each species in each
segment, and then computed the similarity of these
frequency distributions along each axis using a
measure of niche overlap given by Colwell and
Futuyma (1971:573, equation 23). Since each
discriminant axis is independent, I multiplied the
overlap values for each species-pair on each axis
to derive an index of total overlap on all axes
(May 1975). Cluster analysis (UPGMA, Sneath and
Sokal 1973) was used to reveal possible groups of
species using nests with similar characteristics
based on these total overlap values.

For comparison, I also computed the Euclidian
distance C^j^j) between each pair of species i and

j in discriminant space, using the formula:

— —21/2 —
D. . = [Z (d., - d ., ) ] , where d., is the value
ij ^ ik Jk ik

of the mean discriminant score on the kth axis for

^Personal communication with L.A. Marascuilo,
statistics Professor, Department of Education,
University of California, Berkeley.

the ith species, and d^^^ is the value for the jth

species on the same axis ^^^j is equivalent to the

square root of the Mahalanobis distance
statistic). Like the overlap values described
above, these Euclidian distances are less affected
by sample size than the t-test.

To compare the relative importance of the
nest tree and nest stand variables as
discriminators between nest sites of the species,
I performed three additional discriminant
analyses. For the first analysis, I included only
the seven nest stand variables and computed the
proportion of the total variance explained by all
functions (sum of squares between groups divided
by the total sum of squares). Second, I used a
stepwise analysis to determine the best seven of
the ten nest tree variables and calculated the
proportion of the total variance explained by
these seven variables. Third, I used a stepwise
analysis including both the nest tree and nest
stand variables and calculated the proportion of
the total variance explained by the first seven
variables entered of either type. I then compared
these values of explained variance from each
analysis to assess the relative importance of the
tree and stand variables.

RESULTS AND DISCUSSION

Interpretation of Functions

I located a total of 306 nests of 9 excavator
species (table 1). These species formed nine
groups resulting in eight discriminant functions
which explained 83% of the total variance in nest
site characteristics (fig. 1). Each of functions
4 through 8 explained less than 5% of the variance
not accounted for by previous functions and was
not considered in subsequent analyses. The first
three functions explained 55%, 11%, and 7%,
respectively, totaling 75% of the variance.

The first function was correlated most highly
with canopy height, live tree basal area, and
burned vs. unburned habitat (table 2). This
function clearly was associated with nest stand
variables and separates species nesting in
unburned forest stands (red-breasted nuthatch and
sapsuckers) from those nesting in burned stands
(pygmy nuthatch, and Lewis and white-headed
woodpeckers) (fig. 2A ) . The three remaining
species nest in both burned and unburned habitats
and their mean discriminant scores were located at
intermediate locations on the first discriminant
axis (fig. 2A).

The second discriminant function was most
correlated with nest tree variables, particularly
species (the two fir categories) and tree size
(height and diameter) (table 2). Species nesting
in red fir and smaller size trees had the lowest
discriminant scores (black-backed and white-headed
woodpeckers) and species nesting in larger size,
white fir trees had the highest scores (Lewis

144

Table 1. Bird species (hole excavators) nesting
in the Sagehen Creek study area.

100 r

Bird species

Code Sample size

Common flicker

(Colaptes auratus)

Lewis woodpecker
(Asyndesmus lewis)

CF

LW

Red-breasted sapsucker RS
(Sphyrapicus ruber daggetti)

Williamson sapsucker WS
(Sphyrapicus thyroideus)

Hairy woodpecker HW
(Dendrocopos villosus)

White-headed woodpecker WW
(Dendrocopos albolarvatus)

Black-backed-
three-toed woodpecker BW
(Picoides arcticus)

Red-breasted nuthatch RN
(Sitta canadensis)

Pygmy nuthatch FN
(Sitta pygmaea)

68
37
50
50
23
12

8
30
28

TOTAL

306

woodpecker and red-breasted-sapsucker) (fig. 28).
The mean scores of the other five species differed
very little on this axis.

The third discriminant function was also
associated with variables describing the nest tree
(table 2) and it separated the white-headed,
hairy, and black-backed woodpeckers from the
remaining species (fig. 2C). The white-headed
woodpecker nests in large diameter, broken-topped
trees while the black-backed and hairy woodpeckers
nest in smaller diameter, intact-topped trees. As
with the second function, the remaining species
have very similar mean scores on this axis.

Pairwise Comparisons

Results of the 36 t-tests comparing mean
discriminant scores among all possible pairs of
species on each discriminant axis are summarized
in figure 3. The null hypothesis of no difference
between mean scores was rejected on at least one
axis for all but three comparisons (RN vs. WS, BW
vs. HW, WS vs. RS). These results might be used
to suggest that the black-backed and hairy
woodpeckers could be grouped for management

0 12345678
NUMBER OF FUNCTIONS

Figure 1. Cumulative explained variance (% of
total variance) in relation to the number of
discriminant functions considered.

purposes because their nest site characteristics
were statistically indistinguishable. Similarly,
the two sapsucker s could also be grouped. The
pygmy nuthatch is more problematic since its nest
sites are similar to those of the Williamson
sapsucker but not to those of the red-breasted
sapsucker; it might be better, therefore, not to
group pygmy nuthatches with either species.

Because all discriminant axes are orthogonal
(independent), a significant difference between
two species' centroids on any one axis is
sufficient to conclude that the two species use
statistically distinct nest sites. Given that the
null hypothesis is true and that the probability
of at least one significant difference on each
axis is 0.05, the probability of no difference on

any axis is (1.00 - 0.05)^ = 0.86. Thus, for 36
comparisons one could expect approximately
36(0.86) = 31 to be nonsignificant. Clearly,
since only three comparisons were actually
nonsignificant, the null hypothesis is not true.
It is unclear, however, whether the lack of a
significant difference among these three pairs
reflects their inherent ecological similarity or a
chance event.

Interpretation of these pairwise comparisons
is further complicated by the fact that the
comparisons along any one axis are not
independent. When significant differences do
occur among comparisons of this type, they tend to
occur in "bunches" (Lindman 197^*: 82). Therefore,

145

Table 2. Pooled within-groups correlations
between discriminant functions and discrim-
inating variables.

Variable

Correlation with
Discriminant Function:

Nest Stand Variables

scores but their score distributions could overlap
to such an extent that they might justifiably be
grouped for management purposes. To examine such
a possibility I computed overlaps between all
pairs of species on each of the three discriminant
axes and multiplied the three overlap values for
each species-pair to derive an index of nest site
similarity (table 3). The greatest overlap values
are found between the two sapsuckers (0.42) and
between the common flicker and pygmy nuthatch
(0.40) .

Canopy height

0.83

0. 12

-0.03

Live tree basal area

0.70

0.07

-0. 16

Burned or unburned

-0.59

0. 10

-0.03

Shrub cover

-0.26

-0. 19

0.12

Snags < 23 cm DBH

0.02

-0. 18

0.05

Snags 23-38 cm DBH

-0. 14

-0.06

0.05

Snags > 38 cm DBH

-0. 15

0.01

-0.07

Nest Tree Variables

Height

0.51

0.

43

0.20

Diameter

0.31

0.

32

-0.53

Foliage-bearing twigs

0.52

0.

19

0. 14

Bark cover

0.29

0.

17

-0.27

Top condition

0.23

-0.

17

0.38

Jeffrey pine

-0.26

0.

04

-0.21

Lodgepole pine

0.21

-0.

06

0.31

White fir

-0.08

0.

28

0.00

Red fir

0. 08

-0.

47

-0. 15

Tree condition

-0.44

-0.

10

0.01

one cannot predict, a priori, the expected number
of significant differences on one, two, or three
discriminant axes. For these reasons, pairwise
comparisons of species' mean discriminant scores
appear to have limited utility in assessing
habitat similarity among the species. Two other
measures, involving overlap of score distributions
and distance between mean scores, seem better
suited for such a purpose.

Overlap of Discriminant Scores

The range and standard deviation of
discriminant scores on the discriminant axes show
considerable overlap among species (fig. 2). Two
species may have statistically different mean

I used a cluster analysis based on this
similarity matrix (overlap values) to produce a
dendrogram (fig. 4) revealing patterns of nest
site similarity among the bird species. At the
0.4 level, two groups are recognized, one composed
of the two sapsuckers and another containing the
pygmy nuthatch and common flicker. At the 0.2
level, the Lewis woodpecker links with the
flicker-pygmy group and the red-breasted nuthatch
joins the sapsucker group. At this level of
similarity, these groups can be indentified as
burn and unburned specialists, respectively. The
remaining species link at successively lower
overlap values. These results suggest that nest
sites of the two sapsuckers may be similar enough
to permit their management in common and,
likewise, those of the pygmy nuthatch and common
flicker. The pairwise t-test also identified the
sapsucker group, but the t-tests indicated the
hairy and black-backed woodpeckers as a second
possible group rather than the nuthatch and
flicker .

Euclidian Distance

Another measure of overall nest site
similarity is the Euclidian distance between
species' mean discriminant scores. These values
(table 3) measure the separation of the species in
discriminant space, and may be less affected by
sample size than the overlap values. For example,
the overlap of the black-backed woodpecker with
the other species ranges from 0.01 to 0.02 (table
3). These values are lower than the average of
any other species, probably reflecting the small
sample size of black-backed nests (n = 8). In
contrast, the Euclidian distances between
black-backed nests and those of the other species
range from 1.30 to 3.24 and these values are
spread throughout the range exhibited by the other
species-pairs.

Euclidian distances, however, do not directly
convey any information about the dispersion of
each species about its centroid. The overlap
values do convey such information, since it is
precisely the relative patterns of dispersion of
any two species that define their overlap. But, a
linear regression of discriminant score overlap
with Euclidian distance between mean scores in
three-dimensional discriminant space showed that
the two measures were highly correlated (r =
-0.89, P < 0.001, excluding the black-backed
woodpecker). Thus, overlap can be predicted from
Euclidian distance. Given this relationship, and

146

RED-BREASTED NUTHATCH
RED-BREASTED SAPSUCKER

WILLIAMSON SAPSUCKER

BLACK-BACKED
THREE-TOED WOODPECKER

COMMON FLICKER

HAIRY WOODPECKER

WHITE-HEADED WOODPECKER
PYGMY NUTHATCH

LEWIS WOODPECKER

DISCRIMINANT SCORE 1

CANOPY HEIGHT >

LIVE TREE BASAL AREA ►

PROPORTION UNBURNED ►

LEWIS WOODPECKER

RED-BREASTED
SAPSUCKER

WILLIAMSON SAPSUCKER
COMMON FLICKER

PYGMY NUTHATCH
HAIRY WOODPECKER

RED-BREASTED
NUTHATCH

BLACK-BACKED
THREE-TOED WOODPECKER

WHITE-HEADED WOODPECKER

-2 -1 0

DISCRIMINANT SCORE 2

NEST TREE SPECIES (FIR)

NEST TREE HEIGHT ►

NEST TREE DIAMETER ►

HAIRY WOODPECKER

, BLACK-BACKED
THREE-TOED WOODPECKER

PYGMY NUTHATCH
COMMON FLICKER

RED-BREASTED SAPSUCKER

WILLIAMSON SAPSUCKER

RED-BREASTED NUTHATCH

LEWIS WOODPECKER

WHITE-HEADED WOODPECKER

DISCRIMINANT SCORE 3

■• NEST TREE DIAMETER

TOP PRESENCE ►

NEST TREE SPECIES (PINE)

Figure 2. Mean (vertical lines), standard deviation (heavy bars), and range (horizontal lines) of
discriminant scores of each bird species on the first (A), second (B), and third (C) discriminant
functions. Arrows indicate direction of increased values of variables most highly correlated with the
discriminant scores on each function.

147

RN
BW
HW

CF
LW
WW
WS

RS

PN

O

o

o

o

©

RN

DISCRIMINANT
SYMBOL FUNCTION

0

O

©

©

BW

O

©

HW

1
2
3

©

0

©

CF

©

o

o

LW

©

o

WW

©

©

WS

Figure 3. Results of t-tests comparing mean
discriminant scores among all possible pairs of
species on each function. Symbols denote
significant differences (see text). Species
codes are given in table 1 .

the observation that Euclidian distances are
apparently less dependent on sample size than the
t-tests or overlap values, I recommend using
Euclidian distance as a measure of similarity when
sample sizes of some groups are small. When
sample sizes are large, the overlap value is
preferred since it is directly interpretable as a
measure of shared discriminant score distribution.

Relative Importance of Stand and Tree Variables

Figure 3 shows that eight pairs of species
differ only on the basis of stand characteristics
(significant differences occur only on the first
discriminant axis) . These species nest in similar
trees located within otherwise different stands.
In contrast, 13 species-pairs nest in trees with
significantly different characteristics
(significant differences occur only on functions 2
and/or 3). They nest in differing trees located
in otherwise similar stands. The remaining 12
pairs of species nest in both different trees and
stands.

Three additional discriminant analyses, each
using seven variables, were used to assess the
relative discriminating power of the tree and

Table 3. Relative similarity of excavator nest sites. Values to right of
diagonals are the overlaps of species' discriminant score distributions
(larger values indicate more similar nest sites). Values to left of
diagonals are Euclidean distances between species' mean discriminant
scores in three-dimensional discriminant space (larger values indicate
less similar nest sites) .

Bird

species '

PN

RN

BW

HW

CF

LW

WW

WS

RS

PN

0

.02

0.03

0.19

0.40

0.22

0.11

0.05

0.09

RN

2.72

0.02

0.06

0.17

0.01

0.01

0.24

0.21

BW

2.21

1

.90

0.02

0.02

0.01

0.02

0.01

0.02

HW

1 .30

2

.36

1.33

0.22

0.04

0.02

0.10

0.13

CF

0.91

1

.93

1.85

1 .08

0.26

0.09

0.28

0.32

LW

1.15

3

.24

3.24

2.20

1.48

0.01

0.02

0.04

WW

2.23

2

.91

2.75

2.93

2.48

2.87

0.02

0.02

WS

2.47

0

.69

2.09

2.15

1.59

2.82

3.12

0.42

RS

2.59

1

.25

2.40

2.21

1.68

2.79

3.59

0.60

'See table 1 for bird species codes.

148

OVERLAP OF DISCRIMINANT SCORE
DISTRIBUTIONS AMONG SPECIES

-fh

-//-

PYGMY NUTHATCH
COMMON FLICKER
LEWIS WOODPECKER
HAIRY WOODPECKER
RED-BREASTED NUTHATCH
WILLIAMSON SAPSUCKER
RED-BREASTED SAPSUCKER

WHITE-HEADED WOODPECKER

BLACK-BACKED
THREE-TOED WOODPECKER

0.1

0.2 0.3 0.4

OVERLAP INDEX

0.5

-/A

1.0

Figure 4. Dendrogram showing similarity of nest sites of bird species based on cluster analysis using
overlap of discriminant score distributions along three discriminant axes. Larger index values indicate
more similar nest sites.

stand variables (see methods). When only the
seven stand variables were included in the
analysis, the variance explained by the set of
discriminant functions was 60? of the total
variance in the system. The variance explained
using only the best seven nest tree variables
(lodgepole pine, white fir, and foliage bearing
twigs were excluded) is slightly higher, 63%.
Finally, the optimum set of both nest and stand
variables (canopy height, tree height, tree
diameter, red fir, live tree basal area, top
condition, and tree condition listed in order of
entry in stepwise analysis) explained 74? . While
the nest tree variables are slightly better than
the stand variables as discriminators between
species' nest sites, it is apparent that both
types of variables should be included to
adequately characterize nest sites. Consideration
of both types resulted in a substantially greater
separation of species' nest sites.

CONCLUSION

Discriminant analysis is a useful method for
quantifying the similarity of habitat
characteristics among wildlife species. It offers
several options including planned or post-hoc
comparisons of mean scores, analysis of score
distributions in discriminant space, and measures
of the separation of species such as Euclidian
distance. Of these options, Euclidian distance
appears to have the greatest generality since it
is affected less by variation in sample size among
the species analyzed.

Using discriminant analysis to compare the
characteristics of the nest sites of nine
sympatric primary cavity nesting bird species, I
demonstrated that both nest tree and nest stand
characteristics should be included in the suite of
variables used by managers to prescribe management

149

practices to provide the habitat requirements of
these species. Five species used distinct nest
sites and require unique management prescriptions,
but the four other species can be combined for
management purposes into two groups, each
containing pairs of species using very similar
nest sites.

As a final note, researchers (and managers)
must recognize that interspecific comparisons of
habitat characteristics are not sufficient to
define the most important variables involved in
habitat selection. For example, nests of all
species considered in this study were surrounded
by a relatively large number of snags greater than
23 cm DBH. But because the magnitude of this
variable does not vary significantly among
species, snag density is not a good discriminator
between species' nest sites. When snag density is
compared between a species' nest sites and a set
of random plots, however, it becomes one of the
best discriminators (Raphael 1980). Thus,
variables important in habitat selection should be
identified through a series of single species
analyses comparing use and availability of habitat
components rather than by comparing habitat
characteristics among all species of interest.

ACKNOWLEDGMENTS

I thank the U.S. Forest Service, Region 5,
and my wife, Susan, for financial support.
Marshall White and Vernon Hawthorne provided field
facilities at the Sagehen Creek Field Station.
Computer time was provided by grants from the
Department of Forestry and Resource Management,
U.C. Berkeley. I thank David Capen and Jake Rice
for their useful technical and editorial review of
the manuscript. I also express my appreciation to
Nobu Asami for typing several drafts of the
manuscript .

LITERATURE CITED

Colwell, R.K., and D.J. Futuyma. 1971. On the

measurement of niche breadth and overlap.

Ecology 52:567-576.
Dunn, O.J. 1961. Multiple comparisons among

means. Journal of the American Statistical

Association 56:52-64.
Klecka, W.R. 1975. Discriminant analysis. p.

434-467. In Nie, N.H., C.H. Hull, J.G.

Jenkins, K. Steinbrenner , and D.H. Bent.

Statistical package for the social sciences.

Second edition. 675 p. McGraw-Hill, New

York, N.Y.

Lindman, H.R. 1974. Analysis of variance in
complex experimental designs. 352 p. W.H.
Freeman, San Francisco, Calif.

May, R.M. 1975. Some notes on estimating the
competition matrix, a. Ecology 56:737-741.

Morrison, D.F. 1976. Multivariate statistical
methods. Second edition. 415 p. McGraw-
Hill, New York, N.Y.

Nie, N.H., C.H. Hull. J.G. Jenkins, K.
Steinbrenner, and D.H. Bent. 1975.
Statistical package for the social sciences.
Second edition. 675 p. McGraw-Hill, New
York, N.Y.

Raphael, M.G. 1980. Utilization of standing dead
trees by breeding birds at Sagehen Creek,
California. Ph.D. dissertation. 195 p.
University of California, Berkeley.

Raphael, M.G.. and M. White. 1978. Snags,
wildlife, and forest management in the Sierra
Nevada. Cal-Neva Wildlife 1978:23-41.

Sneath, P.H.A. and R.R. Sokal. 1973. Numerical
taxonomy. 573 p. W.H. Freeman, San
Francisco, Calif.

Thomas, J.W., R.G. Anderson, C. Maser, and E.L.
Bull. 1979. Snags. p. 60-67. In Thomas,
J.W., editor. Wildlife habitats in managed
forests — the Blue Mountains of Oregon and
Washington. USDA Forest Service, Agriculture
Handbook No. 533. 512 p. Washington, D.C.

DISCUSSION

DONALD McCRIMMON: Please reiterate the method
used to interpret discriminant axes.

MARTIN RAPHAEL: I calculated the correlations
between each original variable and the
discriminant score for each function (structure
matrix) and interpreted the functions by
considering the most highly correlated variables.

PAUL GEISSLER: I think it would be more
appropriate to use a multivariate analysis of
variance (MANOVA) to test for differences among
species, possibly with orthorgonal contrasts based
on taxonomic relationships. The criteria in
MANOVA are functions of the eigenvalues associated
with the eigenvectors which define the
discriminant functions. It has been suggested
that plots of the treatment means in the space of
the discrimination functions be used to interpret
MANOVA results.

MARTIN RAPHAEL: I believe that a one-way MANOVA
and the multi-group discriminant analysis are the
same procedure. I would not want to limit the
contrasts to taxonomic relations. For example, I
found that the common flicker and pygmy nuthatch —
two very distantly related species — were highly
similar in nest site selection.

KEN WILLIAMS: Did you check for covariance
hetergeneity? Without this characteristic it is
difficult to test your stated hypothesis. Also,
it assures some degree of overlap distortion as
displayed by the discriminant functions.

MARTIN RAPHAEL: I did test for variance/
covariance heterogeneity, and the variance/
covariance matrices were not equal. I could find
no method to assess the consequences of the
failure to meet this assumption, and decided to
proceed with the analysis since some authors state
that the procedure is robust against such a
failure. My understanding is that this failure
has more severe implications on the interpretation
of discriminant coefficients. It is quite
possible that distortions occurred in the pattern
of overlaps among the species; I would ask how
severe such distortion might be: are they so

150

severe that the results are wrong? No one, as
yet, can answer the latter question.

LESLIE MARCUS: I suggest that you plot points or
bivariate ellipse projections into canonical
variate space, to see how covariance-structure
looked in bivariate space. You could use the term
"Mahalanobis distance" rather than "Euclidean
distance in discriminant space" — as one always
knows what it means.

MARTIN RAPHAEL: To answer the first point, I
cannot visualize the advantage of plotting such
projections since any distortions caused by the
covariance structure would still affect the
bivariate axes. I would like to know more about
such a technique if it really would allow an
evaluation of distortions. On the second point, I
suppose it is a matter of personal taste since the
two measures are so closely related. It would be
useful to agree on the use of either measure to
facilitate communications.

151

ROBUST CANONICAL CORRELATION OF SAGE GROUSE
HABITAT-
Mark S. Boyce^

Abstract. — The distribution patterns of sage grouse
( Centrooercus ur ophasi anus Bonaparte) were studied on
Atlantic Richfield Company's Coal Creek coal surface mine
site in northeastern Wyoming. Fecal droppings were
periodically removed from randomly located belt transects
within a 36 km^ study area. Using multiple regression and
canonical correlation the seasonal distribution of droppings
was related to several habitat variables which were
hypothesized to be important to sage grouse. By canonical
correlation, approximately 90% of the variance in a fecal
count variate can be attributed to a habitat variate. The
correlations between each variate and the original variables
were found to be very stable when habitat variables were
rotated. Similarly, the robustness of the loadings was
confirmed by systematically eliminating cases (transects)
from the analysis. Although canonical correlation provides
little insight here over that obtained from multiple
regression analysis, it does provide a succinct summary of a
large and complex set of interrelationships among variables.

Key words: Canonical correlation; habitat; mine
reclamation; multiple regression; patchiness; sage grouse;
Wyoming ,

INTRODUCTION

Applications of canonical correlation in
ecology are relatively uncommon (Smith 1980).
This may be due in part to unstable results or
difficult interpretation (Gauch and Wentworth
1976, Johnston'). These problems are often
attributable to assumption violations in the data,
e.g., variables which are not normally distributed
(Harris 1975), multicollinearity among variables
(Cohen et al . 1979), or nonlinearity (Gauch and

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Assistant Professor, Department of Zoology
and Physiology, University of Wyoming, Laramie, WY
82071.

'Personal communication with R.F. Johnston,
Professor, University of Kansas.

Wentworth 1976). However, if these assumptions
can be approximated, the potential is great for
application of canonical correlation in habitat
studies. Canonical correlation searches for the
maximum correlation between linear combinations of
two sets of variables; for example, one set of
variables may consist of measures of organism
distribution and/or abundance, and another set may
consist of variables characterizing habitat.

In this paper I describe an application of
canonical correlation to an analysis of sage
grouse habitat. One set of variables consists of
the number of sage grouse fecal droppings
deposited in different seasons, whereas the second
set of variables consists of selected habitat
variables which were hypothesized to be important
factors influencing sage grouse distribution.
Correlations between the original variables and
the derived distribution and habitat variates were
found to be robust (contra Cohen et al. 1979),
even though the sample size was relatively small

152

(n = 20). I attribute the robustness of this
application of canonical correlation to thorough
screening for nonlinearity and normality followed
by transformation or elimination of variables
before conducting the analysis.

METHODS AND MATERIALS

The study area was a 36 km^ coal lease (T46N,
R70W) in Campbell County, Wyoming which is
scheduled for development as a surface mine. An
objective of this research was to characterize the
habitat components which are most important to
sage grouse and to develop guidelines for
reclamation of mined lands for sage grouse.

Sage grouse fecal droppings are easily
identified and may last for up to 3 years before
deteriorating. A sampling scheme was designed to
assess seasonal distribution of sage grouse by
periodically checking permanent belt transects for
fresh droppings, and removing all droppings
deposited since transects were last searched.
Twenty 2 m x 1000 m belt transects were randomly
located on the study area, and checked at
irregular intervals.

An independent consulting team was employed
to map and characterize all vegetation types on
the study area (Keammerer and Keammerer 1975). In
addition, detailed data were collected on the
composition of sagebrush density, height and cover
along each transect by employing systematic
point-quarter procedures (Seber 1973).

The total number of fecal pellets found
during a particular season was summed over the
three years of our study, yielding three seasonal
distribution variables, i.e., number of droppings
found in 1) winter, 2) spring, and 3) summer
through early fall. Habitat use patterns are
known to be very similar in summer and early fall
in non-migratory sage grouse populations
(Patterson 1952). Habitat and distribution
variables employed in this analysis are listed in
table 1.

RESULTS

An average of 360 droppings was found on each
transect. All variables were screened for
skewness and kurtosis, and transformations
conducted where appropriate. Seven variables were
either eliminated or combined with other variables
because their distribution contained an excessive
proportion of zeros. Next, because of the
sensitivity of canonical correlation to
non-linearity (Gauch and Wentworth 1976),
bivariate plots were constructed between the
distribution variables and each habitat variable.
When nonlinear patterns appeared, the respective
habitat variable was eliminated from the analysis.
For example, the total droppings vs. the area
within each belt transect in the big sagebrush
habitat type appears nonlinear (fig. 1). The
availability of at least some big sagebrush is

Table 1. Reduced set of variables and appropriate
transformations for variables used in this
analysis .

Distribution variables

TOTAL = log-iQ (total droppings found on each
transect over 3 years, plus 1.0).

SUMMER- = ■'•°8.,Q (sum over 3 years of all

FALL droppings which accumulated between
June and October, plus 1.0).

WINTER = log-iQ (droppings accumulated during
winter months, plus 1.0).

SPRING = log-iQ (sum over 2 years of droppings
accumulated during spring months
through mid-May, plus 1.0).

DIST = fecal droppings canonical variate.

Habitat variables

LEK = distance between the nearest sage
grouse strutting ground (lek) and the
closest point on each belt transect
in km.

FORBS = summed area within each belt transect
within f orb-producing habitats,
including riparian areas, in m^ .

DIV = habitat diversity = -Zp.logp. where
p. is the proportion ^of t^ie ith
habitat type within each belt
transect .

PATCH = patchiness index equal to the number
of times the belt transect crosses
from one habitat type to another,
plus 1.0.

SPRAY = log-jQ (area within belt transect of
big sagebrush habitat which had been
sprayed with 2,4-D, plus 1.0).

COVER = average sagebrush cover along
transect estimated with 21
point-quarter measurements (Seber
1973).

VACOV = standard deviation of COVER.

DENS = average density of sagebrush plants
per m^ estimated from 21 point-
quarter measurements (Seber 1973).

VADEN = standard deviation of DENS.

SAGE = area within belt transect of big
sagebrush habitat type in m^ .

HABITAT = canonical variate comprised of
habitat variables.

153

essential, and transects with no big sagebrush
have no sage grouse utilization. However, large
homogeneous stands of sagebrush tend to have low
forb availability and are not preferred habitat.
Therefore, SAGE was eliminated from further
analysis.

A multiple regression analysis was conducted
using each distribution variable as a dependent
variable. This permitted explicit hypothesis
testing regarding the importance of various
habitat variables. Several interesting patterns
emerge; for example, habitat patchiness and the
proportion of forb-producing habitats were both
positively correlated with the number of droppings
found on each belt transect (fig. 2). Different
habitat variables are important at different times
of the year due to the seasonal changes in food
habita and behavior of the grouse. A few of the
more interesting multiply regression models are
summarized in table 2.

In an attempt to summarize the complex
interrelationships between the seasonal
distribution patterns and habitat, I conducted a
canonical correlation analysis of the
"distribution" set of variables and a selected set
of habitat variables. Due to the small sample
size (n = 20), I began with a small number of
variables: three distribution variables, and
three of the most important habitat variables. As
presented in figure 3A, a habitat variate accounts
for almost 90% of the variance in a distribution
variate and the relationship is highly significant
(P << 0.001). Furthermore the correlations
between the first pair of canonical variates and
the original variables are all intuitively
satisfying and parallel the results of multiple

O

(/) 30-

Q.
Q.
O

_J
o

1.0-

o
o

O
_)

r =0.038

— I —

200

600

1000

1400

^ 30.

O
Q

2.0-

O
— 1.0-

o
o

r =0.847

—

200

600

1000

1400

SAGEBRUSH HABITAT (m^)

FORB PRODUCING HABITAT (m'=)

Figure 2. Total droppings over there years as a
function of the area within each belt transect
in forb-producing habitat types.

regression. The second canonical correlation was

not statistically significant (P < 0.05), and

loadings of the second pair of variates were not
readily interpretable .

Since the 3x3 analysis worked so well, the
number of habitat variables in the model was
increased to assess whether a better synopsis of
the interrelationships between habitat and
distribution could be achieved. The analysis was
attempted with five habitat variables (fig. 3B)
and then with nine habitat variables (fig. 3C).
The high stability of the loadings was impressive,
i.e., the correlations between the first pair of
canonical variates and the original variables
remained relatively constant as additional habitat
variables were added to the analysis. Even more
exciting, however, the loadings were intuitively
reasonable, and provided an exceptionally good
representation of the interrelationships between
variables which I had discovered through a tedious
and detailed multiple regression analysis. All
distribution variables loaded positively into a
distribution variate, thus clearly the
distribution variate reflected intensity of use by
sage grouse. Correlations between original
variables and the habitat variate all reflected
quality of sage grouse habitat, i.e., variables
which were positively correlated tend to be
preferred habitat components and variables which
were inversely correlated with the habitat variate
were usually inversely correlated with the number
of droppings found on transects at various times
of year.

Figure 1. Total droppings over three years plotted
as a function of the area within each belt
transect in the big sagebrush habitat type.

To test the robustness of the canonical
correlations and variable loadings, the 3x9
analysis was rerun 20 times, each time leaving out
one case (transect) from the analysis (similar to

154

Table 2. A sample of multiple regression models with droppings counts as dependent
variables and habitat attributes as independent variables. All models listed account for
a significant proportion of the variance in dependent variables (P < 0.05).

SPRING

= 2.

674 - 0.

452(LEK) - 0.056(SPRAY)

R =

0.780

SPRING

= 1.

603 - 0.

359(LEK) + 0.002(FORBS)

R =

0.841

SPRING

= 0.

002(FORBS) + 10.6(COVER) - 0.413

R =

0.818

SPRING

= 3.

367(DIV)

+ 13.0(COVER) - 0.686

R =

0.745

SPRING

= 0.

936 - 0.

251(LEK) + 0.002(FORBS) + 5.8(COVER)

R =

0.864

SPRING

= 2.

706 - 0.

481 (LEK)

R =

0.779

SPRING

= 0.

003(FORBS) - 0. 163

R =

0.680

SPRING

= 0.

313 + 14

.56 (COVER)

R =

0.671

SUMMER

-FALL

= 0.826

+ 0.002(FORBS) - 0.414(SPRAY)

R :

0.896

SUMMER

-FALL

= 0.467

+ 0. 116(PATCH) + 0.002(FORBS) - 0.416(SPRAY)

R =

0. 912

SUMMER

-FALL

= 0.664

+ 2.3(DIV) - 0.265(LEK) + 0.002(FORBS)

R =

0.879

SUMMER

-FALL

= 0.812

+ 2.28(DIV) - 0.133(LEK) + O.OOI(FORBS) - 0.325(SPRAY)

R =

0.923

SUMMER

-FALL

= 0.994

+ 3.54(DIV) - 0.165(LEK) - 0.371(SPRAY)

R =

0.910

SUMMER

-FALL

= 0.623

+ 3.56(DIV) - 0.528(SPRAY)

R =

0.887

SUMMER

-FALL

= 0.995

- 0.241(LEK) + 0.002(FORBS)

R =

0.861

SUMMER

-FALL

= 1.88

- 0.624(SPRAY)

R =

0.810

SUMMER

-FALL

= 0. 003(FORBS) - 0. 191

R =

0.782

WINTER

= 1.

747 - 2.

5(DIV) + 0.002(FORBS) - 0.423(SPRAY) - 0.238(VADEN)

R =

0.680

WINTER

= 1.

234 + 0.

OOKFORBS) - 0.42(SPRAY) - 0.216(VADEN)

R =

0.649

WINTER

= 1.

935 - 0.

555(SPRAY) - 0.207(VADEN)

R =

0.583

WINTER

= 1.

184 - 0.

314(SPRAY)

R =

0.437

TOTAL

= 1.42 + 4.798(DIV) - 0.374(LEK)

R =

0. 823

TOTAL

: 0.718 + 0.003(FORBS) - 0.26(SPRAY)

R =

0.881

TOTAL

= 2.415 - 0.598(SPRAY)

R =

0.691

a jackknife procedure). Results for the first
pair of canonical variates were impressively
stable, with one exception. When transect number
16 was eliminated from the analysis, the loadings
for the first pair of canonical variates were not
interpretable (table 3)- However, upon inspection
I discovered that the second pair of canonical
variates were loaded precisely in the same pattern
as the first pair in all of the other rotations.
Thus, the same patterns were present in all
trials, but for some reason the second orthogonal
pair of variates switched with the first when
transect number 16 was deleted from the analysis.

The second canonical correlation was statistically
significant (P < 0.05) in this case whereas it was
not in any of the other analyses.

The results of the robustness assessment are
summarized in table 4 where the means and standard
deviations of the canonical correlations and
variable loadings are presented for 20 rotations.
For the analysis where transect 16 was eliminated,
the second pair of canonical variates was used,
but for all other cases the results were averaged
over the first pair of canonical variates.

155

A.

Figure 3. Path diagram of the first pair of
canonical variates with three distribution
variables and three (A), five (B), and nine (C)
habitat variables. The significance levels
based upon Bartlett's test are P = 0.00003,
0.00006 and 0.004 for the 3 x 3, 3 x 5, and 3 x
9 analyses respectively. Definitions of
variables are listed in table 1.

To further assess the robustness of the 3x9
canonical correlation analysis, I systematically
eliminated one of nine habitat variables from the
analysis. The canonical correlations and variable
loadings proved very stable for the first pair of
canonical variates. In this experiment as well as
during the rotation of cases, loadings into the
second pair of canonical variates fluctuated
dramatically. As can be seen in table 4, however,
variation in canonical correlations and variable
loadings for the first pair of canonical variates
is impressively small.

The importance of distance to a lek is not
clear from these results. It seems plausible that
leks are located in good habitats but this may
obscure results since grouse social behavior may
override habitat quality in determining their
distribution. My first attempt to better
understand the importance of leks entailed using
regression analysis to remove variation
attributable to LEK from each of three
distribution variables. Then canonical

correlation analysis was conducted between the
residuals of the distribution variables and the
eight original habitat variables with LEK (sensu
Boyce 1978). However, the canonical correlation
was . not statistically significant, and variable
loadings were not interpretable . Further multiple
regression analysis of the three distribution
variables was conducted by studying residuals
after removing variation attributable to the area
in sprayed sagebrush habitat (SPRAY). The partial
correlation between the fecal dropping variables
and LEK was discovered to be significant only in
the springtime (r = -0.639, P < 0.01). Since the
leks are only actively used during March, April,
and May, it seems plausible that only in spring
should there be any unique contribution to
distribution determined by the proximity to a lek.

DISCUSSION

Sample size guidelines are lacking for
applications of many multivariate procedures.
Although Bartlett's test for canonical correlation
allows one to assess the overall significance of
the relationship between two sets of variables,
there is presently no means by which significance
of individual variable loadings can be evaluated.
Nevertheless, the most useful applications of
procedures such as canonical correlation may be
heuristic ones, thus inference procedures may not
be necessary or even desirable. However,
assessing the goodness of such descriptive tools
does seem important, and robust procedures such as
jackknifing provide such an assessment. If
canonical correlations and variable loadings prove
to be robust, as they were in this study, the
extremely large sample sizes recommended by
Thorndike (1978) should not be necessary. The
impracticality of collecting extremely large data
sets in wildife habitat studies should not
persuade the investigator that applications of
multivariate procedures, such as canonical
correlation, are necessarily inappropriate.

156

Table 3. Correlation between original variables and the first two pairs of derived
canonical variates when transect number 16 was eliminated from the analysis.
Note the similarity between figure 3C and the loadings for the second pair of
canonical variates. The probability levels are from Bartlett's test.

Distribution

variate

Habitat

variate

First Canonical

SUMMER -FA a

-0.542

LEK

0.090

r = 0.973

WINTER

-0.548

FORBS

-0 . 566

(P = 0.0002)

SPRING

-0.040

DIV

-0.229

PATCH

-0.052

SPRAY

0.544

COVER

0,053

VACOV

-0.041

DENS

0. 072

VADEN

-0. 082

Second Canonical

SUMMER-FALL

0.827

LEK

-0,858

r = 0.931

WINTER

0.424

FORBS

0.647

(P = 0.048)

SPRING

0.924

DIV

0,589

PATCH

0.573

SPRAY

-0.655

COVER

0.707

VACOV

0.567

DENS

0,551

VADEN

0,425

Cohen et al. (1979) claim that canonical
weights (loadings) are necessarily unstable in
canonical correlation because each variable
accounts for a unique portion of the variability
in the other set of variables. However, we have
seen in this analysis that unstable loadings or
"bouncing betas" need not occur and that removal
or addition of variables need not influence the
qualitative interpretation. It seems more likely
that nonlinearity or multicollinearity are behind
many applications where "bouncing betas" appear to
be a serious problem (see Gauch and Wentworth
1976).

The results depicted in figure 3 confirm many
of the subjective impressions which were developed
based upon field obsevations of the major habitat
components important to sage grouse. During
winter, sage grouse feed exclusively on leaves of
big sagebrush (Artemisia tridentata) . However,
between April and October, forbs such as dandelion
(Taraxacum of f icianale) , curly-cup gumweed
(Grindelia squarrosa) , sweet clover (Melilotus
off icianalis) , false dandelion (Agoseris glauca) ,
alfalfa (Medicago sativa) , and salsify (Tragopogon
dubius) constitute a major portion of the diet for
both young and adult birds (Patterson 1952).
Although sagebrush provides important visual cover
from aerial predators, sagebrush plants compete
with forbs for nutrients and water. Thus optimal
habitats are often patchy where forbs and
sagebrush cover occur in close proximity.

Each of the variables which we selected to
measure habitat patchiness (DIV, PATCH, VACOV,

VADEN) is positively correlated with the number of
droppings found on the transects. Similarly,
these same variables are positively correlated
with the "habitat quality" variate. SPRAY is
clearly an important variable which is negatively
loaded into the "habitat quality" variate. Since
spraying with herbicides such as 2,4-D kills both
forbs and sagebrush, it is easy to appreciate the
devastating impact of herbicide spraying on the
use of areas by sage grouse.

It is important to caution against cavelier
interpretations of canonical correlation analysis.
As noted earlier, explicit hypothesis testing
regarding any single variable is not possible with
canonical correlation but may be accomplished with
linear regression techniques. Some patterns can
be masked by simple reliance upon the overall
trends summarized by canonical correlation as I
demonstrated above for LEK, Also, to achieve a
robust canonical model, variables must often be
eliminated for statistical reasons even though
some of these variables may be biologically
important, e.g,, SAGE. I strongly recommend that
careful attention be given to any variables of
postulated biological significance but which must
be eliminated on statistical grounds. Various
statistical procedures may provide insight into
the behavior of these variables, e.g., nonlinear
regression, ridge regression, discriminant
analysis, or nonparametric techniques.

In summary, my application of canonical
correlation does not provide much insight into the

rlil«i
V

Sir

Imil

lilj

<

157

Table 4. Robust estimates of canonical correlations and variable loadings. The
top portion of the table lists means and standard deviations (in parentheses)
of values where one case (transect) was systematically eliminated from the
analysis (n = 20). The bottom portion lists means and standard deviations
when one habitat variable was eliminated (n = 9).

Distribution variate

Habitat variate

Rotation of Cases

Canonical

r_= 0.950(0.011)

(P = 0.0077L0.005])

SUMMER-FALL

0.

957(0.

036)

LEK

-0

827(0

051)

WINTER

0

488(0,

063)

FORBS

0

826(0

047)

SPRING

0.

864(0.

062)

DIV

0.

625(0

043)

PATCH

0.

498(0

059)

SPRAY

-0.

811(0

048)

COVER

0.

646(0.

068)

VACOV

0.

564(0,

065)

DENS

0.

539(0.

08)

VADEN

0.

502(0.

075)

Rotation of Habitat Variables

£anonical

r_= 0.940(0.003)

(P = 0.007C0.014])

SUMMER-FALL

WINTER

SPRING

0. 969(0

024)

LEK

-0.837(0.019)

0.506(0

065)

FORBS

0.839(0.014)

0.846(0

069)

DIV

0. 630(0. 012)

PATCH

0.497(0.012)

SPRAY

-0.838(0.015)

COVER

0.634(0.061)

VACOV

0.558(0.046)

DENS

0.527(0.073)

VADEN

0. 495(0.060)

interactions between sage grouse distribution and
habitat that I could not glean from multiple
regression analysis. This is not surprising since
canonical correlation is somewhat of a
generalization of multiple regression (Blackith
and Reyment 1971). However, canonical correlation
does provide a succinct and synoptic way to
summarize a bulky and unwieldly data set. This is
certainly one of the most important functions of
multivariate statistical analysis.

ACKNOWLEDGMENTS

This work was supported by a contract from
ARCO Coal Co. under the helpful supervision of
James Tate, Jr. Several people have provided
useful assistance in the field including B.
Colenso, D. Rothenmaier, B. Holz, and J. Tate, Jr.
I thank L. McDonald, K.G. Smith, and J.R. Karr for
useful discussions about canonical correlation;
and D.E. Capen and J. Rice for suggesting revision
in the manuscript.

LITERATURE CITED

Blackith, R.E., and R.A. Reyment. 1971.
Multivariate morphometries. 412 p. Academic
Press, New York, N.Y.

Boyce, M.S. 1978. Climatic variability and body
size variation in the muskrats ( Ondatra
zibethicus) of North America. Oecologia
36:1-19.

Cohen, P., E. Gaughran, and J. Cohen. 1979. Age
patterns of childbearing: a canonical
analysis. Multivariate Behavioral Research
14:75-89.

Cooley, W.W., and P.R. Lohnes. 1971.

Multivariate data analysis. 364 p. John

Wiley and Sons, New York, N.Y.
Gauch, H.G., Jr., and T.R. Wentworth. 1976.

Canonical correlation analysis as an

ordination technique. Vegetatio 33:17-22.
Harris, R.J. 1975. A primer of multivariate

statistics. 332 p. Academic Press, New

York, N.Y.

Keammerer, W.R., and D.H. Keammerer. 1975.
Floristic studies and vegetation mapping of
the Coal Creek lease area. 55 p.
Stoecker-Keammer er and Assoc., Boulder,
Colo .

Patterson, R.L. 1952. The sage grouse in
Wyoming. 341 p. Sage Books, Denver, Colo.

Seber, G.A.F. 1973. The estimation of animal
abundance and related parameters. 506 p.
Griffin, London.

158

Smith, K.G. 1981. Canonical correlation analysis
and its use in wildlife habitat studies. In
Capen, D.E., editor. The use of multivariate
statistics in studies of wildlife habitat:
Proceedings of a workshop [Burlington, Vt.,
April 23-25, 1980]. USDA Forest Service
General Technical Report. Rocky Mountain
Forest and Range Experiment Station, Fort
Collins, Colo. (In press).

Thorndike, R.M. 1978. Correlational procedures
for research. 3'*0 p. Halsted Press, New
York, N.Y.

159

ECOLOGICAL RELATIONSHIPS OF GRASSLAND BIRDS
TO HABITAT AND FOOD SUPPLY IN EAST AFRICA'
L. Joseph Folse. Jr.^

Abstract. — Canonical correlation analysis was used to
evaluate relationships between 17 species of grassland birds,
structural parameters of the grassland habitat, and biomasses
of 25 age-taxonomic categories of arthropods in a study 'of
ecological relationships of grassland birds in the Serengeti
National Park, Tanzania, East Africa. The bird-habitat
analysis resulted in correlations that were interpretable
both in terms of identifiable characteristics of the habitat
and in terms of identifying specific groups of birds which
exploited the habitat in similar ways. The bird-arthropod
analysis produced correlations which were not easily
interpretable relative to arthropod characteristics nor in
terms of identifying groups of birds based on their
association with the arthropod resource. A canonical
correlation of birds with habitat plus arthropod variables
had results similar to that of the bird-habitat analysis and
dissimilar to the bird-arthropod analysis. These results
suggest that mechanisms affecting the guild structure of
grassland birds in the Serengeti are more likely associated
with habitat structure than with food supply.

Key words: Arthropods; canonical correlations;
grassland birds; habitat selection; multivariate analysis;
Serengeti; Tanzania.

INTRODUCTION

The use of multivariate statis'tics in
evaluating ecological relationships of birds has
expanded considerably in recent years. One
problem of importance that is often approached
with multivariate methods is that of evaluating
the structure of bird communities relative to
their exploitation of habitat and/or food
resources. The multivariate technique which is
most suited to exploring relationships between two

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.
05401 .

^Assistant Professor, Department of Wildlife
and Fisheries Sciences, Texas A&M University,
College Station, TX 77843.

such groups of variables, e.g., community
composition and niche parameters, is that of
canonical correlation (Gittens 1979, Kshirsagar
1978, Morrison 1976). However, this technique
seems to have been used little in ornithology,
perhaps due to lack of adequate data from field
studies.

This study illustrates an attempt to use
canonical correlation analysis to identify and
evaluate some relationships between grassland
birds and their habitat and food supply on the
Serengeti plains, Tanzania. At the outset of the
study, little was known about the community
composition or densities of the birds, nor about
their habitat relations. Consequently, this
analysis was designed to treat the following
objectives: 1) to determine if there is a
structural organization of the avifauna that is
related to characteristics of the habitat and

160

arthropod fauna, 2) to define recognizable groups
of bird species whose members are similarly
associated with habitat and arthropod
characteristics, and 3) to identify specific
characteristics of the habitat and arthropod fauna
which may potentially affect this avifaunal
structure. Objective 1 can be met by determining
if there are significant canonical correlations
between the birds and the environmental
parameters. If groups of bird species are
recognizable based on their correlations with
significant canonical variates, then these may be
considered as guilds due to the similarity of
these species in exploiting resources, and this
would satisfy Objective 2. Objective 3 may be met
if the canonical variates are interpretable in
terms of their associations with the habitat and
arthropod variables of the analysis.

STUDY AREAS

The field study took place from May 1975 to
April 1976 in the Serengeti National Park,
Tanzania. Five 1-km^ study sites were established
to represent a range of grassland vegetation
conditions in the plains portion of the park (fig.
1). Detailed descriptions of each site are
available in Folse (1978). Precipitation in the
Serengeti was seasonal and highly variable, both
in time and space; there was an increasing
gradient in precipitation from the southeast to
the northwest (with 500 mm to 700 mm per anum on
the plains). This variation in precipitation,
coupled with intensive seasonal grazing by
wildebeests, zebra, gazelle, and others, and dry
season fires, resulted in a great degree of
variation in vegetation structure among the
different sites. Within each of the sites,
however , the structure of the vegetation was
reasonably uniform with major variation taking
place through time (seasonally).

METHODS

The data used in these analyses were based on
concurrent samples of numbers of birds of each
species in the community, structural parameters of
the vegetation, and numbers of arthropods in
several age-taxonomic classes. Details of the
sampling procedures are described in Folse (1978).
These samples were taken at monthly intervals at
each of the study sites (two samples were missing
resulting in 58 observations) . Bird numbers were
sampled at each site with nine systematic
transects (in groups of three). Each transect was
1 km long and 20 m wide; they were run in a
vehicle and the number and identity of each bird
species flushed from the transect was recorded.
The sampling intensity was 18% of each study area.
The bird data were converted to biomass estimates
based on sampled weights and were then log-
transformed prior to analysis; preliminary
analyses indicated that mean values and variances
of untransf ormed biomasses (of all biomass
variables in the study) were proportional,
suggesting the need for such a variance
stabilizing transformation prior to analysis.

Figure 1. Location of the study sites (A through
E) in the plains portion of the Serengeti
National Park, Tanzania.

Arthropod data were collected on four
sweep-net transects placed systematically on each
study site. Fifty sweeps were made on each
transect. Size distributions of 25 age-taxonomic
categories of arthropods were available for each
transect, and these were converted to biomass
estimates based on sampled length-weight
relationships for each group. Arthropod biomass
data were also log-transformed prior to analysis.

Habitat data were collected from four
transects associated with the arthropod samples.
Six sample units at 20-pace intervals were used
for each transect. At each sample unit, four
sample points were placed in a square with 2 m
diagonals. At each sample point, a thin wire was
placed vertically, and the number of contacts of
vegetation (both total and green) with the wire
was recorded in 10 cm intervals above the ground.
In addition, maximum height of emergent vegetation
within a 1 m radius of the center of the sample
unit was recorded. From these samples, I
calculated indices of total biomass, green
biomass, percent cover, percent vegetation below
10 cm (an index of vertical vegetation structure),

161

and vegetation height. Two additional variables
were created by using values of total and green
biomass lagged by one month; these variables were
included to evaluate possible delayed response of
birds to changes in the vegetation biomass
variables. Accumulated precipitation (monthly
intervals) data were available from storage gauges
near each site. All biomass data were log-
transformed and all the percentage data were
transformed with the arcsin square root
transformation (Sokal and Rohlf 1969).

Canonical correlations were run on
appropriate subsamples of the data set (with
monthly mean values) with the CANCORR procedure of
SAS (Barr et al . 1979). I shall refer to a linear
compound of original variables produced by the
procedure CANCORR as a canonical variate, and
coefficients of the variates associated with each
original variable as fac^iors. In two-group
canonical analysis, pairs of canonical variates
are produced. Each variate pair contains one
variate of bird factors and one variate of
environmental (habitat and/or arthropod) factors.
The first pair has a maximum possible correlation.
The second pair is maximally correlated, given
that each is orthogonal to its corresponding
variate in the first pair, and so on. Usually,
only the "significant" correlations and their
corresponding variate pairs are considered in
evaluation of the relationships. In the usual
approach to canonical analysis, the observations
are plotted in variate space using either set of
variates, and these observations are then
clustered to look for relationships among the
observations in the variate space. Thus variates
are used to establish relationships among
observations, and canonical correlations are used
as a measure of how well the two sets of variates
provide the same set of relationships among
observations (site-time combinations in the
present case) . Interpretation of what each
variate "means" is usually accomplished by
considering simple correlations of each of the
variables with the variate.

In the present study, emphasis was on the
bird community rather than on the observations
(site-time combinations); thus I wished to use
canonical correlation as a technique of
classification or ordination based directly on
relationships of the bird species to environmental
variables rather than indirectly through
"association" or "distance" relationships among
the birds themselves. Consequently, rather than
group observations based on variate factors (as is
usually done) , I sought to group species of birds
based on their simple correlations with the
variates, using the variates as a means of
establishing structural relationships between
birds and environmental variables (fig. 2).

In addition to the analyses described above,
I tested the homoscedasticity of the variance-
covariance matrices with Wilk's generalized
likelihood ratio test described by Morrison
(1976:250, eq. M). A canonical correlation also
was run with all data with the variables
standardized .

Figure 2. Relationships among bird species,
environmental variables, and site-time (sites)
structure of the data set. Environmental
variables potentially affect composition of the
bird fauna via conditions at the sites. Numbers
indicate number of variables for birds and
environmental characteristics and number of
observations for sites.

RESULTS

Seventeen species of grassland birds were
considered as study species (table 1). These were
fairly small birds which depended on the grassland
habitat for most of their requirements and were
not dependent on specialized microhabitats such as
trees, waterholes, etc. The energy requirements
of these birds were satisfied mainly by arthropods
(estimated 81%, Folse 1978), thus I considered
arthropods as the primary food resource of the
birds used in this analysis (although one species
is almost wholly granivorous) .

The first canonical analysis involved 17
species of birds versus all environmental
variables (8 habitat and 25 arthropod). Six
"significant" canonical correlations resulted with
bird variates that explained 57% of the variance
in the bird portion of the data set; ^5% of that
variance was explained by the first three
correlations (table 2). Simple correlations of
each bird variable with bird variates were used to
determine the coordinates of each species in the
correlation space associatied with the first three
bird variates (fig. 3). This produced six
recognizable groups of birds, each consisting of
species which were associated with the
environmental variables in similar ways (fig. 3).

Interpretation of the environmental variates
produced by the first canonical analysis may be
accomplished by examining correlations of
environmental variables with environmental
variates (table 3). The first variate had
negative associations with high vegetation
biomass, high vegetation cover, tall vegetation,
and with vertical vegetation structure (the latter
indicated by the positive association with percent
vegetation below 10 cm). It also had negative
correlations with shorthorned grasshopppers( ACRA ,
ACRN), spiders (ARAU), crickets (GRYU), bugs
(HEMA), and caterpillars (LEPN). This variate may
be interpreted as a gradient between short grass,
low vegetation biomass versus tall grass, high
vegetation biomass with corresponding low versus

162

Table 1. Common names (scientific name) of bird
species considered in this study.

1.

Crowned lapwing

Vanellus coronatus

2.

Caspian plover

Charadrius asiaticus

Q

J •

IWO— DalJUcU COUroci

V-fUoUl J.UO ai I J.L.ailUo

4.

Northern white-tailed
bush-lark

Mirafra albicauda

5.

Rufous-naped lark

Mirafra africana

6.

Flappet lark

Mirafra rufocinnamomea

7.

Fawn-colored lark

Mirafra africanoides

8. Red-capped lark

9. Fisher's sparrow-
lark

10. Short-tailed lark

11. Richard's pipit _

12. Sandy plain-backed pipit

13. Yellow-throated longclaw

14. Rosy-breasted longclaw

15. Capped wheatear

16. Rattling cisticola

17. Wing-snapping cisticola
Zitting Cisticola

Calandrella cinerea

Eremopterix leucopareia
Pseudalaemon fremantlii

Anthus novaeseelandiae

Anthus vaalensis

Macronyx croceus

Macronyx ameliae

Oenanthe pileata

Cisticola chiniana

Cisticola ayresii
Cisticola juncidis 1

'These two species had broadly overlapping ranges
and were indistinguishable in flight during the
sample censuses.

high biomass gradient of these arthropods. The
second variate was associated with low cover, low
vegetation biomass (lagged) versus high cover,
high vegetation biomass which was independent of
vegetation height. It had no strong associations
with arthropods. The third variate had a weak
positive association with grasshoppers (ACRA,
ACRN, TTRU) but no strong association with any of
the other environmental variables. Precipitation
was unimportant relative to these variates and
green vegetation biomass was weakly associated
with the first variate.

The second canonical analysis involved birds
versus eight habitat variables. This resulted in
three "significant" correlations in which the bird
variates explained 39^5 of the variance in the bird
portion of the data set (table 2). A plot of bird
correlations with bird variates resulted in a
group structure similar to that of the first

Figure 3. Bird species group structure resulting
from simple correlations of bird species with
the first three bird canonical variates from the
complete canonical analysis (with all
environmental variables included). Points below
the horizontal plane of the three-dimensional
plot have dashed lines and open circles. Groups
indicated are potential guilds whose members are
associated with environmental characteristics in
similar ways. Numbers correspond to those in
table 1. Group A is the "woodland" group; B the
"tall-grass" group; C the "clumped-grass" group;
D the "low-cover" group; E the "plains" group;
and F the "short-grass-plains" group.

analysis with all environmental variables (fig.
U) . Bird species 12, the sandy plain-backed
pipit, was less strongly associated with the
"woodland" group and the "short-grass-plains"
group (species 2, the Caspian plover, and species
15, the capped wheatear) was less distinctly
defined. Interpretation of environmental variates
was similar to that of the complete analysis
(without arthropods).

The third canonical analysis involved birds
versus arthropod variables (habitat variables
excluded) . This resulted in four "significant"
correlations in which bird variates explained 37?
of variance in the bird portion of the data set
(31% for the first three variates, table 2). A
plot of bird correlations with bird variates (fig.
5) resulted in a completely different organization
of bird species with a much less definite group
structure. I can recognize 6 groups (fig. 5),
only one of which occurred in the complete
analysis (the "tall-grass" group — species 5, the
rufous-naped lark, and species 17, the
wing-snapping/zitting cistiocolas) .

Canonical correlation of the full model with
standardized variables resulted in different
canonical variates, but identical canonical

163

Table 2. Canonical correlations and percent variance' of bird biomasses accounted for by
each bird canonical variate.

Birds vs. Birds vs. Birds vs.

Habitat + Arthropods Habitat Arthropods

Canonical Canonical Variance Canonical Variance Canonical Variance

variate correlation % correlation % correlation %

1

0.999

19.1

0.984

19.6

0.987

17.9

2

0.995

14. 1

0.904

13.4

0.969

5.9

3

0.989

11.9

0.802

6.4

0.954

7.3

i»

0.972

3.2

0.939

5.6

5

0.963

4.6

6

0.950

4.5

57.4% 39.4% 36.7%

'Percent variance = (1/k)(s • where r^j is the correlation of the j^*^ species with

th

the i bird canonical variate and k is the number of bird species (17 in this case).

Figure 4. Simple correlations of birds species
with the first three bird canonical variates
with habitat variates only. See figure 3 for a
detailed description. Note that group structure
is similar to that of figure 3.

Figure 5. Simple correlations of bird species
with the first three bird canonical variates
with anthropod variables only. See figure 3 for
a detailed description. Note that group
structure is much different from that of figures
3 and 4.

164

Table 3. Simple correlations of environmental
variables with the first three environmental
canonical variates.

Canonical variate
I II III

Habitat :

Tn1"pl vpff f^t" at i on

-0 72

_0 . 43

_o . 09

\J 1 CCli V c o a U X tufll

-0. 41

0 17

0. 09

Percent cover

-0. 51

-0. 58

0. 00

Percent veg.< 10cm

0.89

-0.04

0.20

Height

-0. 89

-0.21

-0.23

Precipitation

-0,15

0.07

0.23

Tot. veg. (lagged)

-0.69

-0.50

-0. 18

Gr . veg. (lagged)

-0.41

0. 10

-0. 12

ihropods' :

ACRA

_0. 53

-0. 38

0.41

ACRN

-0.52

-0. 32

0. 46

ARAL!

_0. 63

-0. 31

-0. 08

BLAA

-0.22

0. 1 9

-0. 16

BLAN

-0. 29

0. 15

-0.03

COLA

-0.37

-0. 15

0. 19

DIPA

-0. 47

-0. 17

0. 13

FORA

-0.17

0. 16

-0. 15

GRYU

-0.67

-0.20

-0. 11

HEMA

-0.52

-0.17

0.08

HEMN

-0.32

-0.25

-0. 06

HOMA

-0.56

0.21

-0.03

HOMN

-0.34

-0.09

-0.11

HYMA

-0.28

0.03

0.03

ISOA

-0.14

0. 17

-0. 17

LEPA

-0. 1 1

0.02

0.07

LEPN

-0.55

-0.23

0.09

MANA

-0.08

0.22

-0. 10

MANN

-0.47

-0.22

0.03

NEUA

-0.08

-0.32

-0. 19

OTHU

-0. 05

0.08

-0.08

PHSU

-0.36

-0.02

-0.21

TETA

-0.27

-0. 14

-0.02

TETN

-0.34

-0.12

0.03

TTRU

-0.33

0.04

0.63

'First three letters of each name symbol
correspond to arthropod order or family name;
last letter refers to age category: A = alate
(winged adult), N = nonalate, and U = unknown
(both A and N). Thus ACRA stands for Acrididae
alate. OTHU is "other".

correlations and identical simple correlations of
the standardized variables (as the original
variables) with their corresponding variates (as
expected) . Results of tests of homoscedasticity
of variance-covariance matrices showed that all
matrices were not homoscedastic.

DISCUSSION

The question of whether or not there is a
structural organization to the avifauna related to

characteristics of habitat and arthropod fauna
(Objective 1) can be answered affirmatively;
results of all the canonical correlation analyses
demonstrated it. The bird with habitat and the
bird with habitat-arthropod analyses produced the
same recognizable groups of bird species
(Objective 2). The bird with arthropod analysis,
however, produced a different organization of bird
species with no clear group structure. In the
complete analysis, the group structure was due
primarily to the habitat characteristics rather
than the arthropod characteristics, since removing
arthropods from the analysis produced very little
change while removing habitat characteristics led
to great change in the resulting structure.

The complete analysis led to six groups of
birds. The "woodland" group consisted of six
species which occurred primarily at site E on the
woodland-plains border. The "low-cover" group had
four species which occurred throughout the
Serengeti plains but were usualy in areas with
extensive open ground and little ground-level
obstruction. Tall grass could occur in these
areas, but it was usually sparsely distributed.
The "tall-grass" group (two species) occurred
throughout the plains but always in association
with tall grass (they were rare at site A). The
greater the vertical vegetation structure and
vegetation density, the greater the density of
these species. The "clumped-grass" group (two
species) occurred mostly at sites C and D, and
were always associated with fairly dense clumps of
vegetation. The "short-grass-plains" group (two
species) occurred mainly at site A with little or
no vertical vegetation structure, but with
reasonably high percent cover. The "plains" group
(two species) had a broad range throughout the
Serengeti plains, but were usually associated with
areas of short vegetation, low cover, and fairly
reduced vegetation biomass. Three of these groups
(woodland, clumped-grass, and short-grass-plains)
were fairly restricted in the range of sites they
used while the remaining groups (low-cover,
tall-grass, and plains) each had quite broad
ranges with respect to sites. However, each used
specific types of microhabitat within sites and/or
used a site at times of the year when habitat
conditions were suitable. Each of these six
groups is a good candidate for a guild (a group of
species which exploit their environmental
resources in similar ways) . Objective 2 seems to
be well satisfied.

Objective 3 has been partially satisfied by
the fact that habitat variables and not arthropod
variables are most important in the organization
of the avifaunal group structure. Consideration
of simple correlations of habitat and arthropod
variables with the corresponding canonical
variates suggested that the variables of most
importance were vegetaion biomass, percent cover,
height, and vertical structure of the vegetation.
Green vegetation biomass and lagged values of
vegetation biomass were also of importance. The
importance of the lagged values suggests that
there is some degree of inertia in the response of
the birds to changes in habitat conditions.

165

Precipitation, per se, was unimportant. The most
important arthropod variables, although minor,
were grasshoppers, spiders, crickets, bugs and
caterpillars.

The canonical correlation analyses of these
data seemed to be very satisfactory relative to
the original objectives, but were they
appropriate? I had a total of 50 variables (worst
case) and only 58 observations. While the number
of observations was sufficient to allow the
analysis to be completed technically, it was
"data-poor." Thus, as an estimation technique for
population parameters, the analysis would be weak
but to establish relationships within my data set,
the sample size was adequate, provided that
"significance" of the resulting canonical
correlations is regarded with caution. Since this
was an exploratory analysis to look for potential
relationships among bird species for further
analysis and study, the technique seemed useful.

Williams (1981) pointed out that hetero-
scedasticity of the variance-covariance matrix
used in canonical correlation analysis can lead to
biased estimates of factors (coefficients of the
canonical variates) and to different relationships
of observations when they are plotted in canonical
space versus observation space. However, these
problems are much less important when considering
group structure of one variable set based on
simple correlations with resulting variates. This
is evident from the fact that both unstandardized
and standardized variables resulted in the same
correlation structure whereas the corresponding
canonical variates were different. However,
standardizing normalized variables individually
does not necessarily produce homoscedasticity
since no consideration is taken of the covariance
structure of the data. Although the variance-
covariance matrices used in these analyses were
heteroscedastic , the fact that most variables were
transformed with traditionally normalizing
transformations, and the correlations rather than
variate factors were used in establishing avian
groups, should have led to a fairly stable
analysis. Dropping out half of the variables for
the habitat analysis (the arthropod variables)
resulted in little change in the avifaunal group
structure, which suggests that heteroscedasticity
characteristics of the arthropod portion of the
variance-covariance matrix had little effect in
biasing the results.

ACKNOWLEDGMENTS

I thank William B. Smith for discussions on
methodology, the Caesar Kleberg Program in

Wildlife Ecology for sponsoring my research in
Africa, and T. Mcharo and the Director and
trustees of Tanzania National Parks for permission
to live and work in the Serengeti National Park.
This is contribution number TA16128 of the Texas
Agricultural Experiment Station.

LITERATURE CITED

Barr, A.J., J.H. Goodnight, J. P. Sail, W.H. Blair,
and D.M. Chilko. 1979. SAS User's Guide.
'<94 p. SAS Institute, Raleigh, N. C.

Folse, L.J., Jr. 1978. Avifauna-resource
relationships on the Serengeti Plains. Ph.D.
Dissertation. 107 p. Texas A&M University,
College Station, Tex.

Gittens, R. 1979. Ecological applications of
canonical analysis, p. 309-535. In Orloci,
L., C.R. Rao, and W.M. Stiteler, editors.
Multivariate methods in ecological work.
550 p. International Co-operative Publishing
House, Fairland, Md.

Kshirsagar, A.M. 1978. Multivariate analysis.
53** p. Dekker, New York, N.Y.

Morrison, D.F. 1976. Multivariate statistical
methods. Second edition. 415 p. McGraw-
Hill, New York, N.Y.

Sokal, R. R., and F.J. Rohlf. 1969. Biometry.
776 p. Freeman, San Francisco.

Williams, B.K. 1981. Discriminant analysis in
wildife research: theory and applications.
In Capen, D.E., editor. The use of
multivariate statistics in studies of
wildlife habitat: Proceedings of a workshop
[Burlington, Vt., April 23-25, 1980]. USDA
Forest Service General Technical Report.
Rocky Mountain Forest and Range Experiment
Station, Fort Collins, Colo. (In press).

DISCUSSION

NOVA SILVY: Could arthropods be correlated with
the vegetation structure in a manner such that
they contributed nothing new to the analysis when
they were added into the data sets?

L. JOSEPH FOLSE, JR.: The arthropod distribution
is very much determined by the vegetation
distribution; however, the relationships do not
seem to be concordant with the bird-habitat
relationships. If they were, the bird-arthropod
canonical correlation would not result in a
completely different organizational structure.

166

HABITAT ASSOCIATIONS OF BIRDS BREEDING IN CLEARCUT

DECIDUOUS FORESTS IN WEST VIRGINIA'
Brian A. Maurer^, Laurence B. McArthur^, and Robert C. Whitmore''

Abstract. — Associations between vegetation structure and
34 bird species in four forested areas of various stages of
clearcut regrowth were examined using principal components
analysis. Relative frequencies for each bird species were
determined during three breeding seasons and used to weight
habitat variables. The resulting data matrix (34 species x 8
habitat variables) was subjected to principal components
analysis using a standardized covariance matrix.

The first principal component was negatively correlated
with percent and mean height of low vegetation, and
positively correlated with percent litter and the number,
height, and percent of canopy layers. The first principal
component separated early successional species from late
successional species. The second principal component was
positively correlated with percent slash and the number of
trees less than 12.7 cm dbh. This component separated
mid- successional species from earlier and later successional
species. The first two components explained 90% of the
variation and thus seemed to be an adequate description of
the habitat associations of most species. The third
component, however, was useful in separating a few of the
mid-successional species, with species that foraged mainly on
the ground having higher values than species that foraged on
small trees and shrubs. The third component was positively
correlated with percent litter and negatively correlated with
the number of trees less than 12.7 cm dbh. Field methods
used in this study appear to be most applicable where it is
impractical to use more conventional methods of collecting
habitat association data, e.g., territory mapping, or use of
male singing perches.

Key words: Clearcut; deciduous forests; habitat
associations; habitat ordination; nongame management;
passerine birds; principal components analysis; West
Virginia .

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Graduate Research Assistant, Division of
Forestry, West Virginia University, Morgantown, W

26505. Present address: Center for Quantitative
Studies, University of Arizona, Tucson, AZ 85721.

^Wildlife Biologist, Division of Natural
Resources, P. 0. Box 67, Elkins, WV 26241.

Associate Professor, Division of Forestry,
West Virginia University, Morgantown, WV 26506.

167

INTRODUCTION

Current field techniques for assessing
habitat relationships are based on measuring
habitat variables in activity centers of
individual organisms. In the case of birds, male
singing perches are often used to determine
activity centers. This paper presents another
method of obtaining habitat association data that
can be used to compare overall habitat preferences
of bird species breeding in several different
habitats. The procedure analyzes variation
between species by obtaining a mean value for each
species for each of several habitat variables, and
subjecting the resulting data matrix to a
principal components analysis.

STUDY AREAS

Data on bird populations and habitat
structure were collected on four watersheds in the
Fernow Experimental Forest, ^.8 km southeast of
Parsons, Tucker Co., West Virginia. The
watersheds have been used by the USDA Forest
Service to assess impacts of various logging and
herbicide treatments on forest hydrology. Logging
and herbicide treatments have created several
different habitats which attract different bird
communities .

Watershed ^ (WS4), a 39-ha area used as a
control in the hydrology experiments, was logged
in 191 1 , and has remained relatively untouched
since then. Trees killed by chestnut blight were
salvaged during the early 1940*3, Tree species
common in this area included sugar maple, red
maple, red oak, chestnut oak, black birch, black
cherry, American beech, and yellow poplar
(scientific names of plants and birds are given in
appendix I ) .

Watershed 3 (WS3), 34 ha in size, was
clearcut between July 1969 and May 1970, except
for a strip of trees along the main drainage.
Remaining trees were removed during the winter of
1972-1973 1 and the area has revegetated naturally
since then. Tree species common in this area
included saplings of many mature forest species,
as well as sassafras and pin cherry.

Watershed 7 (V/S7), a 24-ha watershed, was
partially clearcut in 1964, and regrowth was
suppressed with herbicide treatments until 1967,
when the rest of the watershed was clearcut. The
area was kept barren for another 2 years before
allowing regrowth to resume. Saplings of several
mature forest trees were common in this area.
Sassafras and pin cherry were also present,
however, staghorn sumac was common in the
watershed .

Watershed 6 (WS6), 22 ha, was treated with
herbicides in a manner similar to WS 7. The
watershed has been further treated with herbicides
several times since 1971 to discourage natural
hardwood and herbaceous re vegetation , and to
encourage establishment of Norway spruce. This

Table 1 . Three-year means for vegetation
structure variables in each watershed.

Variable WS4 WS3 WS7 WS6

Litter (%)

87

.3

83

.6

68

.7

59

7

Slash {%)

7

.6

17

.2

11

.7

11

3

Herbaceous

vegetation (?)

42

.2

24

.7

78

.9

85

7

X height

herb. veg. (cm)

27

.9

29

.7

61

.7

59

3

Canopy layers (no.)

2

. 1

1

.4

1

. 1

0

2

Max. canopy
height (m)

21

.3

6

.7

4

.3

2

1

Canopy cover (%)

96

.8

90

.9

77

.8

4

9

Trees <12.7 cm (no. )

0

.6

2

.3

1

. 1

0

2

area had a dominant ground cover of ferns, with
several species of saplings and blackberry
occurring sporadically throughout the watershed.

METHODS

Avian populations were censused during the
breeding seasons of 1977, 1978, and 1979 using
30-m belt transects established in each area. All
areas were visited about the same number of times
each year. Transects were walked between sunrise
and 0730 EDT at a predetermined pace, and the
number of singing males of each bird species was
recorded .

On 1m X 1m plots located randomly within the
belt transects, 13 vegetation characteristics were
measured in each watershed during July. In
statistical analyses, only eight variables (table
1) were actually used to minimize duplication of
information .

We assumed that vegetation characteristics
prevalent in a watershed during a given year
directly or indirectly influenced the relative
abundance of bird species in that watershed.
Making this assumption, we weighted the mean value
of each habitat variable by relative frequency of
a bird species in a watershed for each year.
Years in which a species was more common in a
watershed were weighted more than years in which a
species was relatively less common. This was
accomplished by dividing relative frequency of a
species for a given watershed-year combination by
the sum of all relative frequencies for that
species over the 12 watershed-year combinations.
This procedure is illustrated for the red-eyed
vireo in table 2.

168

Table 2. Relative frequency of £ed-eyed vireos
(RF.) and mean % canopy cover (X^) for 3 years
in Your watersheds. The_procedure of obtaining
weighted mean values (X ) used in principal
components analysis is illustrated by these
data .

Year Watershed RF. X.

1 1

1977

4

0.32

92.7

3

0.06

84,6

7

0.02

66.0

6

0

6.0

1978

4

0,49

98,7

3

0.23

94,2

7

0.15

84.2

6

0.02

2.5

1979

U

0.23

99.4

3

0.28

95.6

7

0,19

88.0

6

0.04

5,0

Z RF. = 2,03

_ 12 _
X^ = Z RF.X. = 91.3%
i=1

2,03

Entries in the data matrix (34 species x 8
habitat variables) to be used for a principal
components analysis were weighted mean values for
each habitat variable for each species.
Calculations described above were done using the
PROC MATRIX procedure of Statistical Analysis
System (SAS, Helwig and Council 1979). A
correlation matrix for the habitat variables was
calculated and a principal components analysis
(PCA) was done on PROC MATRIX using a program
written by the second author. Output included
eight eigenvalues and eigenvectors, correlations
of original variables with each principal
component, scores for each bird species on the
principal components, and a plot of species scores
on the components. Copies of the PCA program are
available from the authors.

RESULTS

Means for all habitat variables are given in
table 1 . Inspection of these data were useful in
obtaining an overall impression of the structural
characteristics of each watershed.

Results of the PCA (table 3) indicated that
the first three principal components accounted for
most variation between bird species in their
associations with habitat variables. Correlations
of original variables with the first three

Table 3. Results of principal components analysis
using weighted averages of 8 habitat variables
for 34 bird species.

Component 1 2 3

Variation explained 67.46% 23.47% 5.50%
Cumulative variation 67.46% 90.93% 96.43%

Variable Correlations with original variables

Litter

0.85»*

0.

23

0.

42«*

Slash

-0.38*

0.

86»*

0.

32

Herb. veg.

-0.93**

-0.

29

0.

07

X ht. herb. veg.

-0.96»»

-0.

01

-0,

04

Canopy layers

0.98»»

-0.

14

0.

01

Max. canopy ht.

0.90«»

-0.

41*

0,

08

Canopy cover

0.95**

0.

11

-0.

20

Trees < 12.7 cm

0.27

0.

90**

-0.

33*

« P < 0.05, »» P < 0.01

components demonstrated that variables which
tended to have high values in WS4 (mature forested
watershed) were positively correlated with the
first component, and variables with low values for
WS4 were negatively correlated with the first
component (table 3). The second principal
component was significantly correlated with
variables that either had high values or low
values for WS3, a mid-successional area.

A plot of species scores on the first two
principal axes indicated that species were
separated into three distinct groups (fig. 1).
The first group had low scores on the first two
components and was representative of species which
preferred early-successional or open habitats.
Some of these species, such as song sparrows and
prairie warblers, were exclusively limited to WS6
and WS7 (McArthur 1980). The second group had
high scores for the first principal component, and
low scores on the second principal component. The
group was composed of species which were primarily
mature forest species. The third group had high
values for both components and included species
typical of mid-successional habitats.

The third principal component explained about
6% of the variation. This component separated
some of the early to mid-successional birds from
each other. Species which forage primarily from
trees and shrubs, such as Canada warblers (Bent

169

1953) had negative scores on this axis, while
ground foraging birds, such as brown thrashers and
rufous-sided towhees (Bent 1968), had positive
scores. The third principal component was
postively correlated with litter and negatively
correlated with number of small trees.

DISCUSSION

Previous methods of collecting habitat
association data for birds have been to locate an
activity center of an individual and measure
habitat variables at that spot. In several
studies, habitat variables have been measured on
small plots (0.05 ha) centered around perches
where singing males were observed (James 1971;
Whitmore 1975, 1977; Smith 1977). However, the
physical structure of some habitats restricts the
ability of observers to approach perches of
singing males, and could" introduce a bias during
data collection. For example, in eastern
deciduous forests, early stages of regrowth
produce dense stands of saplings laced with
greenbriar and other deterrents to movement. WS3
in the present study was such an area. Though we
could easily hear singing males, the problems in
setting up even a small plot were formidable.
Another method of obtaining habitat association
data has been to locate territories (McArthur
1980, Rice 1978) or nests (Wray and Whitmore 1979)
and measure vegetation in these areas. Again, in

some habitats the physical structure of vegetation
may impede efforts to collect data. The method we
have presented alleviates many logistical
difficulties that might be encounterd while
collecting habitat association data in areas such
as WS3.

Limitations of our analysis are as follows.
First, the resolution obtained may not be as fine
as might be desired in some situations. From a
theoretical viewpoint, small differences in
habitat preference between ecologically similar
species may not be distinguishable. Also, though
the method is suggestive, it is not strictly
predictive. In addition, the problem of measuring
the appropriate variables is always present. This
problem can be partially alleviated by using
literature to obtain ideas as to which habitat
variables might be important. Finally, this
method does not analyze within-species variation.

This method of assessing habitat associations
should be useful in habitat management. Specific
information is summarized by such an analysis on
types of habitat characteristics a given bird
species is associated with. Though this does not
necessarily imply a cause-effect relationship
between habitat variables and bird species
abundanace, information provided by the PCA can
give at least a rough estimate of how species will
respond to alterations of habitat due to land use
practices.

No. saplings
% slosh

be

••bl

eb

kw

•bw

hw

•gl

b.

bb

/o herb.veg.
« hi. herb. veg.

Component 1

3

No. con. layer
Max. can. hi.
7» can. cov.
V. litter

Figure 1. Ordination of 3^* bird species on the
first two principal components obtained from a
PCA of 8 habitat variables. Species codes given
in appendix.

ACKNOWLEDGMENTS

We wish to thank B. Griffin, J. Hamilton, S.
Barman, R. Kidwell, R. Manna, B. Parker, and G.
Seidel for their assistance in the field. The
Timber and Watershed Laboratory, USDA Forest
Service in Parsons, West Virginia, kindly provided
housing arrangements and served as a source of
information about the study areas. E.J. Harner
provided statistical advice and encouragement.
This research was supported, in part, by Mclntire-
Stennis funds from USDA (SEA) and is approved by
the Director, West Virginia Agriculture and
Forestry Experiment Station as Scientific Article
No. 1661.

LITERATURE CITED

Bent, A.C. 1953. Life histories of North

American wood warblers. United States

National Museum Bulletin 203:1-73'J.
Bent, A.C. 1968. Life histories of North

American cardinals, grosbeaks, buntings,

towhees, finches, sparrows, and allies.

United States National Museum Bulletin

237: 1-602.
Helwig, J.T., and K.A. Council,

SAS user's guide. p.

Raliegh, N. C.
James, F.C. 1971. Ordinations of habitat

relationships among breeding birds. Wilson

Bulletin 83:215-236.

editors. 1979.
SAS Institute,

170

McArthur, L.B. 1980, The impact of various

forest management practices on passerine

community structure. Ph.D. Dissertation.

West Virginia University, Morgantown, W. Va.
Rice, J.R. 1978. Ecological relationships of two

interspecif ically territorial vireos.

Ecology 59:526-538.
Smith, K.G. 1977. Distribution of summer birds

along a forest moisture gradient in an Ozark

watershed. Ecology 58:810-819.
Whitmore, R.C. 1975. Habitat ordination of

passerine birds of the Virgin River Valley,

southeast Utah. Wilson Bulletin 87:65-74.
Whitmore, R.C. 1977. Habitat partitioning in a

community of passerine birds. Wilson

Bulletin 89:253-265.
Wray, T., Jr., and R.C. Whitmore. 1979. Effects

of vegetation on nesting success of vesper

sparrows. Auk 96:802-805.

Appendix I

Scientific names of bird and plant species
mentioned in the text. Codes for birds given in
parentheses .

Birds

Great-crested flycatcher (gf) Myiarchus crinitus

Acadian flycatcher (ac) Empidonax virescens

Eastern wood pewee (ew) Contopus vlrens

Black-capped chickadee (be) Parus atricapillus

White-breasted nuthatch (wn) Sitta carolinensis
Gray catbird (cb) Dumetella carolinensis

Brown thrasher (bt)
Wood thrush (wt)
Veery (ve)
Cedar waxwing (cw)
Solitary vireo (sv)
White-eyed vireo (wv)
Red-eyed vireo (rv)
Black-and-white warbler (bw)

Toxostoma ruf rum
Hylocichla mustelina
Catharus fuscescens
Bombycilla cedrorum
Vireo solitarius
V. grlseus
V , olivaceus
Mniotilta varia

Golden-winged warbler (gw) Vermivora chrysoptera

Dendroica virens

Black-throated green
warbler (bg)

Prairie warbler (pw)

Ovenbird (ob)

Common yellowthroat (yt)

Yellow-breasted chat (yc)

Kentucky warbler (kw)

Hooded warbler (hw)

Canada warbler (ca)

American redstart (ar)

Scarlet tanager (st)

Cardinal (cl)

Rose-breasted
grosbeak (rg)

Indigo bunting (ib)

American goldfinch (gf)

Rufous-sided towhee (rt)

Field sparrow (fs)

Song sparrow (ss)

Plants
Red maple
Sugar maple
Black birch
American beech
Yellow poplar
Norway spruce
Pin cherry
Black cherry
Chestnut oak
Red oak

Staghorn sumac
Blackberry
Sassafras
Greenbriar

Black-throated blue
warbler (bb)

D. discolor
Seiurus aurocapillus
Geothlypis trichas
Icteria virens
Oporonis formosus
Wilsonia cltrina
W. canadensis
Setophaga ruticilla
Piranga olivacea
Cardinalis cardinalis

Pheucticus ludovicianus
Passerina cyanea
Spinus tristus
Pipilo erythropthalmus
Spizella pusilla
Melospiza melodia

Acer rubrum
A, saccharum
Betula lenta
Fagus grandifolia
Liriodendron tulipfera
Picea abies
Prunus pensylvanica
?_. serotina
Quercus prinus
Q. rubra
Rhus typhina
Rubus spp.
Sassafras albidum
Smilax spp.

D. caerulescens

Chestnut-sided warbler (cs)

D. pensylvanica

171

DISCUSSION

BARRY NOON: Did your principal components
analysis give you any additional insights into the
species-habitat associations that were not
apparent from a simple list of species by habitat
type?

BRIAN MAURER: Yes, the PCA helped to identify
specific relationships between bird species and
habitat variables. For example, both chestnut-
sided warblers and common yellowthroats were found
on three areas. The PCA showed that yellowthroats
were more strongly associated with variables
dealing with the abundance and height of
herbaceous vegetation, while chestnut-sided
warblers were not. These types of relationships
are not apparent by simply examining species

lists. In the example just given, since both
species would appear on lists for the same
habitats there would be no way to further
differentiate between their specific habitat
requirements, other than drawing speculative
conclusions from the life history literature.

KEN MORRISON: Why did you use the matrix
procedure of SAS rather than PROC FACTOR with
METHOD=PRINT to obtain your principal components?

BRIAN MAURER: It is easier to get the factor
scores from PROC MATRIX, because you do not need
to call an additional procedure. Also, the
researcher can easily modify a PROC MATRIX program
to fit the particular needs of the data.

172

PRINCIPAL COMPONENTS ANALYSIS OF DEER HARVEST-
LAND USE RELATIONSHIPS IN MASSACHUSETTS'
Philip J. Sczerzenie^

Abstract. — Relationships of deer harvests with land use
and forest cover types in Massachusetts' 351 townships in
1951 and 1971 were investigated using principal components
(PC) techniques. PC analysis and PC regression methods are
described and their value in reducing the dimensionality of
the land-use and forest-type data is emphasized. Components
on softwood vs. urban land, stand-age and hardwoods +
farmland accounted for over 40% of the X data variability.
Increasing softwood composition, decreasing age, and
increased association of hardwoods and farmlands were
significant positive effects when related to deer harvest
levels. PC regression, under well-defined criteria for
component deletion, allowed estimation of effects in the
original beta-space with reduced variances on those effects
while maintaining the statistical integrity of the data sets.

Key words: Forest types; land use; multiple regression;
principal components; white-tailed deer.

INTRODUCTION

Massachusetts' legal deer harvests began in
1910 under supervision of the state's Division of
Fisheries and Wildlife. Harvest levels increased,
except during depression and war years, to a
record take of 4,887 deer in 1958. Harvests
declined precipitously after 1958, until in 1966
mandatory deer checking, and in 1967 anterless
deer permit systems, were instituted.

Two concurrent factors appeared to have
caused the decline in harvests. First, either-sex
hunting from 1910 to I966 resulted in an
overharvest of does, so the reproductive portion

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Graduate Research Assistant, Dept. Forestry
and Wildlife Management, Holdsworth Hall,
University of Massachusetts, Amherst, MA 01003.
Present address: Ketron, Inc., 18th Floor Rosslyn
Center, 1700 North Moore Street, Arlington, VA
22209.

of the herd was unable to replenish itself.
Second, farm abandonment and secondary succession
that led to buildup of the Massachusetts herds in
the late 1800's no longer provided enough
productive habitat. Instead, a decline in forage
and cover was occurring due to urbanization and
forest succession. The antler less permit system
was established to remedy the former problem, and
while it was successful in limiting doe take and
increasing buck harvests (McDonough and Pottie
1979), it had no effect on the second problem.

The present study was undertaken to quantify
effects of different land uses and forest types on
deer harvests so that alternative approaches to
habitat management could allocate management
effort optimally. The 20-year period between
samples would tend to strengthen conclusions drawn
on the basis of effects that remained the same.

DATA DESCRIPTION

The dependent variable, deer harvest, was the
mean 5-year kill in each of 351 townships in
Massachusetts from 1949-53 and 1969-73. As

173

recommended by Labisky et al. (196^*), kill was

2

transformed using log^^CI + kill/mi ). Deer

harvest averages centered on the years 1951 and
1971 to coincide with land-use surveys of the
state based on aerial photographs (MacConnell
1975). Aerial photos were land-use typed to
approximately 3.5 acres; this information was
transferred to topographic sheets; and the area of
each of 104 types computed and recorded for each
township.

I used 35 types as independent variables:
abandoned fields (AF), pasture (PS), urban land
(UL), and 32 forest types (table 1). Acreage of
each type was transformed to arcsin square root
percent (Steel and Torrie 1950:158). Total
acreage of the 35 types was 4,453,360 in 1951 and
4,480,327 in 1971 representing 85.9% and 86,2% of
the state's total surface area of 5.2 million
acres in respective years.

PRINCIPAL COMPONENT ANALYSIS
Method

Principal component analysis (PCA) has been
used by ecologists as one of a number of
multivariate techniques to describe habitat
preferences of various species of North American
breeding birds (James 1971, Smith 1977) and to
describe site differences in relation to
vegetation communities (Austin 1958, Page 1976).

PCA is helpful because it reduces the
dimensionality of a data set, such as an array of
environmental variables assumed to influence the
species under consideration. A more parsimonious
description of habitat influence is possible,
therefore, assuming the principal components have
some reasonable biological interpretation.

The original data (X), expressed as a matrix
of simple correlations (by standardizing X and
premultiplying by X'), is transformed into a
matrix of principal component scores, z^^ ,

(regressors in PC regression). This procedure is
accomplished by creating a set of eigenvectors,
a^^, and eigenvalues, X^.

Each a^^ is a linear combination of loadings

between -1 and +1 on each X., constructed so that,

1

in sequence ( i= 1 , 2, . . . ,k) , the maximum possible
amount of variability in the X data is accounted
for, with the stipulation that each eigenvector be
orthogonal with every other a_^ ( a.^^ ' • = 0).

Each eigenvalue, , represents the portion

of variation explained by its corresponding
eigenvector. The following equations apply:

X a . = z . ( 1 )

z'. . z. = X. = a' . X'X a. (2)

and the resultant matrix of orthogonal eigenvalues
is

When based on the correlation matrix (X'X),
IX^ = k, where k is the number of independent (X)

variables .

Table 1. Classification of 32 forest types used as independent
variables (MacConnell 1973).

Forest composition:

H: hardwoods HS:

S: softwoods SH:

hardwood dominant mixed forest
softwood dominant mixed forest

Forest height classes:

1: one to 20 feet 2: 21 to 40 feet 3: 'il to 60 feet
4: 51 to 80 feet 6: uneven heights (3 or more classes

present)

Crown closure classes: '

A: 81 to 100% crown closure B: 30 to 80% crown closure

'Height classes 1 and 5 have no crown closure associated.

174

Table 2. Eigenvalues and cumulative porportion of X variation of 1951 and
1971 data on 35 land uses.

1951

Eigenvalues

98896
35130
69'»'»0

45877
277ng

, 16296

it. 16814
1. 16983
0.65197
0.40060
0.25885
0, 16006

3.29646
1. 06800
0.62429
0. 36848
0.22551
0. 14834

2.63986
0.84554
0.59965
0.35173
0.21778
0. 11799

1.66862
0.82070
0.52160
0.34111
0. 19392
0.09221

1.41153
0.73461
0.49797
0.29181
0. 17896

Cumulative proportion of total variance of independent variable

22826
64357
79594
89178
94981

0.98518

0.34735
0.67699
0.81457
0.90322
0.95721
0.98976

0.44153
0.70751
0.83241
0.91375
0.96365
0.99399

0.51695
0.73166
0.84954
0. 92380
0.96987
0.99737

0.56463
0.75511
0.86444
0.93355
0.95741
1.00000

0.60496
0.77610
0.87867
0.94188
0.98053

1971

Eigenvalues

6.82717
1.24683
0.56755
0.39114
0.27362
0. 15844

4.99291
1.07258
0.51426
0.36751
0.23534
0. 14815

4.24402
0.97177
0.50080
0.31664
0.22413
0. 12635

2,84598
0.79921
0.46745
0. 30569
0.20672

0. 11795

2.43800
0.75289
0.42616
0.28716

0. 17137

0.08863

1.39437
0.65541
0.41565
0.28383
0. 16534

Cumulative portion of total variance of independent variable

0. 19503

0.68538
0.82308
0. 90066

0.93507
0.98625

0.33679
0.71602
0.83777
O.9III6
0.95980
0.99049

45895
74379
85208
92021
96620
99410

0.54026
0.76662

0.86543
0.92824

0. 9721 1

0.99747

O.6O992
0.78813
0.87761
0.93715
0.97700
1.00000

0.64976
0.80686
0.88949
0. 94526
0.98173

The first few principal components normally
account for the bulk of X variability while the
last few explain a negligible portion and can, in
theory, be disregarded. This reduces the
dimensionality of X to z^, less than k, and, if

can be given some interpretation (usually by
inspection of eigenvector loadings) , a theory
about the underlying structure of X can be
expounded. With respect to a dependent variable,
the relationship between Y and each may be

examined, in many cases graphically, to determine
how the underlying X structure affects the species
of interest. For a more complete treatment of
principal component analysis the reader may
consult Johnston (1972) or Nichols (1977).

Results

Table 2 lists eigenvalues of the 1951 and
1971 X data (land uses and forest types) in order
of importance and the cumulative proportion of
total X variance explained by successive
eigenvalues. In both cases, the first four
eigenvalues account for more than half the
variability, leaving the remaining 31 components
to explain less than 50%. Figure 1 illustrates
the striking similarities between eigenvectors a^

each year, eigenvectors each year, and a^^ in

1951 and in 1971. Each pair of eigenvectors

measures the same relationship in X in both years
although figure lA and IC are mirror images due to
pecularities in computation of a^^.

175

1951 r = -.570

I

ftPU
FSL

r

HARDWOOD
1 2 3 4 6
A BAB A B

SOFTWOOD
12 3 4
ABA BAB

HS MIX
12 3 4 .
AB ABAB

SH MIX
12 3 4 .
A B ABAB

ilillliiiUiiiiililiiil

1. 1 III

1. Ill

t 1 PI

|l 'II >'!!

APU HARDWOOO' SOFTWOOD HS MIX SH MIX
FS L 1 2 3 4 611 2 3 4 6 1 2 3 4 6 1 2 3„ 4 6
I ABABAB I AbABAB | ABABAB ABABAB

Ji

I 1B71 r=.235

T

1951 r = .15S

APU
FSL

HARDWOOiySOF T WOOD
l^^3 4_6 1^_3.4.6

SH MIX
1 2 3 4 6
AB ABAB

J li Im-Jil 4

1971 r=-.097

period .

Figure 1 . Eigenvectors of 35 Massachusetts land
types: (A) eigenvectors (B) eigenvectors

a^, (C) eigenvectors a.^,3_^-

All forests with a softwood component have
relatively heavy loadings in while urban land

is weighted in the opposite direction. This can
be termed the softwood vs. urban land (SUL)
effect. It accounts for about 20% of X
variability in each year. The second eigenvector
is loaded positively for young forest of all
types, less positively and eventually negatively
for stands of intermediate age, and negatively for
older age forest stands. This effect of stand age
(STA) remains the same over the 20-year sampling

Eigenvectors a^ of 1951 and a^ of 1971

can be described as the hardwood/farmland (HF)
effect since they are loaded for hardwood forest
types and AF and PS in both years.

Three corresponding pairs of eigenvectors;
a^ a^, and a^ of 1951 and a^, a^, and a^ of 1971;

account for 44 and 42 percent of X variability
respectively, therefore, the dimensionality of
land-use and forest type data has been reduced, to
a great extent, to information on three
independent axes of variability that have clear
structures .

Eigenvectors act on X to form principal
component scores, z^. , as follows. Each township
has a given percentage composition of the 35
land-use types that constitute, when transformed,
the observations X.^, X^, . . . for the township in

that year. Each X^ is multiplied by the
corresponding eigenvector element and the products
summed to form z. for the township according to
equation (1). a township has, for example,

large percentages of older hardwoods and urban
land in 1971, its score for would be negative,

for negative and for positive.

Relationship with Deer Harvest

Relationships between principal axes of
land-use and forest-type variation and corres-
ponding township deer kill levels are summarized
in table 3. Component 1 in each year has the
highest eigenvalue, explains the highest
percentage of X variation, and has the highest
simple correlation with deer harvest. Because the
s are mirror images their corresponding r's are

of opposite sign. One can conclude townships with
low amounts of urban land and higher acreage of
softwoods are more productive of deer. Component
2 in each year explains less X variability and has
a lower, positive correlation with deer harvest,
thus, stand-age is an important determinant of a
township's productivity for deer. Those with
older stands, in general, have fewer deer
harvested .

The third PC effect, hardwood/farmland, has a
relatively low correlation with deer harvest and,
therefore, although townships with more hardwoods
generally have higher harvests, one would tend to
manage for softwoods or mixed forests rather than
hardwoods since the latter is a much stronger
relationship and appears to contradict the former
finding. The combination of hardwoods and
farmland, on the other hand, may serve as an
indication of the relative lack of urbanization in
a township and, in this respect, would reasonably
be associated with higher deer harvests.

176

Table 3. Comparison of three important principal components of 1951 and
1971 data on three land-use and 32 forest types.

X Variation

Year Component Eigenvalue explained (%) r b'

1951
1971

7.989
6.826

22.8
19.5

-0.570
0.464

-0.148
0.133

1951
1971

4.168
4.993

11.9
14. 3

0.257
0.235

0.092
0.079

1951
1971

3.296
2.846

9.4
8. 1

0.155 0.063
-0.097 -0.043

'Prinicpal component beta coefficients, P < 0.01.

PRINCIPAL COMPONENTS REGRESSION
Method

Least Squares Model

The preceding PC analysis was done as an
intermediate stage in what many researchers
consider the primary purpose of multivariate
analysis, estimation of coefficients on the
original variable set. Thus, the value of
principal components lies in their usefulness in
solving the general linear model:

^j=^o*^^1j*V2j*--- -Vkj-"j
where Yj are observations on the dependent

variable, ( j=1 , 2, . . . ,n) is the intercept, is

the effect of the i^*^ independent variable on Y,
(i=1,2, . . . ,k) , and u^ is a randomly distributed

error or disturbance term. In matrix notation (3)
becomes

X B + e

(4)

where Y is an n x 1 vector of dependent variable
observations, X is an n x k matrix of independent
variable observations, B is a k x 1 vector of
coefficients on X, and ^ is a residual term
(substituted for u) that is the difference between
Y observed and Y predicted using our estimate of
B.

Using the matrix of k eigenvectors derived
from the matrix X'X of correlations, we form the n
X k matrix of z scores ^ by the k x 1 vector d^ of
coefficients on the scores as

The vector d of PC coefficients is easily solved
for

d = (Z'Z) ^ Z'Y +(Z'Z) ^Z'e

(6)

since

.7-1 -

Z'Z

1/X,

1/X,

The second term on the right in (6) is zero. This

leads to the ordinary least squares (OLS) solution

when d^ is transformed back to the original
beta-space using B = Ad.

Component Deletion

The PC regression (PCR) solution, therefore,
calls for deletion of one or more components to
rid the data structure of "noise" that provides
little in the form of information about X
variation (recall cumulative proportion of
variability from table 2) but adds considerably to
variance about B.

The data structure is partitioned as:

Y = XA^ d^ + XA2^2 - = -1-1 * -2-2 -

and deletion of the second portion is required.
This is equivalent to the restriction XA^d^ = 0

(Fomby and Hill 1978). If the restriction is
true, the estimator remains unbiased.

Test of the restriction is as follows:

Y = XA'A B + e = XAd + e = Zd + e

(5)

177

(SSR - SSR , )/ J

where SSR and SSR , are residual sums of
res ols

squares on the restricted and unrestricted (OLS)
models respectively, J is the number of
restrictions (deleted components) and n and k are
as defined above. Because this is a test of the
truth of XA2d2 = 0, i.e., = 0, the test

statistic u is compared with a centrally
distributed F, a classical F-test.

If one is willing to accept some bias for
further reducing variances on B, a mean square
error (MSE) criterion can be used minimizing Z(var

2

+ bias ). This uses the same test statistic u but
now comparison is with a non-central F
distribution (Goodnight and Wallace 1972). Should
further component deletion be desired, some
eigenvalue size criterion might be employed,
however, statistical properties of the resultant
estimator, as in stepwise regression (Freund 197^)
will be unknown.

Results

The data sets for 1951 and 1971 were analyzed
by principal component regression (tables 4 and
5). In the analysis of 1951 data, one component
was deleted under the classical F criterion and
one additional component deleted under the MSE
criterion. For the 1971 data all components were
retained under the classical F criterion, so the
OLS model is the only unbiased estimator of B in
terras of the specified model assumptions. Under
the MSE criterion a single component was deleted.

Coefficients produced in the study are
consistent from year to year and within
year-period under OLS and both deletion criteria.
Those effects found significant in both years, PS,
UL, H2A, H3A, HS2A, and HS3A remained of the same
sign and magnitude. The close agreement over the
20-year period and under deletion criteria within
each year lead me to conclude that real
relationships have been estimated.

Clearly, results show urban land as a
significant negative effect in 1951 and 1971; this
was to be expected. Other significant negative
effects for both years were pasture, cutover
hardwoods (H2B), older hardwoods (H3A), and a
young mixed type (HS2A).

Table M. Results of the 1951 analyses under OLS,
classical F, and MSE criteria of deer kill
versus 35 land-use types defined in table 1.

Land-use
type

b OLS

b CI "F"

b MSE

AF
PS
UL

H2A

H2B

H3A

H4A

SMA

HS1

HS2A

HS3A

SH2A

1.75 «»

-.269 **

-.378 »»

-0.37 ns

-3.96 »*

-1.95 ns

-3.80 »«

9.31 «

-2.68 »«

-2.23 *

-0.39 ns

1.12 ns

-0

1.81 »»

•2.56 ««

•3.60 »»
ns

•3.60 »*

•0.36 ns

•3.64 »«

9.27 »

•2.18 **

•3.22 «»

0.91 ns

2.18 »»

1.70 **

-2.43 **

-3.45 »*

1.14 »»

-3.59 «*

-1.63 **

-3.05 *

9.12 »

-1.94 «»

-2.97 **

2.17 *«

0.66 ns

* P < 0.05, ** P < 0.01, ns = nonsignificant

Table 5. Results of the 1971 analyses under OLS
and MSE criteria of deer kill versus 35 land-use
types defined in table 1.

Land-use type

b OLS

b MSE

PS

UL

HI

H2A

H2B

H3A

SI

S2A

S3A

S3B

HS2A

HS3A

SH3A

-2.04 «»

-4.63 **

2.01 «

1.72 »

-3.87 »«

-1.72 »*

-4.09 *

4.30 »»

-2.64 »

3.65 *

-1 .62 ns

-0.33 ns

-2.69 ns

-1.85 *

-4.26 »«

2.41 »«

2.27 «»

-4.04 »»

-1.83 »»

-4.03 *

4.84 »«

-2.18 »

3.87 »

-2.52 »

0.88 »»

-2.49 »*

» P < 0.05, »» P < 0.01, ns = nonsignificant

178

Significant positive effects in both periods
were young, dense-canopy hardwoods (H2A) an older,
dense-canopy mixed forest (HS3A). Softwood types
were found significantly positive in both periods
but specific types were also identified as
negative effects in 1971.

Age, canopy-closure and softwood mixture of
hardwood and hardwood-dominant mixed forest types
appear to be governing factors in terras of
individual type effects on deer harvests. With
softwood composition, this accounts for much of
forest type effect on harvests. Young, dense
hardwood stands are used by deer as a major source
of browse while some of the more open softwood
stands may function both as winter cover and
feeding areas.

Those types that were negative (H2B, H3A,
H4A, HSl, HS2A) may provide neither enough food
nor enough escape cover in limited areas to be
acceptable to deer. The negative softwood types
(S3A, SH3A) because of their year-round dense
canopies, likely provide little food or cover at
all. Old softwood types that were positive
effects (S3B, S4A) may function primarily as
wintering sites with enough light penetration
through or underneath the canopy to support an
understory .

It is possible, therefore, to explain why the
land-use types investigated had their calculated
effects on deer harvests, although this may be
viewed as a simplistic causal assumption. There
are, of course, many effects of different land-use
and forest types, both direct and indirect, that
enter into the investigated relationships.
Nevertheless, quantification of these general
relationships may eventually lead to better
definition of specific hypotheses about impacts of
land-uses and forest composition.

CONCLUSIONS

Principal component analysis has shown an
underlying structure in land-use and forest type
composition in Massachusetts that is consistent
over a 20-year period. Relationships between
these axes of variability and deer harvest are
also consistent and are capable of reasonable
biological interpretation.

Principal component regression has allowed
elucidation of individual type effects on deer
harvest by reducing variances on B under criteria
that produce known statistical properties. This
is in direct contrast to data-dredging techniques,
such as stepwise regression, that eliminate
independent variables through mechanical
manipulation and that leave estimators that are
unreliable, at best, because their statistical
properties are not known. PCA and PCR are
therefore seen as valuable multivariate analytical
tools for the ecological investigator.

ACKNOWLEDGMENTS

I wish to thank Dr. Wendell E. Dodge of the
Massachusetts Cooperative Wildlife Research Unit
(U.S. Fish and Wildlife Service, Massachusetts
Division of Fisheries and Wildlife, University of
Massachusetts, Amherst, and The Wildlife
Management Institute), Dr. Bernard Morzuch of the
Department of Food and Resource Economics,
University of Massachusetts, and Chet McCord and
James McDonough of the Massachusetts Division of
Fisheries and Wildlife for their advice and
encouragement in this project.

LITERATURE CITED

Austin, M.P. 1968. An ordination of a chalk
grassland community. Journal of Ecology
56(3):739-757.

Fomby, T.B., and R.C. Hill. 1978. Deletion
criteria for principal components regression
analysis. American Journal of Agricultural
Economics 60 ( 3 ): 524-527 .

Freund, R.J. 1974. On the misuse of significance
tests. American Journal of Agricultural
Economics 56(1): 192.

Goodnight, J., and T.D. Wallace. 1972.
Operational techniques and tables for making
weak MSE tests for restrictions in
regressions. Econometrica 40(4) :699-709.

James, F.C. 1971. Ordinations of habitat
relationships among breeding birds. Wilson
Bulletin 83:215-236.

Johnston, J. 1972. Econometric methods. Second
edition. 437p. McGraw-Hill, New York, N.Y,

Labisky, R.F,, J. A. Harper, and F. Greeley. 1964.
Influence of land-use, calcium and weather on
the distribution and abundance of pheasants
in Illinois. 19p. Illinois Natural History
Survey Biological Notes No. 51.

MacConnell, W.P. 1973. Massachusetts mapdown:
land use and vegetative cover mapping
classification manual for use with
Massachusetts mapdown maps. 19 p.

University of Massachusetts Cooperative
Extension Service Publication 97.

MacConnell, W.P. 1975. Remote sensing 20 years
of change in Massachusetts, 1951/52
1971/72. 79 p. Massachusetts Agricultural
Experiment Station Bulletin 630.

McDonough, J.J., and J.J. Pottie. 1979. A
successful antlerless deer hunting permit
system in Massachusetts. Transactions
Northeast Fish and Wildlife Conference
36: 1 10-1 19.

Nichols, S. 1977. On the interpretation of
principal components analysis in ecological
contexts. Vegetatio 34(3) : 191-197.

Page, G. 1976. Quantitative evaluation of site
potential for spruce and fir in Newfoundland.
Forest Science 22(2) : 131-143.

Smith, K.G. 1977. Distribution of summer birds
along a forest moisture gradient in an Ozark
watershed. Ecology 58 ( 4 ) : 81 0-81 9 .

Steel, R.G.D., and J.H. Torrie. I960. Principles
and procedures of statistics. 481 p.
McGraw-Hill, New York, N.Y.

179

AN APPLICATION OF FACTOR ANALYSIS IN
AN AQUATIC HABITAT STUDY'
T.J. Harshbarger^, and H. Bhattacharyya^

Abstract. — In five small, high-gradient trout streams in
western North Carolina, 18 cover variables were related to
standing crop biomass of wild brook trout (Salvelinus
f ontinalis ) , rainbow trout (Salmo gairdneri ) and brown trout
(Salmo trutta) in randomly selected stream sections. Factor
analysis of the data set showed that only a small number of
factors or variables was needed to explain relations between
variables in the observed set. Key cover factors were area
in debris; turbulent water; vegetation, both in and over
stream; and overhanging banks. Resolutions obtained were
used in stepwise regressions to explore relationships between
standing crop of trout and age of fish. Regressions
containing factors as independent variables explained less
variation in fish standing crop than did regressions
containing equal numbers of original habitat attributes as
independent variables.

Key words: Aquatic habitat; cover; factor analysis;
multivariate analysis; regression analysis; stream fish;
trout.

INTRODUCTION

The presence of a self-sustaining population
of fish usually indicates compatibility between
the aquatic envrionment and the ecological
requirements of the fish. Wild trout are
excellent indicators of current environmental
conditions and their population density in streams
reflects their level of compatibility with a
highly integrated chemical, physical and
biological situation. However, simply realizing
that the trout population reflects its envrionment
is not particularly informative to the resource
manager. Explicit relationships between species
and their environments are needed to assess the

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Aquatic Ecologist, Southeastern Forest
Experiment Station, Asheville, NC 28806.

'Mathematical Statistician, Southeastern
Forest Experiment Station, Research Triangle Park,
NC 27709.

actual and potential capabilities of the habitat.

Functional and correlative approaches have
been used to study factors influencing the
distribution and abundance of a species. The
functional approach is used when factors are known
to influence certain attributes of the species.
Correlative procedures are best suited for
exploratory studies, where the relationship
between a species and its environment are unknown.
This procedure provides little information about
causality, but helps the researcher to make
inference for rigorous testing. Many variables
can potentially influence the distribution and
density of wild trout in a stream. Often the
choice of parameters to measure and analyze is
difficult because environmental variables in lotic
waters are typically correlated and confounded
with one another (Reid 1961).

Attempts to correlate single and multiple
variables to trout populations in streams have met
with varying degrees of success. Boussu (195'4),
Saunders and Smith (1962), and Wickham (1967)
investigated the relationship of a trout

180

population to cover. In two studies (Schuck 19^*3.
Hunt 1969), water depth strongly influenced trout
densities in streams, and current velocity was
similarly implicated in another study (Lewis
1969). Few investigators, notably Lewis (1969),
Stewart (1970), Platts (1974), and Binns and
Eiserman (1979), have studied the simultaneous
effect of several environmental variables on
stream trout populations using multiple regression
and correlation analysis.

The present study was designed to look at the
relationships between various habitat parameters
and trout. Factor analysis was used to delineate
and examine a group of environmental variables
that seemed important to trout, those providing
cover or shelter. Regression analysis was used to
examine relationships between the environmental
factors produced and trout.

STUDY SITES AND METHODS

We selected five small, high-gradient trout
streams in western North Carolina. In each, we
inventoried and measured 18 variables providing
fish cover, in 20 randomly selected 30m stream
sections. Each section was surveyed along line
transects established every 3m across the stream,
perpendicular to its center line. All physical
structures providing shelter or concealment for
trout were located and their cross-sectional areas
parallel to the air-water interface recorded.
Structures included rocks or ledges which afforded
cover to fish, undercut banks, aquatic vegetation,
and logs and brush in the stream. Other cover
situations such as water with sufficient surface
turbulance to prevent visibility of the stream
bottom were also measured and recorded.

Brush and loosely compacted debris in the
stream and streamside vegetation overhanging the
water surface were estimated ocularly. Cover
afforded by brush and debris was recorded as the
surface occupied by solid material and expressed
as a percentage of the water surface it covered.
Cover provided by overhanging bank vegetation was
expressed in two variables: as the percentage of
stream covered by vegetation between the water
surface and 1.0m above, and the percentage between
1.0m and 2.0m above the surface.

Standing crop biomass of wild trout was
estimated by depletion analysis electrof ishing.
All fish were weighed, measured, and returned to
the midpoint of the section under study.

Factor Analysis

Starting with the observed correlation matrix
on the 18 cover variables, we used SAS FACTOR
programs for principal axis, maximum likelihood,
and iterated principal axis factoring (SAS 1979).
Using the principal axis method and keeping only
those factors corresponding to eigenvalues greater
than one, six factors were retained. To make the
factors more conceptually meaningful, several

rotational methods were used, including varimax,
quartimax, equimax and promax (oblique) procedures
(SAS 1979).

All factor and rotation procedures used on
the 18 cover variables produced essentially the
same information about underlying structure.
However, the iterated principal axis and oblique
rotational procedures are intuitively appealing.
These procedures address communalities directly
and recognize that some factors may very likely be
correlated. As such only the interated principal
axis solution using an oblique rotation of the
cover related factors will be presented in this
paper .

Regression Analysis

Relationships between standing crop of trout
and the factors obtained from oblique rotation of
the iterated principal axis solution were explored
using SAS STEPWISE regression procedures and the
maximum R^ improvement technique developed by
Goodnight (SAS 1979). Relationships between the
habitat variables themselves and standing crop of
trout were similarly explored to determine if
factors entered the stepwise model in about the
same order as the variable they contained.
Coefficients of determination were compared to
assess performance of models containing factors
and variables.

RESULTS

"\

The six factors combined the 18 cover jj'

variables into groups which generally reflected \t
meaningful patterns in relation to the stream

environment. Factors were named after the !f

variable or variables producing greatest n

correlation. Factor loadings for the variables jj

are shown in table 1 . Factors had zero or close ^
to zero projections on most variables, very few

immediate loadings, and two or three high ij

associations. !•

ir

Factor 1 is a measure of debris with high 8

loadings for number of logs and the surface areas, j
parallel to the stream, in logs and in brush.

Factor 2 is a measure of side stream cover and is t

strongly negatively correlated with area in and j

percent cover of vegetation trailing in the water 1

surface. The third factor expresses the \

percentages of cover provided by overstream 1

vegetation 0-1m and 1-2m above the stream. Factor '

4 is highly correlated with area in turbulent I

water and to a lesser degree with the number of I
units of turbulent water. This factor is also

correlated with total cover and can be considered ]
a general cover factor. Rock area and number load
heavily on factor 5, and the sixth factor is
highly correlated with area in overstream
vegetation in the 1- to 2-m zone.

Factor values for each of the 100 stream
section observations were obtained from the
scoring coefficient matrix. Trout, regardless of

181

Table 1. Variables associated with cover factors.

Factor loadings

Variables

1

2

3

4

5

6

Ledge area

(1)

0.

063

0.

241

0.030

0.400

-0.288

0. 106

Rock

Number

(2)

-0 .

075

-0

047

-0 . 009

-0 . 096

0. 850

0.214

Area

(3)

0.

099

0

194

0. 114

0. 109

0.646

-0.191

Turbulent water

Number

(4)

-0.

108

0

304

-0.075

0.491

0.007

0. 156

Area

(5)

-0.

244

-0.

054

-0.059

0.936

-0. 115

-0.014

Logs

f

Number

(6)

0.

913

-0

049

-0.007

-0. 108

-0. 146

-0.003

Area

(7)

0.

969

0

003

-0.060

-0.089

-0.019

0.020

Bank area

(8)

0.

080

-0

539

-0.039

-0. 125

-0 . 1 1 9

-0. 080

Other area

(9)

0.

045

0.

002

0.345

-0.051

0. 112

-0.090

Brush

% Cover

(10)

0.

102

-0

150

0. 282

0. 062

0.033

-0.012

Area

(11)

0.

824

0

076

0.067

-0.006

0.084

0.076

Side-stream

vegetation

% Cover

(12)

-0.

116

-0

772

0.048

-0.087

0.082

0. 198

Area

(13)

0.

001

-0

712

0.006'

0.012

0.01 1

0.015

Over-stream

vegetation

(0-1m)

% Cover

(14)

-0.

045

-0

042

0.896

0.034

-0. 109

-0 . 20 1

Area

(15)

0.

174

0

217

0.111

0.231

-0.333

0.363

Over-stream

vegetation

(1-2m)

% Cover

(16)

-0.

032

0

089

0.844

-0. 110

0.039

0.293

Area

(17)

0.

048

-0

050

-0.062

0.046

0.094

0.999

Total cover

area (18)

0.

365

-0

105

0.062

0.775

0.225

-0.076

species, in each stream section were segregated
into the four age classes represented in each of
the five streams under study. The relationship
between factor values and trout standing crop for
each age group was examined using the stepwise
maximum regression technique. Likewise, the
relationship between the original 18 cover
attributes and trout standing crop was similarly
examined .

The coefficient of determination (R^)
corresponding to the six factors was lower than
that corresponding to the original 18 variables
for each age class of fish (table 2). Further,
the best one-attribute, two-attribute, etc.,
models obtained by the stepwise procedure also
produced higher R^ values when the original
variables were used than when the deduced factors

were used. Six-factor models produced

coefficients (R^) which ranged from 0.09 for the
standing crop of the young-of-the-year trout to
0.53 for trout in age group II. Regressions on
the original variables produced coefficients
between O.31 and 0.71 for 18 variable models and
0.26 and 0.66 for models containing six variables.

For young-of-the-year trout a single
variable, rock area, produced an R^ value (0.10)
equivalent to that obtained from the model
containing all six factors. In age group I, two
habitat variables, numbers of rocks and total
cover, produced a coefficient (0.18) as large as
the six-factor model. A coefficient similar to
that produced by the six-factor model was obtained
for trout in age group II with three variables
(R^=0.56); number of rocks, percent cover provided

182

Table 2. Order in which the habitat attributes shown in table 1 entered stepwise regressions and the
coefficient of determination they produced for each age group of trout.

Step 1 Step 2 Step 3 Step 4 Step 5 Step 6

Age group Attributes Attributes Attributes Attributes Attributes Attributes

(RM (RM (RM (RM (R') (R')

Group 0
Factors

c
0

(0.03)

11 A

(0.05)

1 h. (\
(0.07)

9 ^ U
'i, J, ^, o

(0.08)

1 o •J n
(0,09)

1 P "5 il R ft

(0.09)

Variables

3

(0. 10)

2,3
(0. 16)

2,3,12
(0.23)

3, 12. 15,27
(0.29)

2,3,12
(0.31)

2.3, 10. 12.

(0.32)

Group I
Factors

5

(0.08)

II c

4, b

(0. 12)

(0. 14)

(0. 15)

1 O O )I c

1 , 3, 't, b
(0.15)

(0. 15)

Variables

2

( C\ 111 \

I u. It ;

2, 18

1,2. 10

1,2,3, 10

1,2,3,4.12

\\J.eLo)

1,2.3.4. 10. 12

Group II
Factors

6

(0. 17)

5,6
(0.40)

4.5.6
(0.46)

2,4,5,6
(0.49)

1,2.4.5.6
(0.52)

1.2.3,4,5,6
(0.53)

Variables

2

(0.31

2,3
(0.i»4)

2, 10, 17
(0.55)

2, 10, 15, 17
(0.61)

2, 10, 12, 15, 17
(0.64)

2, 10, 12. 15.
17,18
(0.66)

Group III
Factors

6

(0.16)

5,6
(0.27)

2,5.6
(0.31)

1,2,5,6
(0.35)

1,2,4,5,6
(0.36)

1,2.3,4,5.6
(0.37)

Variables

17
(0.23)

15,17
(0.29)

12. 15, 17
(0.35)

4, 12, 15, 17
(0.38)

2,4, 12, 15, 17
(0.40)

2.4. 10. 12,
15. 17
(0.42)

by instream brush, and area in overstream cover
between 1 and 2 m; and for trout in age group III
with four variables (R^=0.34); number of pockets
of turbulent water, percent cover of side-stream
vegetation, area in overstream cover to 1 m, and
area in overstream cover to 2 m.

Factor 5 (rocks) and factor 6 (area in
overstream cover 2 m and above) and their
equivalent habitat attributes were, in most
instances, the first parameters to enter stepwise
regressions containing factors or variables.
Factor 2 (overstream cover) and factor 4 (general
cover) entered models intermediately, and factor 1
(debris) and 3 (percent overstream cover) entered
models last. Area in and percent cover provided
by side-stream vegetation consistently entered
models intermediately. Other habitat attributes
either entered models intermediately or late in
the stepwise procedure.

DISCUSSION

Factor analysis was useful in defining the
underlying structure of the cover portion of the
habitat for trout. Although 18 variables were
singled out and measured. cover consists

essentially of a six-dimensional space
characterized by six factors: debris, side-stream
cover, percent of overstream vegetation, turbulent
water. rock area. and area of overstream
vegetation in the 1- to 2-m zone.

In examining the relationships between
standing crop of fish and the habitat attributes
by stepwise regression methods, a higher
coefficient of determination (R^) was obtained by
using the 18 original variables than by using the
six derived factors. Moreover, it was found that
the best one-attribute, two-attribute, etc.,
models also resulted in higher R^ values when
original variables were used. This indicates that
when the original variable measurements are
available, there is no reason to form regression
models based on derived factors. With the
exception of 2-year-old trout, no R^ value between
trout and the set of variables exceeded 0.5,
indicating that there is a substantial portion of
variability in fish biomass that is not explained
by the measured variables and hence also not
explained by the derived factors. Part of this
variability may be accounted for by water or flow
related variables. The results of a combined
analysis of water and cover variables will be
reported at a later date.

183

LITERATURE CITED

Boussu, M.F. 1954. Relationship between trout

populations and cover on a small stream.

Journal of Wildlife Management 18:229-239.
Binns, N.A., and F.M. Eiserman. 1979.

Quantification of fluvial trout habitat in

Wyoming. Transactions of the American

Fisheries Society 108:215-228.
Hunt, R.L. 1969. Effects of habitat alteration

on production, standing crops, and yield of

brook trout in Lawrence Creek, Wisconsin.

p. 281-312. In, T.G. Northcote, editor.

Symposium on salmon and trout in streams.

H.R. MacMillan Lectures in Fisheries,

Vancouver, Canada.
Lewis, S.L. 1969. Physical factors influencing

fish populations in pools of a trout stream.

Transactions of the American Fisheries

Society 98: 14-19.
Platts, W.S. 197^*. Geomorphic and aquatic

conditions influencing salmonids and stream

classification — with application to ecosystem

classification. USDA Forest Service,

Intermountain Forest and Range Experiment

Station, Boise, Idaho.
Reid, O.K. 1961. The ecology of inland waters

and estuaries. 375 p. Reinhold, New York,

N.Y.

SAS. 1979. Users guide. 494 p. SAS Institute,
Inc., Raleigh, N.C.

Saunders, J.W., and M.W. Smith. 1962. Physical
alteration of stream habitat to improve brook
trout production. Transactions of the
American Fisheries Society 91:185-188.

Schuck, H.A. 1943. Survival, population density,
growth, and movement of the wild brown trout
in Crystal Creek. Transactions of the
American Fisheries Society 73:209-230.

Stewart, P. A. 1970. Physical factors influencing
trout density in a small stream. Ph.D.
Dissertation, Colorado State University, Fort
Collins, Colo.

Wickham, M.G. 1967. Physical microhabitat of
trout. M.S.^ Thesis, Colorado State
University, Fort Collins, Colo.

DISCUSSION

TERRY LARSON: Most papers given here which
involved multiple regression analysis used
stepwise techniques. Why did you not use an all
possible subsets technique like BMDP-9R? This
program is not costly to run and will pick up
subsets that stepwise techniques miss.

HELEN BHATTACHARYYA: The SAS STEPWISE procedure
with MAXR option was used. You are quite right
that all possible regression (SAS RSQUARE
procedure) may pick out combinations not covered
by stepwise. However, STEPWISE/MXR is almost as
good (see SAS writeup) and has the advantage of
printing all other regression statistics, slope,
intercept, mean squares, etc., besides just the R^
value .

184

New Approaches to Analysis and Interpretation

185

BIRD COMMUNITY USE OF RIPARIAN HABITATS:
THE IMPORTANCE OF TEMPORAL SCALE IN
INTERPRETING DISCRIMINANT ANALYSIS'

Jake Rice2, Robert D. Ohmart^, and Bertin Anderson^

Abstract. — Discriminant functions analyses were used to
differentiate habitats used from habitats not used by every
bird species in the lower Colorado River riparian areas,
during the breeding season. Most of these DFA's produced
statistically significant results, but the mean percent of
transects correctly classified into the species-present or
species-absent groups, across all species, was less than 75%.
Patterns of errors of classification were examined and found
to be random with regard to vegetation community, foliage
structure or avian species.

When we considered a second year's census data, for 18
of 21 species a significant number of errors in predicted
occurrences were "corrected" with the new avian distribu-
tions. The habitat DFA's were then repeated between the
group of transects used both years and those not used both
years. Across all species and all seasons, transects could
be correctly classified with an accuracy of 89%. We also
found biologically meaningful patterns of seasonal variation
1) in habitat selectivity of the avian community, 2) in
differences among vegetational communities and 3) of
transects with irregular occurrences of the avian species.

Key words: Birds; Colorado River; community ecology;
discriminant function analysis; habitat use; riparian;
species turnover.

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop. April 23-25, 1980, Burlington, Vt.

^Research Professor, Department of Zoology,
Arizona State University, Tempe , AZ 85281 and
Assistant Professor, Biology Department, Memorial
University of Newfoundland, Canada, A1B 2X8.

'Associate Professor, Department of Zoology
and the Center for Environmental Studies, Arizona
State University, Tempe, AZ 85281.

■•Research Associate, Department of Zoology
and the Center for Environmental Studies, Arizona
State University, Tempe, AZ 85281.

INTRODUCTION

During the past decade there has been a major
upswing in the use of multivariate statistics in
the study of ecology. In avian studies the uses
have been largely to quantitatively describe
community structure through various types of
ordinations (James 1971, Whitmore 1975, Conner and
Adkisson 1977), or to document niche partitioning
among groups of species (e.g., Cody 1968,
Hespenheide 1971, Whitmore 1977). This
predisposition toward each of these goals can be
understood, given current areas of emphasis of
theory in community organization and evolutionary
ecology in general. The relationships among bird

186

species, and between bird species and character-
istics of their habitats, have proven reconcilable
with theory, and have often even led to enlight-
ening extensions of our understanding of
ecological processes.

On the other hand, a large number of these
studies can be, and often have been, criticized
(if rarely in print, frequently in discussions)
for taking a very casual attitude toward several
aspects of the original data. There are a variety
of potential sources of variation that can be
reflected in ecological data, and valid, useful,
management studies require research designs that
take adequate account of these kinds of
variability. Specifically, relatively little
attention has been given to verifying that the
basic data sets on which the multivariate
techniques are used: 1) sample the true and
complete range of habitats acceptable to species
included in an area; 2) cover the within-species
variability in habitat use across communities; and
3) cover the season-to-season and year-to-year
variation in bird-vegetation relationships,
although a few papers considering some of these
points can be found (e.g.. Smith 1977, Rotenberry
1978, Rotenberry et al. 1979).

A comprehensive understanding of community
ecology will require detailed investigation of
each of those sources of variation. Additionally,
when investigations move from the theoretical
realm to the practical, all of these
considerations become even more important. If
habitat management plans are actually going to be
developed and implemented based on the results of
sophisticated quantitative studies, it is
essential that the findings truly reflect the
species-habitat relationships and are not just
statistically significant or consistent with one
of many diverse theoretical expectations.

As specific examples, the questions of which
spatial scale and which temporal scale to sample
become of paramount importance. Furthermore, the
errors that occur, for example, in the
classification step of a discriminant function
analysis, are no longer simply inconveniences or
embarrassments, but they become real problems
affecting the potential success of any habitat
management plan. This paper presents discriminant
functions investigations into the habitat use
patterns of an entire avifauna; looking
particularly at what the errors of classification
truly represent, at least in our system, and what
temporal scale is suitable for bird habitat
studies .

Riparian habitats in the desert Southwest are
rich ecological oases for many species of birds
and mammals. These areas are also subjected to
intense competing land and water use demands for
agricultural lands, municipal and industrial
purposes, river channel and flood control, and
recreational use, in addition to their value for
wildlife. For several years the Colorado River
Project has been quantifying wildlife densities
and use of all riparian habitats along the lower

Colorado River. We are currently using these data
to develop a predictive model of bird-vegetation
(and soon mammal-vegetation) relationships within
the system. This model will be used by state,
federal and private concerns in actual land use
decision making and will be a major tool in
assessing and planning habitat mitigation in these
riparian areas.

Because of the intended use of our findings
we had to know not simply how species X and Y
differed in habitat preferences in an area, or
even what major gradients we could uncover in
community structure. Rather, we had to know, out
of the complete range of riparian habitats
available, which habitat factors determined or
allowed the occurrence of each avian species. Our
answers had to be valid over the entire year,
because management decisions directed toward a
single season will nonetheless have year-round
ramifications. Correspondingly, our results had
to be valid for several years, not for just
whatever special conditions reigned for any single
year .

METHODS

It required 72 transects of 1600 m or 900 m
to census every community present in each stage of
development and in the proportion in which each
habitat type occurred along the lower Colorado
River. Each transect was then censused three
times each month using the familiar strip method
(Emlen 1971, 1977). On the basis of climatic and
demographic patterns, we divided the year into
five seasons: Spring (March-April), Summer (May-
June-July), Late Summer (August-September), Fall
(October-November) and Winter (December-January-
February). Censuses within each season were
averaged when we calculated bird species
occurrences and densities.

Every tree by species within 16 m of each
side of each transect was counted. Also foliage
density measures at several heights were taken at
50 m intervals along each transect. The
vegetation density measures were combined to
produce, for each transect, a 12 variable array
describing the foliage density at each stratum,
the relative species composition (transformed with
the ARCSIN square root transformation), and the
foliage height diversity of that transect. These
variables were all habitat attributes which we
believed were both potentially adequate to
characterize habitat use by members of the avian
community and were realistically manageable from
both the standpoints of data analysis and habitat
management.

We used discriminant functions analyses to
quantify habitat use attributes of each species.
For each species, each season, we divided the pool
of 72 transects into two groups: those where the
species in question was recorded, "Present", and
those where the species was not recorded,
"Absent". The Present and the Absent groups of
transects were then differentiated on the array of

187

habitat measures, to quantify both how different
the used and the unused areas were vegetationally ,
and what attributes of the vegetation
characterized the areas used by each species in
the community. Actually, for many species (those
with widespread distributions and widely differing
abundances on different transects), we repeated
the analyses using three or even four groups,
based on increasing densities. The accuracy of
those analyses were comparable to those of the two
group discriminations. The multigroup analyses
only introduce further complications because of
the additional possible axes of discrimination,
and they will not be discussed.

RESULTS AND DISCUSSION

Single-year Discriminations

A synopsis of results of discriminant
function analyses for the 39 species which were
present during the summer is presented in figure
1. The majority of individual species

discriminations showed that habitats were
significantly different between used and unused
areas. However, quite a number were not different
statistically; a matter for possible concern.
Some of these cases may represent strictly
statistical problems, of the sort discussed by
Williams (1981), whereas others might represent
species truly showing no vegetation differences
between used and unused sites.

Rather than go into all possible
investigations of sources of statistical artifacts
or errors, we had previously decided that results
of the classification step of the discriminant

16-1

212-
'Z

alO-
(/)

•5 8-

<.01 <.05 <.10 <.20 <.50 <.90 <,99
P_ Value From Discrimination

Figure 1. The number of species for which the
discriminant analyses of used versus unused
habitats reached specific levels of
significance. Distribution data were from a
single summer.

Table 1. Types of transect classifications
possible and their biological significance.

Status predicted from
classification step of
the discriminant analyses.

PRESENT ABSENT

Actual status PRESENT A B
of species

on transect ABSENT C D

A = Correct prediction of the species presence on
the transect

D = Correct prediction of the species absence on
the transect

B = Suitable habitat identified as unsuitable for
the species

C = Unsuitable habitat identified as suitable for
the species

function analyses would be most appropriate for
our model, and to us it represented the most
important measure of the success of the analyses.
We knew, for example, that necessary assumptions
of homogeneity of variances and covariances
between the Present and Absent groups of transects
would often be violated by species with either
very widespread or restricted occurrences in the
riparian vegetation. Such violations would affect
the statistical significance of the analyses.
However, regardless of the statistical
significance of any given discrimination, if we
were able to correctly identify areas that were
commonly used by a particular species on the basis
of vegetation attributes, we felt we would have a
useful management tool.

Statistically, the classification step of a
discriminant function analysis could produce two
different types of errors, illustrated in table 1.
From a management standpoint, the two possible
types of classification errors are quite different
in their importance, and use determines which
errors are most serious. To include some unused
areas in the used group (C errors, table 1) would
be a conservative error for habitat preservation
activities. It would lead to preservation of some
unsuitable habitats as well as all suitable ones.
The other type of error (B errors, table 1) would
be more serious; one would reject areas which
were, in fact, good habitats for the species in
question. Conversely, if one were engaged in
management activities to create or modify
habitats, the seriousness of the errors would be
reversed. To make type B errors would be to

188

develop only some of the range of habitats
suitable for a species, whereas to make type C
errors would be to spend time and resources
developing habitats which would turn out to be
unsuitable for the species.

Analysis of Errors

Figure 2 shows the distribution of the
percent of correct classifications of transects
for the single year discriminant analyses.
Although we found few gross failures in the
classification steps, there were also
correspondingly few species for which we had great
success at predicting transect suitability.
Rather than continue to use this equivocal tool
with data from other seasons, we decided to look
in detail at what sorts of errors were occurring,
in hope of isolating specific problems of the
approach. Possible sources of errors in this
study (and correspondingly, other similar studies)
included: 1) uneven variability in either the
suitability of habitats or the distribution of the
species on a scale small enough to affect our
findings; 2) inadequate habitat measures, i.e., we
had not measured plant community traits important
to avian habitat selection; and 3) low habitat
selectivity by the bird species, i.e., the used
and unused areas truly did not differ. To be
tractable, we chose 21 of 39 avian species for
detailed investigation, arbitrarily selecting the
first 21 species from an alphabetical listing of
all the species present.

We looked first at the habitat variability
problem; that is, were our discriminations poor
because some specific habitats were particularly
good or bad at supporting avian species? This is
an aspect of the question posed by several
ecologists, such Colwell and Futuyma (1971) and

9-

8-

M. 4-
e

3-

.s Z—
E

1 1-

100 95 90 85 80 75 70 65 60 55-^
Percent Correct Classification

Figure 2. The number of species for which the
classification functions of the discriminant
analyses were able to correctly identify
transects as used or not used at various
accuracy rates.

■ Observed

Numbers of Errors of Classification

Figure 3. Fit of number of errors of classifi-
cation of species suitabilities per transect to
a Poisson distribution.

Willson (197^); that is, are all equal mensural
discontinuities along a resource continuum of
equal biological importance?

We investigated this problem in a simple way.
Across the 72 transects there was a mean of 4.64
species misclassif ied per transect. If these
errors were random, we would expect the number of
transects with no species misclassif ied, with one
species misclassif ied , with two species
misclassif ied , and so on, to be distributed as a
Poisson (random pattern) with a mean of 4.64. If
specific types of habitats were either better or
worse than average in terms of their ability to
support species, relative to other habitats nearby
on the vegetation continua represented by the J;
discriminant functions, the distribution of errors I
per transect would deviate from the expected t
values. There was an excellent fit to the 'I
predicted distribution (fig. 3). From this we j
concluded that our error rate was not tied, at •
least primarily, to differential habitat j
attractiveness, beyond those differences captured j
in our vegetation measurements on each transect.

I

Initially we had intended to use data from '
subsequent years to test the effectiveness of the
model. However, looking at the distributional
data from the next year might shed light on the
degree of consistency of habitat selection of the
species in the avian community. From the
classification step, we knew for each species: 1)
the number of transects where the species was
absent, yet were classified by the analysis as
suitable habitat, and 2) the number of transects
where the species was present, yet were classified
by the analysis as unsuitable habitat. From the
occurrence data for the next summer across the
same 72 transects, we also determined: 3) the
number of transects missing each species in the
first year but supporting it in the second; and 4)
the number of transects supporting each species in
the first year but missing it in the second. All
four of these counts can be converted into
probabilities simply by dividing by 72 (the number
of transects) . If errors of classification were
true errors with no biological relationship to

189

species occurrences, the product of 1 and 3 and of
2 and 4 would give the expected number of new
species occurrences that were "corrections" of
previous type C errors and the number of new
species absences that were "corrections" of
previous type B errors.

We found a surprisingly high rate of species
turnover. New appearances occurred on 18% of all
possible species-transect combinations, and new
absences occurred in 16? of the species-transect
combinations. On a species-by-species basis, the
predicted number of independent "corrections" due
to these species turnovers was usually too low for
statistical comparison (commonly one to three
expected "correct" new transect appearances or
absences per species). However for 18 of the 21
species, the actual number of correct new
appearances was greater than the predicted number,
and for 16 of 21 species^ the observed number of
correct new absences was also greater than the
predicted number. Using a binomial test, we
determined that both of these divisions were
statistically significant. Therefore, a

significant number of what appeared to be errors
of classification based on one year's
distributional data were actually valid
predictions of future distributions of the
species .

Adding data from a second year resulted in a
trade of one problem for another. In addition to
some new occurrences being in areas previously

Table 2. Rates of species turnover per transect,
and changes which were "corrections" or new
errors of status relative to discriminant
analysis classification.

Overall mean probability of status change from:
Year 1 to Year 2

Absent Present 0.183

Present Absent 0.164

Number of observed cases > number of predicted
cases from independence of species turnover rates
and errors of classification:

Correct new presences: 18 of 21 species

Binomial "P" = 0.0012

Correct new absences: 16 of 21 species

Binomial "P" = 0.018

Number of new correct presences = 87

Number of new errors of presence = 108

Number of correct new absences = 49

Number of new errors of absence = 87

Table 3. The mean percent of transects which
were correctly classified as supporting or not
supporting each species by season:

Group

Season Present Absent Total

Summer

89.

7

84

6

86.5

Late Summer

94

8

89

1

91.3

Fall

91

1

87

1

88.5

Winter

87

7

86

7

87. 1

Spring

89.

3

91

2

90.0

Total

90

8

88

0

89.0

identified as suitable, other new occurrences of
each species were in transects previously
classified correctly as unsuitable for that
species. Absences in the second year were also
noted in areas which previously had been
classified correctly as suitable with data from
only one year. Although these new errors were
usually less frequent than expected by chance,'
they still outnumbered the "corrections" for both
kinds of changes in the status of bird species
between the two years (table 2). Clearly, the
major problem affecting our ability to define
quantitatively and precisely the range of
acceptable habitats for each species was the high
rate of species turnover from year to year on the
same transects.

Two-year Discriminations

In light of the high rate of species turnover
on the transects, for each species each season we
regrouped the transects on a new criterion;
consistency of species occurrence. Three groups
were formed: 1) transects where species X was
recorded both years, 2) transects where species X
was absent both years, and 3) transects where
species X was present in one of the years but
absent in the other. Discriminant function
analyses were then conducted between groups 1 and
2 for each species each season.

Results of these analyses showed a marked
improvement in ability to predict correctly areas
which would or would not be used by each species.
Regardless of season, the mean percent of
transects classified correctly was always high,
and the important groups of transects used
consistently were even more frequently identified
correctly (table 3). This high rate of correct
transect classification also eliminates the two
other possible sources of error in the one-year
discriminations. When investigated at an
appropriate temporal scale, the bird species

190

Summer

Late Summer

Fall

Winter

Spring

■J — L

— T 1 1 1 1 1 1 1

100 95 90 85 80 75 70 65 60
Percent of Correct Classifications

(fig. 4) through late summer, fall, and winter is
important for selecting a scale in which to
undertake habitat management investigations. In
summer nearly all species showed a fairly high
degree of habitat selectivity, but almost none
occupied areas perfectly or nearly perfectly
discriminable from unoccupied areas. At other
seasons the number of species with perfectly
discriminable habitat-use patterns made up a
substantial fraction of the community, whereas the
habitat discriminability for other avian species
in the community is markedly lower. The structure
of the community changed from one of essentially
all moderate habitat specialists in summer to a
mixture of some extreme habitat specialists and
other species showing quite weak habitat
specificity through the fall, winter, and spring.

Considerable variability was still observed
in the degree of habitat selectivity reflected by
individual species. For resident species there
was more variability between seasons in the
discriminability of habitats used by each species
(as reflected in the mean range of percent correct
classifications) than between the rest of the
community and a randomly selected species each
season (table 4). Furthermore, the criteria of
habitat selection shown by resident species, as
reflected by variables with high loadings on the
discriminant functions each season, changed as
much as did the degree of habitat selectivity.
Only 4 of 20 year-round residents had any one
variable significantly different between used and
unused areas for all five seasons (table 5).
Information of this type is obviously of great
value to persons involved in applied ecological
work, and there are even a few substantial
theoretical implications of these findings.

Figure 4. The number of species each season for
which the classification procedures were able to
correctly identify transects as used or not used
with various accuracy rates. Distributional
data were from two years.

generally did show habitat selectivity, and our
habitat measures were adequate to detect this
selectivity. The spread of the percent of species
distribution predicted correctly by season (fig.
4) supports the impressions from table 3 and
additionally underscores several points of general
ecological significance.

One point apparent from figure 4 was the
difference in the distribution of transect
identifications by species between the summer and
other seasons. The summer season had both the
lowest mean transect identif lability and a
distribution markedly more unimodal than the
distribution of the transect identifications by
species for the other seasons. This is, in
itself, a noteworthy insight into the system we
are studying.

The growing bimodality of the distribution

Remaining Errors

What of the errors of classification which
remain: that 10? of the classifications which are
still in error; and what of the distribution of
the irregular transects on the discriminant axes
of each bird species? Space does not allow an
in-depth examination of both of these points, but
we can consider the biologically most important
one: areas used consistently by a species but
nonetheless classified unsuitable for occupancy.
In a management context, making such errors would
involve the loss of habitat of high quality to the
species of concern.

We again fitted the observed number of
transects with no unpredicted species present,
with one unpredicted species present, with two,
three, and so on to the expected Poisson
distribution for each season (fig. 5). In every
season except winter the observed distribution
deviated significantly from the predicted one, and
in winter the fit was marginal. The preponderance
of transects with no errors implied that the
habitat suitability predictors, that is, the
discriminant functions, were very accurate most of
the time. Even more noteworthy, when we looked at
those transects with three or four unpredicted

191

Table 4. Measures of the amount of variability of transects correctly
classified for resident species.

Within species:
Mean maximum
range among
seasons

X = 18. 16?
s.d.= 7.25

Between species:

Mean range around a randomly selected species
each season

Summer

Late Summer Fall Winter Spring

11.58?
7.22

13.75?
5.62

n.30

8.57?
6.39

8.81?
3.89

Table 5. Criteria of habitat selection as reflected by variables significant in stepwise discriminant
analyses of present vs. absent transects for resident species all seasons.

Species

Summer

Late Summer

Fall

Winter

Spring

Verdin

(Auriparus flaviceps)

None

Cactus wren HM+,SC,HMt
(Campy lorhynchus brunneicapillus)

House finch W,HM+HMt
(Carpodacus mexicanus)

Gila woodpecker FHD,W,C
(Melanerpes uropygialis)

Common flicker W.FHD.C

(Colaptes auratus) SM+

Gambel's quail SC,HM+, W,

(Lophortyx gambelii) V15,C,HMt

Ladder-backed woodpecker FHD,V6,V15,

(Picoides scalaris) HM+

Roadrunner V5
(Geococcyx californianus)

Loggerhead shrike SC,HM+,HMt,

(Lanius ludovicianus) SMtV15

Song sparrow C.FHD.W,

(Melospiza melodia ) V15

Mockingbird HM+,HMt,SC
(Mimus polyglottos)

Ash-throated flycatcher FHD,SC,HM+,

(Myiarchus cinerascens ) O.HMt

Phainopepla HMt,HM+,SC
(Phainopepla nitens)

None
FHD.SC

C.HMt.FHD, SC

FHD,HM+,W

FHD,HM+,0

FHD

None

FHD

C.FHD

FHD.HMt.W,

C

HMt

None

HM +

None

SM+,SC

None

HMt

V5,SC

SC.FHD

None

None

V15,V6,V5,
HMt.O

None

HMt

C,SM+,SC

FHD,O.HM+, FHD,HM+,

HMt, 0
W.HM+

V15

None

None

FHD.W

FHD.C.SC.W, FHD.C.W,
HMt.SMt.O HMt

HMt

HMt

HMt,FHD,V15. None
0,V5,V6,SM+

HMt,SM+,V15 HMt.SM+

None

SC

None

FHD,HM+,C,
HMt.SC.W,
V5,TV

TV

V15,V5,
FHD.C,
SM+,SC

V5

HMt

FHD

HMt

V6,TV,C

HMt

192

Abert's towhee
(Pi'pllo aberti )

Black-tailed gnatcatcher
(Polioptila melanura)

FHD

SC.HM+.C.W,
HMt,V15,V6,
TV

None

FHD. SC. W

None

FHD.W

None

FHD

None

TV,V6,C

Rough-winged swallow W,C None

(Stelgidopteryx ruficollis)

Crissal thrasher SC,HM+,HMt FHD

(Toxostoma dorsale)

Western kingbird W None

(Tyrannus verticalis)

Mourning dove FHD,HMt,HM+ None

(Zenaida macroura)

White-crowned sparrow HM+ None

(Zonotrichia leucophrys)

None

None

FHD, W. VI 5, FHD
V5

None

HMt

None

FHD

None

V15,SC,C
FHD,V6

C,SM+,SMt

V6,TV,V5.
V15

V15.TV.HMt,
W,HM+,SC

Variable symbols:

V6 = Foliage volume 0.1 m - 0.6 m (0.5 ft - 2 ft)

V5 = Foliage volume 1 .6 m - 3. 1 m (5 ft - 10 ft)
V15 = Foliage volume 4.6 m and greater (15 ft)

TV = Total foliage volume
FHD = Foliage height diversity

HMt = Total proportion of honey mesquite (Prosopis glandulosa)

HM+ = Total proportion of honey mesquite with mistletoe (Phoradendron californicum)
SMt = Total proportion of screwbean mesquite (Prosopis pubescens)
SM+ = Total proportion of screwbean mesquite with mistletoe
SC = Total proportion of salt cedar (Tamarlx chinensis)

W = Total proportion of willow (Salix gooddingii )

C = Total proportion of cottonwood (Populus fremontii )

0 = Total proportion of mixed other species

species present, we found that certain types of
communities displayed greater than chance
frequency (table 6). Specifically, honey mesquite
and/or screwbean mesquite were often classed as
not suitable for species which did occur; i.e.,
their value to wildife is underestimated, often by
as many as four or more species. This knowledge
can be incorporated readily into habitat
management plans.

CONCLUSION: TEMPORAL SCALE

To return to one of the major questions we
initially posed, the discriminant analyses from
one year's occurrence data provided a frequently
significant but nonetheless weak ability to
predict habitat use by each species in the
riparian summer community. Merely incorporating a
second year's data on distribution improved the
accuracy of the predictive system markedly. For
ecologists interested in the habitat attributes of
species or communities, we would strongly
recommend that however the initial habitat
suitability judgments are formed, be they from
line transects, singing male perches, from spot
mapping, or whatever, a single year's data are not
adequate for the study. The habitat-use patterns

of avian species simply show too much year-to-year
variability. Furthermore, the marked changes in
community organization overall, and in habitat
selectivity and selection criteria of individual
species between seasons, indicate that a temporal
scale smaller than an entire year is also
inappropriate for a complete study of wildlife-
habitat relationships.

In terms of extending the use of multivariate
analyses, a stated purpose of this conference, we
emphasize that many of the points made here did
not come from a consideration of statistical
significance of the analyses, nor from a
consideration of the relative contributions of
various habitat measures to the discriminant
functions (the two things commonly looked at
first). Rather they came from a careful
consideration of errors made in the classification
steps and attention to the adequacy of data to
truly represent the system under study. As with
many other lines of scientific inquiry, it was
through attention to failure of the initial
approach that subsequent insights arose.

193

45-1

30-

15-

45-1

30-

15-

145-1

30-

15-

Number of Errors: Unexpected Species Presences

Figure 5. Fit of numbers of errors of
classification of species occurrences per
transect over two years to a Poisson
distribution by season.

Table 6. Habitats where suitability for avian
species is significantly often erroneously
predicted by season.

Season Suitability significantly often:
Underestimated Overestimated

Summer

Scr'^wbean mesquite

None

Late
summer

Honey mesquite and
screwbean mesquite

Salt cedar

Fall

Honey mesquite and
structure type IV

None

Winter

None

None

Spring

Honey mesquite and
screwbean mesquite
(P = 0.063)

None

ACKNOWLEDGMENTS

We thank J. Anderson for criticisms of the
manuscript. Cindy D. Zisner typed the final draft
and prepared some figures. We are grateful to the
many field biologists who helped collect the data.
Kurt Webb carried out the computer analyses. This
work was funded through Water and Power Resources
Service Grant #7-07-30-V0009 .

LITERATURE CITED

Cody, M.L. 1968. On the methods of resource
division in grassland bird communities.
American Naturalist 102:107-147.
Colwell, R.K., and D.J. Futuyma. 1971. On the
measurement of niche breadth and overlap.
Ecology 52:567-576.
Conner, R.N., and C.S. Adkisson. 1977. Principal
component analysis of woodpecker nesting
habitat. Wilson Bulletin 89:122-129.
Emlen, J.T. 1971. Population densities of birds
derived from transect counts. Auk
88:323-342.
Emlen, J.T. 1977.
bird densities
94:455-468.
Hespenheide, H.A.

Estimating breeding season
from transect counts. Auk

1971. Flycatcher habitat
selection in the eastern deciduous forest.
Auk 88:61-74.
James, F.C. 1971. Ordinations of habitat
relationships among breeding birds. Wilson
Bulletin 83:215-236.
Rice, J,.C. 1978. Ecological relationships of two
interspecif ically territorial vireos.
Ecology 59:526-538.
Rotenberry, J.T. 1978. Components of avian
diversity along a multifactorial climatic
gradient. Ecology 59:693-699.
Rotenberry, J.T., R.E. Fitzner, and W.H. Rickard.
1979. Seasonal variation in avian community
structure: Differences in mechanisms
regulating diversity. Auk 96:499-505.
Smith, K.G. 1977. Distribution of summer birds
along a forest moisture gradient in an Ozark
watershed. Ecology 58:810-819.
Whitmore, R.C. 1975. Habitat ordination of
passerine birds of the Virgin River valley,
southwestern Utah. Wilson Bulletin 87:65-74.
Whitmore, R.C. 1977. Habitat partitioning in a
community of passerine birds. Wilson
Bulletin 80:253-265.
Williams, B.K. 1981. Discriminant analysis in
wildlife research: theory and applications.
Iri Capen, D.E., editor. The use of multi-
variate statistics in studies of wildlife
habitat: Proceedings of a workshop [April
23-25, 1980, Burlington, Vt.]. USDA Forest
Service General Technical Report. Rocky
Mountain Forest and Range Experiment Station,
Fort Collins, Colo. (In press).
Willson, M.F. 1974. Avian community organization
and habitat structure. Ecology 55:1017-1029.

194

DISCUSSION

B.K. WILLIAMS: I want to compliment you on this
focus of discriminant analysis on its
classification capabilities. I would suggest that
we should generally shift our perspective on this
methodology more toward classification and away
from group mean separation.

JAKE RICE: Thanks, and I agree with your feelings
about a change in emphasis.

MARTIN RAPHAEL: Did you use equal or prior
probabilities in your analysis of classification
success?

JAKE RICE: Priors.

BOB CLARK: Once you have determined that a
species is present, can you use DFA to classify
densities (low, moderate, high) or is the system
too variable? How does your food availability
data tie into presence/absence on "predicted"
suitable habitat?

JAKE RICE: We have had limited success with
density classification; about comparable to that
of the initial one-year discriminations. As you
suggest, year-to-year variability in abundance is
the major problem. V/e are including abundance
predictions in the model being constructed with
these DFA results, but as a step after predicting
species composition for a locality. We are using
a regression approach to abundance predictions,
and not surprisingly, the confidence intervals are
large, because of the great year-to-year
variability in abundance at the same sites.

As for food availability, we have the
necessary data but the analyses are not yet far
enough to provide useful answers to the problem of
bird distributions. Not surprisingly, insect and
seed abundances are at least as variable,
seasonally and yearly, as are bird distributions.
Insects and seeds are also spatially highly
variable, and the variation is asynchronous
between sites. Tying together two such variable
systems is going to be a long, slow process.

MARK BOYCE: Since your characterization of
habitat is based solely on vegetation
characteristics, I am concerned that other
components, especially insect abundance, may vary
temporally, thus possibly invalidating your
remarks regarding generalists vs. specialists.

JAKE RICE: My comments on generalists and
specialists were meant to apply solely in terms of
habitat selection attributes. Empirically the
figure shows that in late summer, fall and winter
some species in the community have their used
habitats clearly differentiated from areas not
used, whereas other species show little habitat
differentiation. It was the species showing
little differentiation that I was calling

generalists. Much current ecological theory would
predict that these habitat generalists would be
specialists on some other criterion. EVERY
species could be (and probably is) a specialist on
some ecological attribute, but such an approach to
ecology (i.e., studying every species until one
found SOMETHING on which it specialized) would
provide only a limited insight into community
organization, overlooking as it would all
ecological attributes where a species was more
generalized as being uninteresting or
"invalidated" by the finding that it was a
specialist on something.

JAMES DUNN: Will you clarify your statement that
stepwise variable selection often resulted in
entirely different variables from one season to
another. Is this mainly because your habitat
variables are sensitive to seasonal and/or yearly
change? Or does habitat preference actually
change seasonally?

JAKE RICE: To a small extent, values of the
habitat measures do change seasonally. That is
the case only for foliage volume measures, of
course, and not tree species composition measures.
The changes in significant variables reflect, in
very large part, changing distributions of the
species by transect, and inferentially , changing
criteria of habitat selection.

JAMES DUNN: Your attack on the problem suggests
that you believe that site suitability for a
species is not a simple yes/no question, but
rather has a range of probabilities. If so, then
why not use a classification method which works by
assigning a probability for each site, e.g., the
multivariate model as proposed by S.H. V/alker and
D.B. Duncan (1967. Estimation of the probability
of an event as a function of several independent
variables. Biometrika 5^*: 167-179) .

JAKE RICE: We have been moving in precisely that
direction with our model; predicting which species
will be found in a specified area with certainty,
with high probability, often, etc. The classes
are pretty rough, but adequate for our users.
Thank you for the reference.

PAUL GEISSLER: You have suggested sampling the
same transect in several years. Resampling
transects provided very valuable information for
your study. However, for the different objective
of determining bird habitat relationships, I think
it would be advantageous to take a new sample of
routes each year to provide protection against the
effect of some unmeasured and possibly
unmeasurable habitat effect being confounded with
the effects of measured habitat variables. To put
it another way, the measurements on the same route
have correlated residuals.

JAKE RICE: I do not really see how "determining
bird habitat relationships" is a "different
objective" from what we are attempting. The major

195

point here is that, from our data (and we think
our findings are pretty general; surprisingly few
data are available on the consistency of species
occurrences and densities over several years at
same sites) a single year's censuses are not
reliable indicators of bird distributions, and
therefore also are not reliable indicators of a
species "habitat preferences" (or "optimal
situations," if you prefer). You need multiple
year's data to even get a good idea of a species'
distribution pattern. Statistical considerations
like correlated residuals are important, but come
secondary to getting a reliable measure of the

phenomenon one is trying to explain. Our point is
that one year's data will not provide a reliable
measure to predict, to discriminate or otherwise
to statistically manipulate.

E. JAMES HARNER: Just a comment. The
probabilities of misclassif ication tend to be
underestimated in discriminant analysis. Bias can
be estimated by a leave-one-out strategy or the
bootstrap method (Efron, B. 1979. Bootstrap
methods: another look at the jackknife. Annuls
of Statistics 7(l):l-26).

196

A SYNTHETIC APPROACH TO PRINCIPAL COMPONENT
ANALYSIS OF BIRD/HABITAT RELATIONSHIPS'
John T. Rotenberry^ and John A. Wiens^

Abstract. — The application of principal components
analysis (PCA) to bird/habitat relationships has essentially
followed two paths: 1) species are ordinated based on PCA of
average habitat values for individuals; 2) plots ordinated
based on their average habitat values, and species'
abundances on those plots correlated with the resulting
component axes (which presumably reflect underlying
environmental gradients) .

Our proposed synthetic method is plot-based, but
requires that sample points within plots be classified as
lying within or outside of each individual species' area of
use. As in (2), the total environmental variation, or
multidemensional "habitat space", is defined by PCA of plot
habitat values. However, rather than subjecting habitat
values for each species to an independent PCA as in (1), a
simple methodology may be used to map each species in the
habitat space described by the plot PCA.

Several advantages accrue to mapping species and plots
in the same environmental space. By graphing contours of
species densities in this multidimensional space, patterns of
abundance/habitat relationships that are not apparent from
simple correlational analysis may emerge. Comparisons of
plot means with values for individual species within a plot
may reveal active habitat selection, or even consistent
patterns of within-plot habitat partitioning between two
species. Comparisons of density contours may suggest the
presence of biological interactions, such as competition or
ecological replacement, between two or more species.

The use of this technique is illustrated by analysis of
22 structural habitat variables collected at 26 North
American grassland and shrubsteppe sites.

Key words: Birds; density contours; gradient analysis;
principal components analysis; shrubsteppe; vegetation
structure .

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Research Associate, Shrubsteppe Habitat
Investigation Team, Department of Biology,
University of New Mexico, Albuquerque, NM 87131.

Present address: Assistant Professor, Department
of Biological Sciences, Bowling Green State
University, Bowling Green, OH 43403.

^Professor, Shrubsteppe Habitat Investigation
Team, Department of Biology, University of New
Mexico, Albuquerque, NM 87131.

197

INTRODUCTION

The use of principal components analysis
(PCA) in describing patterns of avian habitat
occupancy has been most fruitful, both for
consideration of the adaptations of individual
species (e.g., James 1971) and analysis of
community composition (Rotenberry and Wiens 1980).
Indeed, PCA is now routinely performed as a matter
of course on a wide variety of habitat parameters,
not only for birds, but also for organisms
spanning a large taxonomic range [e.g., Miracle
1974 (plankton), Johnson 1977a, b (bog plants)].

The application of PCA to bird/habitat
relationships has essentially followed two paths
that differ, to a certain degree, in their
conceptual orientations. We propose a different,
somewhat synthetic approach to PCA of avian
habitats that comBines the virtues of both
previous approaches, yet retains a relatively
simple and straightforward methodology. Results
from the methodology are compatible with the
concept of gradient analysis (V/hittaker 1967,
Terborgh 1971), and can be further interpreted in
such a context.

RATIONALE AND METHODS

Previous Approaches

Bird Mean Habitat Vector

The first approach to bird habitat analysis
can be called the bird mean habitat vector method
(James 1971, Anderson and Shugart 1971, Whitmore
1975). In this method a sampling point is usually
determined by the presence of a singing male bird.
Once a point is located, a variety of habitat
variables, generally assumed to be of ecological
relevance, are measured in the immediate vicinity.
Samples are taken for a number of different
species at a number of different points, the
average value of each variable is calculated for
each species, and this set of averages then
determines the mean habitat vector. These vectors
are combined into Y, the matrix of standardized
variables for all species, and this matrix is
analyzed using standard principal component
techniques (e.g., Barr et al . 1976). Resulting
orthogonal components are interpreted in light of
their factor loadings (the correlation between new
components and original variables). Habitat
relationships among species are then reconstructed
by plotting the location of each species in the
newly defined component space, using their factor
scores as coordinates. These scores for bird
species are given by

(1 )

where R is the correlation matrix of the original
variables, and S is the factor structure matrix
containing factor loadings (notation based on that
of Thorndike 1978). The subscript b denotes bird
species data.

While such a technique clearly has
considerable heuristic value (as, for example, in
describing relative positions of species in
multivariate habitat space), it also has two
potential shortcomings: 1) by focusing mainly on
the simple presence of a species, the advantages
of a community approach are sacrificed, and
observations concerning species diversity,
resource partitioning, or relative widths of
ecological habitat niches are not possible; and 2)
because data are collected only on the basis of a
species' presence, there is no informataion
concerning variation in the species' numerical
abundance as habitat changes.

Site Mean Habitat Vector

The usual alternative approach is in fact
community oriented, although not without drawbacks
of its own (e.g., Cody 1975, Rotenberry and Wiens
1980). The method begins with selection of a
large series of plots or transects, generally
chosen to be representative of variation in either
some set of habitat variables or some set of bird
species. Habitat variables of interest are
measured at a number of randomly selected points
on each plot and their average taken to yield what
may now be called the mean habitat vector for the
site. All plots are combined in Z, the set of
standardized variables for plots, and PCA is
performed. The factor structure matrix again
provides interpretation of the components and
sites are ordered in this multidimensional space
by their factor scores, given by

F = S '
-s — s

R

— s

-1

(2)

where S and ^ are the factor structure matrix and
correlation matrix for site habitat data (denoted
by the s subscript) .

Because these are plot-based samples, each
site also has associated with it species
abundances, diversities, or any other attribute of
the avian community one cares to calculate, and
these can then be correlated with site factor
scores. Significant individual correlations are
generally interpreted as representing species'
responses to the multidimensional habitat spectrum
represented by the site ordination. If species
diversity seems to vary in some meaningful manner
after the sites have been ordered, then one can
speculate about the nature of community
organization along the gradient. These sorts of
interpretations are not possible under the bird
mean habitat vector method, but unfortunately
their gain is offset by loss of some information
about individual species. Because a species
cannot be considered separately from a site, the
point for that site represents all species
simultaneously. If there is any within-plot
habitat selection or partitioning by species, for
example, this will be completely obscured, and any
generalizations that one might want to make about
species' relationships could be compromised.

198

Synthetic Approach

The technique we propose combines what we
think are useful elements of both site and species
ordinations. Although the methodology is largely
based on site-oriented sampling, slight
modifications of traditional methodology yields
data that are compatible with the species-oriented
approach as well.

As in the site-oriented approach, a series of
sites are selected that encompass some
environmental range in which one is interested,
and attributes of both habitat and bird
populations on these sites are measured. In the
course of estimating bird densities, it is
generally possible to estimate microhabitat or
within-plot use by the individuals of a species as
well. If, for example, individuals are
territorial, this becomes a simple exercise in
mapping territorial boundaries. Superimposed upon
this are locations of the random points at which
habitat variables are measured. These points may
be characterized as lying within or outside of the
area used by a species (e.g., Wiens 1959), and
those that lie within the use-area can be used to
create a mean habitat vector for that species at
that site. These vectors will likely differ for
different species at the same site (depending on
the degree of within-plot spatial or habitat
overlap between them), or for the same species at
different sites. To the extent that a species is
nonrandomly selecting habitat within the plot, its
vector will differ from that of the plot as a
whole. The plot or site mean vector, of course,
is determined by all points.

The multidimensional environmental space in
which all samples have been taken is defined by a
PCA of the matrix of standardized variables for
the sites, Z^, and as such does not differ from
that presented above (equation 2). PCA is still
picking out the major patterns of covariation in
habitat variables that are latent within our
overall selection of plots. One elects, of
course, to concentrate on those components that
are relatively strong (as evidenced by the
relative magnitude of their eigenvalues) and
meaningful, in that the patterns of their factor
loadings appear to make some sort of ecological
sense. Because habitat variables are in fact
environmental measures, the components can be
interpreted as representing real-world ecological
gradients, and one can begin to apply concepts of
gradient analysis to the sites, and now to the
species as well.

The next step is to plot species along the
same gradients. To do this, new factor scores are
calculated using the factor structure matrix and
correlation matrix from the habitat variables at
sites, and the matrix of standardized variables
from birds. Thus,

where all matrices are as in equations (1) and
(2). Resulting factor scores thus map the habitat

selection of birds (Y, ) onto the multidimensional

—V

environmental gradients defined by sites (S^ and

R^). In other words, sites are used regardless of
— s

their species composition to determine the habitat
patterns, then the distribution of the birds is
plotted along these patterns, regardless of the
sites on which they occur.

There are several advantages to this
methodology. First is that as data are collectd
from relatively large plots or transects, the
community-oriented attributes that are lacking in
the species-specific approach are retained, as
well as estimates of individual species
abundances. At the same time, however, single
species may be ordinated or otherwise related to
one another on the basis of species-specific, not
site-specific, responses to the multivariate
habitat gradient. The second major advantage is
that although species at a site are analyzed in
the same environmental space as the site, they
need not be equated with that site's mean values.
This means, for example, that within-plot habitat
selection will not be obscured (fig. 1). Under
normal site-based analysis, the point that
represents the plot also represents, in this case,
three species. By this synthetic method, however,
if a species occupies a distinctly different
subset of habitat than the average for a site on
which it occurs, such will be readily apparent by
the degree of departure of the species' position
from the site's position in the PCA-space (fig.
1). Because the point for the site represents all
species, any sort of within-plot habitat
partitioning between them that may be occurring
will be obscured using the site mean habitat
vector approach. Such partitioning may be
detected under the synthetic methodology, however,
if there are consistent patterns of displacement
of two species that otherwise co-occur at a number
of sites (e.g., species 1 and species 2 in fig.
1).

Perhaps one of the most interesting
properties of site-oriented analyses arises from
the fact that each site-specific point for a
species can be associated with that species'
density at the site as well. By plotting these
densities on the derived component axes, one can
begin to build a picture of the quantitative
distribution of species with respect to
environmental gradients represented by these axes.
If there are a sufficient number of sample points,
contours of a species' abundance patterns may be
plotted as well. The contours shown in figure 2
present a considerably stylized example, but they
can be used to demonstrate some patterns that
might arise from an analysis of this sort. This
set of contours, for example, shows this species'
numerical response to the derived habitat
gradients, a response that is unlikely to be
detected through any correlational analysis
because of its intrinsic nonlinearity . Further,
projection of contours onto each of the axes
separately indicates that this species is rather

199

O^Sp I

^OSpZ

OSpl
Sp 3 0-»

^Sp3

>Sp2

PC I

-OSp 2

b Sp2

O Species
• Sites

Figure 1. Hypothetical species and sites plotted
in environmental space defined by the first two
principal components (PCI and PCII) of
site-based environmental variables. Lines
connect species to sites on which they occurred.

Figure 2. Hypothetical contours of species
abundance patterns plotted in environmental
space defined by the first two principal
components (PCI and PCII) of site-based
environmental measures. Contours represent
isopleths of density (individuals/km^).

Figure 3. Hypothetical contours of species abundance patterns. Axes and isopleths as in figure 2. Arrows
denote change in site characteristics as a result of habitat alteration. Changes in site characteristics
may effect the following changes in a species' abundance at the site: A-increase; B-decrease; C-local
invasion; D-local extinction.

200

generalized with respect to its distribution along
these gradients. Although still maintaining the
same basic configuration of contours, a different
species, on the other hand, might occupy a
relatively wide range on one axis but a much
narrower range on the other — we can thus identify
what may be called axes of specialization and
generalization.

Perhaps the most interesting aspect of these
contours is their potential in habitat management
predictions. Insofar as one has some idea of how
an environmental alteration will affect a site's
location along each of the gradients — that is to
say, to the degree that one can predict how a site
will "move" in multidimensional habitat space
after some sort of treatment — then one should be
able to estimate the effect of that treatment on a
species' population. For example, any sort of
alteration that caused a plot to move in habitat
space as indicated in figure 3A would likely
result in an increase in abundance of this
hypothetical species, while a different change
(fig. 3B) may be much more likely to result in a
decrease. We might even be able to predict
changes in habiat that would lead to invasion of a
species into the area (fig. 3C) , assuming, of
course, that it were biogeographically feasible
for it to do so. Perhaps most important from a
management standpoint, one might be able to define
a habitat alteration that would lead to a species'
local extinction (fig. 3D).

The sorts of contours depicted in figure 3
are, of course, stylized, and more often than not
those derived from sets of real data are likely to
deviate markedly from such smooth patterns.
Certainly one of the biggest contributors to an
uneven species distribution will be uneven
sampling intensity with respect to the derived
gradients; unfortunately such omissions are
apparent only after the fact and are thus
difficult to control. Other reasons, however, are
more biological in nature. For example, one may
sample at the periphery of habitat that is
suitable for a species and thus may map only a
portion of its contours (fig. 4A) . Alternatively,
a species may be distributed in a non-Gaussian
fashion along both habitat gradients (cf. Colwell
and Futuyma 1971), which would yield a pattern
similar to that of figure 4B. The most extreme
expression of this type of response would be a
species' recognition of an environmental
discontinuity or ecotone on what are otherwise
statistically continuous gradients. Such an
ecotonal response would likely be evidenced by a
very rapid change in a species' abundance over an
apparently short environmental distance (fig. 4C) .
On the other hand, such a pattern might also
correspond with sharp abutment of one species upon
another (fig. MD) . Although such a pattern may
still reflect an ecotonal response, some sort of
biological interaction between the two species,
such as competition, becomes more likely. This
pattern also identifies habitat in which
investigations of such interactions should be
conducted. Experiments or other comparative
analyses conducted in such areas identified by

Figure 4. Hypothetical deviation from the
"normal" species contours of figures 2 and 3-
Axes and isopleths as in figure 2. A-Species
peripheral to habitat samples. B-Non-Gaussian
distribution along both environmental gradients.
C-Species perceives and responds to "ecotone" on
otherwise statistically continuous gradient.
D-Competitive interaction between species
results in habitat displacement.

this technique are likely to be more rewarding
than those conducted in locales randomly chosen
throughout the species' ranges.

If one is inclined to treat the derived
gradients as representing habitat niche axes of
one sort or another [and there seems to be ample
reason to do this for birds (Rotenberry, 1981)],
then there arises yet another set of calculations
that can be made. Niche width, for example, can
be estimated from some measure of dispersion
around the species' centroid in multivariate
space. Niche overlap between species may also be
calculated, incorporating both distances between
two centroids and changes in the species' contours
as well. While we offer no speculation on the
specific form such breadth and overlap statistics
might take, the distributions of species in this
environmental space will likely, we think, provide
a fertile field for the application of many
traditional (and even nontraditional) niche metric
manipulations.

AN EXAMPLE: NORTH AMERICAN STEPPE AVIFAUNA

We would now like to provide a brief demon-
stration of the application of this methodology
and some of its concepts to a real set of data,
although unfortunately not one originally
collected explicitly for this purpose. The data
come from a collection of 26 grassland and
shrubsteppe sites scattered throughout middle and
western North America, representing a wide array
of steppe vegetation types (fig. 5). Sampling
methodology, site locations, species lists and
abundances, descriptions of variables, etc. are
fully detailed elsewhere (Rotenberry and Wiens
1980). At each plot we set up a 10-ha grid,
mapped the territories of all breeding birds, and
measured habitat variables. In this particular

201

STEPPE TYPE

Figure 5. Location of avian censuses used in this
analysis. Numerals keyed to table 1 in
Rotenberry and Wiens (1980). Steppe types
generalized from Kuchler (1964).

study we were interested in the role of spatial
heterogeneity, or patchiness in vegetaion
structure, in determining distribution and
abundance of grassland birds and the structure of
their communities. The 22 variables that were
measured fell into two basic categories: coverage
variables (simply the percent coverage of various
physiognomic classes, such as shrubs, grasses,
forbs, or bare ground) and structural variables.
Structural variables are also physiognomic in
nature, but have the additional property of
"dimension"; that is to say, variation in a
structural measure is generally associated with
variation in either a horizontal or a vertical
plane. Ultimately 10 horizontal heterogeneity
indices, 5 vertical indices, and 7 coverage
classes were considered. The results, given here
in the most basic form (table 1), were somewhat
surprising in that a very distinct separation
between the vertical and horizontal indices were
observed; 9 of 10 horizontal indices loaded high
on the first axis (41? of total variation), while
all 5 vertical indices loaded high on the second
(22% variation). Increasing horizontal

heterogeneity was associated with increasing
coverage of shrubs and bare ground, and decreasing
coverage of grass and litter. Vertical
heterogeneity varied slightly with changing forb
coverage. Together, these two axes accounted for
almost two-thirds of the total variation. We thus
interpreted these components as representing two
largely independent gradients in vegetation
structural heterogeneity — a major axis of
horizontal patchiness and a minor axis in vertical
patchiness .

The sites tended to sort themselves into
broad classes along these axes in horizontal and
vertical heterogeneity (fig. 6A). Tallgrass sites
evidenced substantial vertical heterogeneity, but
relatively little patchiness in a horizontal
plane. Shrubsteppe sites generally showed as much

Table 1. Factor loadings of vegetation principal
components. Only those loadings greater than
0.40 (analogous to a correlation significant at
the 0.05 probability level) are shown; only the
first two of five factors with eigenvalues
greater than 1 are shown. Complete details of
the analysis, including explanations of
variables, are given by Rotenberry and Wiens
(1980) .

Vegetation
variable

Factor: I

Eigenvalue: 9.02

% a: 41 .0

I % a: 41.0

Structural-
horizontal

HIT-10

LITDEP

CVTOTHIT

CVMAXHGT

CVLITDEP

CVTOTHGT

CVHGTDIF

HITS-HI

LIT-HI

DIST-HI

Structural-
vertical

TOTHITS

MAXHGT

EFFHGT

TOTHGT

HGT-HI

Coverage

Grass

Forb

Shrub

Bare

Litter

0.H6
-0.71
0.83
0.84
0.80
0.50
0.68
0.93
0.69

0.56

-0.87

0.80
0.92

-0.82

II

4.94

22.4
63.4

0.44
0.52

0.73

0.56
0.77
0.86
0.88
0.70

0.40

202

B

■ TallgroK
O ShsrlgroM
• Snrubilapi

O Shofiaroii
• Shrubstepp«

SPECIES

•

»
■

■
■

■

•

•

PCI

> •

•

o

0 i

•

o

o

o

o

o

Figure 6. Distribution of steppe bird census sites (A) and bird species means (B) in environmental space
defined by the first two principal components (PCI and PCII) of 22 site-based habitat structure variables.
PCI represents increasing horizontal heterogeneity; PCII represents increasing vertical heterogeneity.

vertical heterogeneity as the tallgrass ones, but
differed markedly in being much patchier
horizontally, Shortgrass sites were intermediate
in horizontal heterogeneity but, as one might
expect, showed very little vertical variability.
Mixed-grass sites were the most varied in their
structure, with some positioned intermediate to
shortgrass and tallgrass plots, while another more
closely resembled shrubsteppe sites in structure.
Montane sites were intermediate as well.

Figure 6B shows bird species plotted in the
same space; each point represents the average of a
species' factor scores for all of its occurrences
combined without regard to its abundance. Not
surprisingly, species that are identifiable as
being largely tallgrass, shortgrass, or
shrubsteppe species are generally found in the

■ Tolgroat
• Shrutntepp.

Figure 7. Distribution of tallgrass and
shrubsteppe bird census sites and individual
bird species from those sites. Axes as in
figure 6. EML = eastern meadowlark, DCK =
dickcissel, GRS = grasshopper sparrow, UPP =
upland plover, WML = western meadowlark, HLK =
horned lark, SGS = sage sparrow, MDV = mourning
dove, RKW = rock wren, VSP = vesper sparrow, SGT
= sage thrasher, BRS = Brewer's sparrow, LGS =
loggerhead shrike.

same areas of PCA-space as their sites. Of more
interest, of course, are some details that go into
generating these means. For example, figure 7
reprsents both sites and birds for four tallgrass
and four shrubsteppe samples. (Scientific names
of all species are given in appendix I.) It is
apparent that a number of shrubsteppe species are
not very close to their site means, and indeed
some sites appear to have a great deal of
dispersion around them. Although tallgrass
species did not appear as dispersed, dickcissels
and grasshopper sparrows did demonstrate a
consistent pattern of within-plot partitioning,
with the latter found in vegetation of less
vertical heterogeneity but greater horizontal
patchiness. The co-occurring eastern meadowlark,
however, seemed to be positioned independently.

Contour intervals for the species show a wide
variety of patterns (figs. 8-15). Our collection
of sites appears to have almost exactly centered
on the habitat distribution of western meadowlarks
(fig. 8), and this pattern, in retrospect, is
perfectly consistent with most of what is known
about the behavior, habitat selection, and

a.

WESTERN UEAOOWLARK

^0 ^ —

— 25 // \ \

-50^ / 1

PC I // /

Figure 8. Distributional pattern of western
meadowlarks. Axes as in figure 6. Isopleths as
individuals/km^ .

203

Figure 9. Distributional pattern of dickcissels,
sage sparrows, and McCown's longspurs. Axes and
isopleths as in figure 8.

geographical distribution of this species (Lanyon
1956, and personal observation).

It also appears that the range of habitats
sample was only peripherally used by several
species. For example, the dickcissel (a tallgrass
prairie species), sage sparrow (a shrubsteppe
bird), and McCown's longspur (abundant in
shortgrass) all seem to have centers of
distribution that lie at the edge or even outside
of the limits of habitat gradients defined here
(fig. 9). Presumably, then, if data were
collected in more extreme shrubsteppe vegetation,
we would be able to define the limits of sage
sparrow distribution more accurately. The most
extreme example of peripheral species are those
that occurred on just one of the 25 plots we
sampled (fig. 10). If a line is graphed that
encloses all the site mean values for these
gradients, it is apparent that these species all
lie beyond that boundary, indicating that even at
those sites on which they did occur they were
occupying peripheral habitat. It likewise is
assumed here that if sampling effort were
increased away from this boundary, more sites at
which these species occur would be encountered.

HOflNED LARK

Figure 11. Distributional pattern of horned
larks. Axes and isopleths as in figure 8.

O Sriorigroii

siNGLE-sire

SPECIES

— -

• LGS

■ hn:
■ BC

a

/

PC t

/ • RKW

• MDV

OCCL~ .
OMTF

" ONHK - - _

Figure 10. Distributional pattern of species that
occurred at only a single census site. Species
associated with montaine meadows have been
omitted. Axes as in figure 6. Dashed line
encloses all site mean values (figure 6A). HNS
= Henslow's sparrow, BOB = bobolink, CGL =
chestnut-collared longspur, MTP = mountain
plover, NHK = common nighthawk, MDV = mourning
dove, RKW = rock wren, LGS = loggerhead shrike.

Horned larks, present in both shrubsteppe and
shortgrass habitats, apparently increase in
abundance as the habitat becomes more uniform in
both dimensions (fig. 11). This is consistent
with our own observations of the species, which
suggest that the lark's ultimate idea of habitat
nirvana is the uniform monotony of a paved parking
lot.

Some patterns of contours are more difficult
to interpret. Brewer's sparrows, for example,
evidenced a rather discontinuous distribution
(fig. 12). While we have searched for another
species that might fit in the gap, none of the
species enounterd in our censuses througout the
North American steppe vegetation showed a
reciprocal distribution in this area. At this
time we can only surmise that there may be some
sort of biogeographical constraint to Brewer's
sparrows filling in the vacancy, or that perhaps

BREWER'S SPARROW

PC I

0 50 '00 150

Figure 12. Distributional pattern of Brewer's
sparrows. Axes and isopleths as in figure 8.

204

Figure 13. Distributional pattern of vesper
sparrows, dickcissels, and lark buntings. Axes
and isopleths as in figure 8.

populations in the two rather disjunct habitat
types represent different ecological races or even
incipient subspecies.

The precise habitat affinities of vesper
sparrows have defied ready generalizations, and
their distribution along these synthetic habitat
gradients (fig. 13), when considered by itself,
adds little to a more precise definition. It is
interesting, however, to plot the vesper sparrow
simultaneously with the dickcissel (a tallgrass
species) and the lark bunting (a shortgrass
species) (fig. 13). While one cannot conclude
from this analysis what mechanism might ultimately
be responsible for these observed distributional
patterns, they certainly appear to be nonrandom.

One may also examine situations where
nonrandom patterns of distribution might be
predicted a priori , such as might result from
competitive interactions. Eastern and western
meadowlarks, for example, are congeneric and are
thought to express a number of competition-induced

EJISTEfIN MEADOWLAKK

Figure 11. Distributional pattern of eastern and
western meadowlarks. Axes and isopleths as in
figure 8.

relationships. Their contours, while largely
exclusive, do show some overlap (fig. 14).
Although no really clear pattern is present in
terms of a sharp abutment or exclusivity, we could
predict that sites that fall within the particular
area where density overlap is greatest are where
interspecific territoriality is likely to be most
intense. Although dickcissels and grasshopper
sparrows appeared to partition individual
tallgrass plots (fig. 7), their pattern of
contours (fig. 15) suggests little partitioning on
a regional scale and instead seems indicative of
independent differential habitat selection.

Finally, one can examine the predicted effect
of some major alteration in some part of the
steppe environment; say, for example, a decision
were made to mow a tallgrass prairie? One expects
a tremendous drop in vertical heterogeneity as the
grasses are all cut flat, with perhaps a slight
increase in horizontal patchiness as new areas of
bare ground open up following removal of forbs and
small shrubs. By superimposing this change upon
the species' contours, changes in species
abundances can be predicted (fig. 16). Clearly
the tallgrass species such as dickcissels (fig.
16A) and eastern meadowlarks (fig. 16B) are likely
to disappear altogether, while species abundant in
habitat with low vertical heterogeneity, like
horned larks (fig. 16C) or lark buntings (fig.
16D) , will probably increase considerably if there
are no biogeographical constraints to their
response to this habitat alteration. Grasshopper
sparrow densities may change (fig. 16E), but it
seems unlikely that the species would disappear
altogether. In the case of the western
meadowlark, however, predictions could be
equivocal (fig. 16F); this will depend both upon
accuracy in estimating how an altered site will
"move" in habitat space and upon proximity of the
new site position to contours representing an area
of fairly rapid change in a species' abundance.

Figure 15. Distributional pattern of dickcissels
and grasshopper sparrows. Axes and isopleths as
in figure 8.

205

CONCLUSIONS

We hope to have made it clear that, even with
a set of data that was not taken with the
methodology in mind, our technique of combining
the individual species approach with one that is
plot-oriented is feasible; indeed, there may be
sets of existing bird/habitat data to which it may
be applied. Analysis certainly need not be
confined to physiognomic variables as we have done
here, but can be extended to whatever environ-
mental or habitat variation one thinks is
important in affecting species distributions and
relationships. Although certain biological
objections can be raised regarding application of
PCA to either species- or plot-oriented data
(Johnson 1981), we believe this new technique,
combined with judicious selection of variables,
appropriate sample sizes, and other attention to
statistical niceties, may ameliorate these
objections substantially. One must bear in mind
that the technique of PCA is descriptive rather
than inferential and that it basicially provides
us a sophisticated "multivariate natural history".
We think that such descriptions will ultimately
prove robust with respect to what is already knoim
about the particular set of species described and
that they will be capable of generating new
insights for those that employ them.

ACKNOWLEDGMENTS

Field data were gathered during studies
supported by National Science Foundation grants
GB-6606 to J. A. Wiens and 08-7824, GB-13096, GB
31862X, GB-31862, and DEB73-02027 A03 to the
Grassland Biome, United States International

Biological Program. The final stages of the
research were supported by NSF grant BMS 75-11898.
Data analysis was supported by the University of
New Mexico Computer Center.

LITERATURE CITED

Anderson, S.H., and H.H. Shugart. 1974. Habitat
selection of breeding birds in an east
Tennessee deciduous forest. Ecology
55:828-837.

Barr, A.J., J.H. Goodnight, J. P. Sail, and J.T.
Helwig. 1976. A user's guide to SAS76. SAS
Institute, Inc., Raleigh, N.C.

Cody, M.L. 1975. Towards a theory of continental
species diversities. p. 214-157 In Cody,
M.L., and J.M. Diamond, editors. Ecology and
evaluation of communities. Belknap Press,
Cambridge, Mass.

Colwell, R.K., and D.J. Futuyma. 1971. On the
measurement of niche breadth and overlap.
Ecology 52:567-576.

James, F.C. 1971. Ordinations of habitat
relations among breeding birds. Wilson
Bulletin 83:215-236.

Johnson, D.H. 1981. How to measure habitat — a
statistical perspective. Ijn Capen, D.E.,
editor. The use of multivariate statistics
in studies of wildlife habitat: Proceedings
of a workshop [Burlington, Vt . , April 23-25,
1980]. USDA Forest Service General Technical
Report. Rocky Mountain Forest and Range
Experiment Station, Fort Collins, Colo. (In
press) .

Johnson, E.A. 1977a. A multivariate analysis of
the niches of plant populations in raised
bogs. I. Niche dimensions. Canadian
Journal of Botany 55:1201-1210.

206

Johnson, E.A. 1977b. A multivariate analysis of
the niches of plant populations in raised
bogs. II. Niche width and overlap.
Canadian Journal of Botany 55:1211-1220.

Kiichler, A.W. 1964. Potential natural vegetation
of the conterminous United States. American
Geographical Society Special Publication No.
36, New York, N.Y.

Lanyon, W.E. 1956. Ecological aspects of the
sympatric distribution of meadowlarks in the
north-central states. Ecology 37:98-108.

Miracle, M.R. 1974. Niche structure in
zooplankton: a principal components
approach. Ecology 55:1306-1316.

Rotenberry, J.T. 1981. Why measure bird habitat?
In Capen, D.E., editor. The use of multi-
variate statistics in studies of wildlife
habitat: Proceedings of a workshop
[Burlington, Vt., April 23-25, 1980]. USDA
Forest Service General Technical Report.
Rocky Mountain Forest and Range Experiment
Station, Fort Collins, Colo. (In press).

Rotenberry, J.T., and J. A. Wiens. 1980. Habitat
structure, patchiness, and avian communities
in North American steppe vegetation: a
multivariate analysis. Ecology 61:1228-1250.

Terborgh, J. 1971. Distribution on environmental
gradients: theory and preliminary

interpretation of distributional patterns in
the avifauna of the Cordillera Vilcabamba,
Peru. Ecology 52:23-40.

Thorndike, R.M. 1978. Correlation procedures for
research. 340 p. Gardner Press, New York,
N.Y.

Whitmore, R.C. 1975. Habitat ordination of
passerine birds in the Virgin River valley,
southwestern Utah. Wilson Bulletin 87:65-74.

Whittaker, R.M. 1967. Gradient analysis of
vegetation. Biological Review 42:207-264.

Wiens, J. A. 1969. An approach to the study of
ecological relationships among grassland
birds. 93 p. Ornithological Monographs 8.

Appendix I. Scientific names of all birds
mentioned in text or figures.

Mountain plover
Upland plover
Mourning dove
Common nighthawk
Horned lark
Rock wren
Sage thrasher
Loggerhead shrike
Bobolink

Eastern meadowlark

Eupoda montana
Bartramia longicauda
Zenaida macroura
Chordelis minor
Eremophila alpestris
Salpinctes obsoletus
Oreoscoptes montanus
Lanius ludovicianus
Dolichonyx oryzlvorus
Sturnella magna

Western meadowlark

Dickci ssel

Grasshopper sparrow

Henslow's sparrow

Lark bunting

Vesper sparrow

Sage sparrow

Brewer's sparrow

McCown's longspur

Chestnut-collared
longspur

Sturnella neglecta
Spiza americana
Ammodramus savannarum
Passerherbulus henslowii
Calamospiza melanocorys
Pooecetes gramineus
Amphispiza belli
Spizella brewer!
Calcarlus mccowni

Calcarius ornatus

DISCUSSION

BOB WHITMORE: How did you compute the density
contours? The smooth curves that you presented

Q

would seem to indicate many OlO ) data points.
How many points per species did you have?

JOHN ROTENBERRY: Not nearly as many as we would
have liked, especially for species whose
distributions take them off the edges of the
habitat space. The curves were initially drawn
using strict linear interpolation between points;
I subsequently smoothed them by eye for this
presentation. To a considerable extent it
represents artistic license tempered by the
interpolated realities and biological intuition.

BOB WHITMORE: You had some species with
"peripheral ranges" on your ordinations. How do
you know that the observed distributions are not
just an artifact of the variables being measured?

JOHN ROTENBERRY: We cannot know for sure, but it
seems likely due to biogeographical considerations
and the distribution of our sites.

E. JAMES HARNER: How did you scale the variables
in your PCA analysis? Did you only use the
correlation matrix?

JOHN ROTENBERRY: The variables were all
normalized using either log or arcsin square root
transformation. I only used the correlation
matrix for PCA, which is analogous to using the
covariance matrix of standardized data. The
component axes were also rotated using varimax.

BOB WHITMORE: Given the overlap problems when
using centroid distances, why not use a
multivariate measure of niche overlap?

207

JOHN ROTENBERRY: I think that would be the best
approach.

JAMES DUNN: How the species counts balance each
other as a function of site properties is an
interesting question. Does your analysis reflect
this? Why not try the GSK model as implemented by
PROC FUNCAT(SAS)? One nice feature of the model
is that it would allow formal tests of the effects
of habitat variation on relative frequencies of
the species. In particular, the total bird count/
plot could be used as a predictor to see if site
productivity affects the relative composition of
the species. The only difficulty I see with the
FUNCAT is zero counts which you may have.
Potentially a maximum likelihood solution could
handle that using "working values."

JOHN ROTENBERRY: In the paper where the data were
originally described, we did test correlations of
species diversity, richness, and evenness with
each of the component axes. Diversity did
significantly increase with incoming vertical
heterogeneity, but varied independently of
horizontal heterogeneity. Richness and evenness
were not significantly correlated with either
axes. With respect to examining relative
frequencies, I think it might be worthwhile, but
we are, in fact, plagued with zero counts
throughout the bird data set. I'm not familiar
with FUNCAT, so I don't know how the use of
"working values" will affect the outcome.

CHARLES SMITH: When making measurements in the
field, how does one differentiate between the
extremes of the "niche-habitat continuum," that
is, are habitat variables distinguished from niche
variables when measurements are taken?

JOHN ROTENBERRY: I suspect that in general we are
not dealing with the extremes, but that instead
most of the variables we chose to measure
reflected both aspects. My feeling is that the
synthetic component axes do represent habitat
gradients, and that (for birds at least) habitat
is a niche dimension, for the reasons I suggested
earlier in this workshop.

PAUL GEISSLER: Looking at your figures, I noted
that many of the frequency ellipses were oriented
at about 45° to the axis. Perhaps the
interpretability of the axes for the birds might
be improved if the axes were rotated parallel to
the orientation of the ellipses.

JOHN ROTENBERRY: There are two problems with
doing that. First, if we rotate the axes with
respect to the birds, we destroy the relationships
among the variables and change the interpretation
of the axes in some unknown way. This would not
be the same thing as varimax or orthomax rotation.
Second, the ellipses clo represent the relations of
the bird species to the derived axes. If a
species shows a response to both horizontal and
vertical heterogeneity, we expect frequency
ellipses to be oriented just this way.

208

ROBUST PRINCIPAL COMPONENT AND DISCRIMINANT
ANALYSIS OF TWO GRASSLAND BIRD SPECIES' HABITAT'
E. James Harner^ and Robert C. Whitmore^

Abstract. — Outliers in multivariate data can have
pronounced effects on the interpretations and conclusions of
statistical analyses. However, data is rarely examined for
outliers, due in part to the difficulty in discovering them
if the dimensionality of the variable space is greater than
two or three. We also do not believe that researchers are
aware of the severity of the problem that outliers create.

Techniques are currently being developed which are
robust to small departures from the assumed model and, in
particular, to outliers. The robust methods summarized in
this paper are simple extensions of maximum likelihood
estimators of the mean vector and covariance matrix assuming
an underlying normal distribution. In essence, they are
weighted versions of maximum likelihood techniques in which
weights are determined iteratively based upon the size of
Mahalanobis distances. Principal components can be
determined from the robust covariance or correlation matrix.
For the discriminant problem, robust estimates are determined
for each group and these are substituted into Fisher's linear
discriminant function.

These techniques are applied to two sparrow species on
which four habitat variables were measured. These data have
some outliers but the data set would not be classified as
being "bad." The principal component analyses for both
species resulted in less emphasis being placed on the first
component for the robust techniques as compared to the
classical method. Fisher's linear discriminant model
depended too heavily on several outlying values. In order to
improve the stability of estimates and the accuracy of future
classifications, Huber's robust discriminant analysis with a
cutoff value of 1.97 was adjudged to be best.

Key words: Discriminant analysis; grassland birds;
maximum likelihood estimation; principal components analysis;
robustness .

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt.

^Associate Professor of Statistics,
Department of Statistics and Computer Science and

Associate Statistician, Agricultural and Forestry
Experiment Station, West Virginia University,
Morgantown , WV 26506

'Associate Professor of Wildlife Ecology,
Division of Forestry, West Virginia University,
Morgantown, WV 26506.

209

INTRODUCTION

M-ESTIMATORS

Multivariate analyses often start with mean
vector and covariance matrix estimates. In
principal components the eigenvectors and
eigenvalues are computed from the covariance
matrix or from scaled versions of it. The mean
vector and covariance matrix are calculated from
each sample in discriminant analysis.
Unfortunately, classical maximum likelihood
estimators based on multivariate normal theory and
least squares estimators are extremely sensitive
to outlying observations.

During the past 20 years statisticians have
been developing new estimators or modifying
existing estimators to make them less sensitive to
certain departures from the assumptions of a
model. Three classes of estimators are being
developed :

1 ) M-estimators or maximum likelihood type
estimators ;

2) R-estimators or rank type estimators; and

3) L-estimators or estimators based on linear
combinations of order statistics.

Estimators of the above types are called robust
estimators. By robustness we mean that the
distribution of an estimator is insensitive
against small deviations in the shape of the
underlying distribution of the assumed model.
This is essentially equivalent to an estimator
being robust to outliers although other
distributional variations are possible. In many
biological sampling situations outliers frequently
arise due to gross errors (bad data points) or
from heavy-tailed distributions. Huber (1977a)
states that "5-1 0% wrong values in a data set seem
to be the rule rather than the exception." Since
our experience with outliers in biological data is
consistent with this statement, we feel robust
methodologies are of great value to researchers.

A general theory of R and L-estimators is
currently being developed for the univariate
linear model ( Hettmansperger and McKean 1977,
Bickel 1973). However it is not clear how to
extend these to the multivariate case. The theory
of M-estimators was begun by Huber (1964). Huber
(1977a, 1977b) and Maronna (1976) developed
M-estimation techniques for the mean vector and
covariance matrix of a single population. They
derived properties of these estimators and
suggested algorithms to compute them. Randies et
al . (1978) studied the performance of these and
other estimators in the two group linear and
quadratic discriminant analysis problem. Other
than these papers, little else has been done.

The intent of this paper is to apply the
estimators developed by Huber and Maronna to bird
habitat data. Both principal component and
discriminant analyses are presented. First an
introduction to M-estimators is given.

Univariate Problem

For intuition the univariate location problem
is considered first. Let f(x-p) represent a
density and p the location parameter (mean) we
want to estimate. X^,X2....,X^ represents a

random sample from this distribution. The maximum
likelihood estimate is that value of u which
maximizes

n

L(y) = n f(x.-u),
i=1

the likelihood function. It is more convenient to
maximize InL(ij) which gives the same solution
since the natural logarithm is a monotonic
transformation. Thus we maximize

n n
InL(p) = z Inf(x.-p) = - £ p(x.-u)
i=1 i=1

with respect to p where p(x-p) = -lnf(x-p).
Differentiating and setting the resulting
expression to zero we get

n

d[lnL(p)]/dp = - z f'(x -p)/f(x -p)
i=1

= Z<t>(x^-p) = 0,

where <ti(x-p) = d[-p( x-p)]/dp.

The maximum likelihood estimate, p, solves the
last equation.

The normal case is given by:

f(x-p) = (l//?^)e[-(x-p)^/2],

2

p(x-p) = (x-p) /2-c, and

(t>(X-p) = x-p.
The solution then is found by solving

Z(ti(x^-p) = z(x^-p) = 0 or

p = zx^/n,

the sample mean.

Maximum likelihood estimators may or may not
be robust depending on the form of f(x-p). How-
ever, the commonly used normal case is decidedly
non-robust. To illustrate this, consider a sample
of four from a normal distribution together with
one outlier. Suppose the values are 2, U, 10,
3.5, 3. The p functions of each x. are given in
figure 1 with c arbitrarily set to zero. The
maximum likelihood estimate, p=4.5, is being
influenced greatly by x, = 10. On the other hand.

210

the median, a robust estimator, has a value of
3.5. The difficulty is that we are minimizing
Zp(x.-u) which is quadratic in the normal case.
For an outlier, say, pCx^-y) is quite large

near the center of the distribution and thus the
solution u is pulled to the right so that pCx^-u)
is not so large.

Clearly this p function does not result in a
robust estimate. How can we robustify p?
Intuitively, it appears that at some distance
symmetric about each x^ we should make p increase
less rapidly, perhaps even level off. Various
functions have been suggested by researchers and
two of these are sketched in figure 2 along with
the maximum likelihood p. Huber's function
increases quadratically along with the maximum
likelihood p until |x-u| = k^. For | x-u | >k^

Huber's function increases linearly. Hampel's
function follows Huber's until | x-y | = k.^ at which

time p(x-u) begins to level off, which it does at
|x-ii| = k^. The values for k^, and k^ can be

chosen by the researcher but reasonable values are
k^ = 1.7, k^ = 3.4, and k^ = 8.5 (Hogg 1979). The

analytic representations of the functions are also
given in Hogg (1979).

Robustness can be characterized more easily
in terms of the ^ functions (fig. 3). To be
robust the 41 function should at least be bounded.

-8 -4 0 4 8

X-M

Figure 2. Normal density, Huber, and Hampel p's
plotted as a function of x-u with constants of

k^ = 1,7, = 3.^, and k^ = 8.5.

211

Note that Hampel's function even redescends to
zero.

Why is the shape of the ^ function so
important? First, the influence an observation
has on the estimators is proportion to ^ for
M-estimators (Hampel 1974). Thus if 4, is
unbounded as is the <ti function in the normal case,
a single "bad" value can have a great influence.
Secondly, M-estimators of this type can be
expressed as weighted means with the weights
proportional to 41. That is

Z(t)(x^-u) = zw^(x^-u)

where w^ = * (x^-y )/( x^-y ) .

y . = X . 6+e. ,
' 1 —1— 1

where y^ is the ith "dependent" observation, x^ is

the ith 1 X p known "independent" vector, 8^ is a
p X 1 vector of unknown parameters, and e. is the
ith random error. We now want to solve \or £ in
the p equations

Zx^j(t)[ (y.-x^£)/a] = 0, j = 1, 2 p,

where x^^ is the jth "independent variable" on the

ith observation. Of course ancillary estimates of
o are required. The system is solved iteratively
as before.

In the normal case w^ = 1 for all i but the Huber
and Hample estimates may have w^ < 1 .

For simplicity we have been examining the
case in which a = 1 (i.e., the scale parameter is
unity) . Actually we must solve

E(t)[ (x^-U )/a]

0

by obtaining an ancillary estimate of a.
Generally the equation is solved iteratively
starting from initial estimates, and 3^. At

stage j the estimate of scale is updated and the
new estimate of location is then computed. The
estimate of scale should be robust, such as the
median absolute deviation (MAD) estimator
(Mosteller and Tukey 1977).

The procedure described above can easily bo
extended to the regression case (e.g., Hogg 1979).
Consider the model

10

Normal

Huber

A Normal Mean
K Huber Mean

10

Figure ^. Example illustrating the effect of a
single outlier on the mean vector and 50 percent
concentration ellipse.

Multivariate Problem

The multivariate estimation problem is much
more complicated than the univariate one. In
particular, outliers in high demensional space are
difficult to detect. Outliers may be apparent
only on new variables which are linear
combinations of the original variables. Although
difficult to detect, their effect may be large.
Figure 4 illustrates the effect of a single
outlier on the concentration ellipse and on the
sample mean vector. Notice that the sample mean
vector is pulled toward the outlier. More
importantly the shape of the concentration
ellipse, which is determined from the sample
covariance matrix, is distorted. This is
important since most multivariate analyses are
based on estimates of covariance matrices. The
robust correlation of 0.904 more adequately
expresses the relation between x and y than does
the standard Pearson product-moment correlation of
0.634.

Clearly robust estimators are needed which
minimize the effect of outliers. This is
particularly true in the multivariate case, since
variances and covariances are extremely sensitive
to outliers.

Maronna (1976) and Huber (1977a, 1977b) have
developed most of the theory for robustly
estimating multivariate location and scale using
M-estimators. Maronna characterizes M-estimators
as being the solutions to the following matrix
equations :

n~' I U [/HTKx.-y^) = 0
i=l

-1 "

n E U^Ed. ](x.-u)(x.-u.)' = I
i =1

where y is the p x 1 mean vector, z_ is the pxp

-1 1 /2 .

covariance matrix, d^ = [ (iij^-u_) ^iLi~iL^ ■'■^

the Mahalanobis distance, and and are weight

212

functions. In the multivariate normal case =

= 1 for all i and the ordinary maximum

likelihood estimators result, that is u. = l\^/n,

and t = [z(x.^-2.) (2Lj^-£.) ' ]/n. Huber's development

of M-estimators is slightly more general than
Maronna ' s .

We used a modified form as suggested by
Randies et al. (1978). Start with x and S, the
ordinary estimates. Compute

d. = [(x.-x) 'S"\x.-x)]^^^, i = 1, 2 n.

Replace x and S by weighted estimates
— «

x^ = (zw^x^i)/z;w^ and

y 2 # # 2

S = (ew. (x.-x )(x.-x )')/zw.
— 1 —1 — —1 — 1

where w. = k/d. if d. > k and w. = 1 if d. < k.
Ill 1 1 —

_«

The procedure is iterated until the change in x

and S is sufficiently small or until a
predetermined number of iterations is reached.

A graph of the weight function is given in
figure 5. Note that the weight is 1 for an
observation unless the Mahalanobis distance is
greater than k. The weight then decreases
inversely with the Mahalanobis distance. If all
observations are sufficiently well-behaved, then
all the weights remain 1 and the ordinary
estimates result.

1.0

o)0.5

0

2 4 6 8 10

Mahalanobis Distance

Figure 5. Weight function for estimating the mean
vector and covariance matrix by Huber's method
with a constant of k = 1.97.

This approach of robust estimation can be
fruitfully applied in principal component
analysis, factor analysis, regression,
discriminant analysis, etc. The application in
this paper will be in principal components and
discriminant analysis.

The robust discriminant problem is discussed
in Randies et al. (1978). They discuss several
procedures for the two-group case. A method that
is suggested naturally is to estimate the mean
vector and covariance matrix separately and
robustly (as above) for each group. These are
then substituted into the linear or quadratic
discriminant function.

Let x-|. ^2' In ^"'^ ^1' ^2' "•' ^

two independent random samples from p-variate
populations. Let z be an observation from one or
the other population. The Huberized linear
discriminant function (HLDF) then is

D» (z)=[z- 1/2(x*+y*)]' S»~^[x«-y»] where x« and
-L-- -- -p-- -

y* are the robust sample mean vectors and S* is

the robust pooled sample covariance matrix. It is
defined by

S« = [(n-1)S»+(m-1)S«]/(n+m-2)
-p -X -y

where S* and S* are the robust sample covariance
-X -y

matrices. The robust sample discriminant
coefficients are given by

— — —

Fisher's linear discriminant function (LDF)
is defined similarly except that the ordinary
estimates are placed in the above HLDF. Huberized
or Fisherian quadratic discriminant functions can
be defined easily as in Randies et al . (1978).

The geometry of robust discriminant analysis
is similar to that of ordinary discriminant
analysis. Concentration ellipsoids are determined
by the equations of the form

(x-x») 'S»~Vx-x*) = c and

(y-y*)'S»"''(y-y*) = c

where c is a positive constant. Probability
statements can be made concerning the ellipsoids
by using a chi-square distribution with p degrees
of freedom as an approximation.

Stability of the Estimators

In order to assess the discriminant model it
is important to assess the stability of the sample
discriminant coefficients. A technique both for
obtaining new estimates and for assessing their
stability is the Jackknife (Hosteller and Tukey
1977). An initial step of the Jackknife procedure

213

called "leave-one-out" is also useful for
validating the model. These procedures are not
applied to the principal component coefficients in
this paper although it would be possible to
jackknife them.

The robust discriminant coefficients are

given by g '

l*. g» The method

starts by recomputing the coefficients each time
after leaving out a single observation. For
simplicity, let j range from 1 to N=n+m whereas
i=l,2,...,p denotes the variables, i.e. is

the coefficient for the ith variable with the jth
observation removed. B*^ is used to predict

the jth observation in the leave-one-out
validation. Note that a prediction is being made
based on values which were not used in building
the model.

by

The jth pseudo-value for variable i is given

Then the jackknife estimator is given by

j=i

A standard deviation estimate for both and 'i*
is given by

^ 2 1/2

^i j=i '^-J^ '

We can then test for significance since B*/s^* or

^ ^i

f*/SoJ* has an approximate t distribution with N-1
' h

degrees of freedom under the null hypothesis that
the coefficient is zero (Tukey 1958).

Methods of measurement are described by Whitmore
(1979). Units of measurement are presented only
in table 1. (Tables follow literature cited.)
Twenty savannah sparrows and 51 grasshopper
sparrows were found on this mine in 1978.
However, in order to make the sample sizes equal
(for convenience) only the first 20 grasshopper
sparrows were used. No attempt was made to screen
the data to pick the "best" observations to
illustrate our techniques.

Principal Component Analyses

The effects of outliers on principal
component analyses depend on their position
relative to the "ellipsoidal" swarm of points in
p-space. The general effect is to shift and
stretch the ellipsoid in the direction of the
outliers. If the outliers are along the major
axes, then the ellipsoid is elongated more than it
should be. On the other hand, if the outliers are
orthogonal to the major axes, the ellipsoid is
contracted more than it should be.

The analyses that follow estimate the mean
vectors and covariance matrices by ordinary
maximum likelihood (ML) adjusted to be unbiased
and by Huber's method at cutoffs of k = 1.97
(H197) and k = 1.56 (H156). The cutoff at 1.97
corresponds to weighting observations less which
fall in the outer 10 percent of the distribution
whereas the 1.56 cutoff corresponds to those
observations in the outer 30 percent of the
distribution. For this data set the 1.97 cutoff
should be adequate to protect against the bad
effects of outliers.

The raw data are presented in table 1 and
weights based on the Huber technique in table 2.
The savannah sparrow data is typical of most data
sets in having a few observations (about 10%)
somewhat distant from the bulk, particularly
observations 16 and 17. Three values are
relatively bad for the grasshopper sparrow data,
observations 1, 3, and 13 (table 2).

SPARROW EXAMPLE

Since 1976 data on sparrow habitats have been
collected on "reclaimed" strip mines in northern
West Virginia. Our purpose is to illustrate the
robust procedures discussed in the previous
section using a small portion of these data. In
particular, habitat data of savannah sparrows
( Passerculus sandwi chensi s ) and grasshopper
sparrows ( Ammodramus savannarum) areused based on
field work done in 1978 on the Great Mine (47.5
ha) in Preston County, West Virginia.

Ten vegetation variables were measured based
on the territories of the sparrows. However,
after preliminary screening, only four variables
are used in this paper. They are bare ground
cover (BGC) , litter depth (LD) , vertical diversity
of grass (VH), and total grass density (TD).

Savannah Sparrow

Summary statistics for habitat variables
measured on savannah sparrows are given in table
3. The robust mean estimates were rather similar
to the ML estimates although some shift is
present. The robust estimate for BGC increased
relative to the ML estimate, whereas the robust
mean estimates for LD, VH and TD decreased (table
3). The standard deviations estimated by HI 97
decreased relative to those estimated by ML,
except for VH. The H156 standard deviations are
consistently larger than the H197 standard
deviations.

The most pronounced effect caused by outlying
values was on the relationships among the
variables. The H197 correlation estimates were
consistently less (in absolute value) than the ML
estimates. The H156 estimates were not as

214

consistent but note that the correlation between
BGC and VH is essentially zero (table 3).

In order to understand better the nature of
the relationships, principal components were
computed on the correlation matrix. The largest
eigenvalue decreased as robustif icaion increased,
whereas the second largest eigenvalue increased
(table M). This indicates that the outlying
observations, principally observations 1, 12, 16,
and 17 (table 2), approximately lie along the
first principal axis which causes the ellipsoid to
be elongated with certain relationships
accentuated and others obscured.

In all three techniques the percent of
variation explained by the first two components
was about 90% (table 4). The change of emphasis,
however, is best seen by examining the
eigenvectors (principal component coefficients).
The first component contrasts BGC to the other
variables in all cases. However, BGC becomes
increasingly less important based on the robust
estimates. For the ML technique, the second
component seems to suggest a contrast between LD
and TD. On the other hand, the robust second
component suggests a linear combination of BGC and
VH is important.

Grasshopper Sparrow

Summary statistics for the four variables
measured on grasshopper sparrow habitat are given
in table 5. Although the mean vector estimates
for grasshopper sparrows were somewhat different
than those for savannah sparrows, a nearly
identical change occurred when the estimates were
robustif ied. The same was true of sample standard
deviations although not always with the same
variables (tables 3 and 5).

For the most part, the grasshopper sparrow ML
correlations were lower in absolute value than the
corresponding Savannah sparrow correlations. The
HI 97 estimates consistently indicated less strong
relationships, particularly between BGC and VH.
In all cases, the correlation between LD and TD
remained about the same.

Results of the principal component analyses
for grasshopper sparrows were analogous to those
for savannah sparrows. The first two components
explained about 85* of the variation with
increasing importance going to the second
component for the robust estimates (table 6).
Again, notice the decreasing importance of BGC in
the first component. In the second component, LD
becomes less important whereas VH becomes more
important. As with the savannah sparrow data, the
robust second component is primarily a linear
combination of BGC and VH (table 6).

Discriminant Analysis

The next phase of the study was to examine
whether or not it is possible to discriminate

between savannah and grasshopper sparrow habitat
based on these four variables. Tests to determine
equality of covariance matrices were not done,
since it is not clear how to do this for the
robust estimates. However, upon inspecting the
covariance matrices, the differences between them
for each estimation technique did not appear to be
large enough to cause concern.

Discriminant coefficients, their standard
errors, and the t-values for all methods are given
in table 7. None of the coefficients were
significant for any of the methods. However, in
all cases the variables in the LDF model were
closer to significance than the corresponding
variables in the HLDF (1.97) or HLDF (1.56)
models. What is the reason for this? For both
species, the outliers lie along the principal axis
which tends to elongate the ellipsoids, perhaps
resulting in an optimistic appraisal of the LDF
model's ability to discriminate. However, this
better discrimination can be deceiving since
estimates of the coefficients are based on
outliers. Outliers in another sample might appear
in totally different regions resulting in a very
different discriminant model.

Predictions for the LDF, HLDF (1.97) and HLDF
(1.56) models are given in table 8. A value
greater or equal to zero predicts the observation
to be associated with a savannah sparrow whereas a
value less than zero predicts the observation to
be of the grasshopper sparrow type. For the LDF
model 75? of the observations are classified
correctly. The corresponding result for the HLDF
(1.97) model is 77.5?, and for the HLDF (1.56)
model it is 65? (table 8). The LDF and HLDF
(1.97) models have rather similar performances
with HLDF (1.56) giving poorer results. More
accurate estimates of misclassif ication
percentages, however, were obtained from the
leave-one-out predictions (table 9). These give
70, 70, and 60 correct classification percentages
for LDF, HLDF (1.97), and HLDF (1.56),
respectively.

We would not recommend the use of the LDF
model, however, due to its dependence on outliers;
the HLDF (1.97) model does as well and is more
stable. As mentioned before, these data are
reasonably typical with respect to outlier
occurrence. Thus, a cutoff at 1.56 probably
targets too many observations as outliers. This
means that the HLDF (1.56) model for this data is
too harsh on judging an observation to be an
outlier and should not be used.

The lack of significance for the terms in the
models together with the reasonably good
classification indicate that models with fewer
variables might discriminate nearly as well. In
fact, a model based on only VH does nearly as
well, but does not illustrate robustness in a
multivariate setting.

Realizing that covariance matrices for each
species may have been too different to justify
linear discriminant analyses, ordinary and robust

215

quadratic discriminant analyses were done. These
models did not improve the probability of
correctly classifying an observation and thus are
not included.

REMARKS

Outliers in multivariate data are difficult
to detect if the dimensionality of the space is
greater than two. It is rare that practitioners
of multivariate methods attempt to adjust their
analyses to minimize these effects. Partly this
is due to both the unavailability of techniques to
deal with multivariate outliers and the lack of
computer programs. Nonetheless, outliers can have
pronounced effects on the conclusions.

We suggest that both classical and robust
analyses be performed on the data. If they give
similar results, then use the results from the
classical analysis and merely note that the robust
analysis was done. If the analyses give different
interpretations, it is important to understand
what observations caused the discrepancies. The
robust analysis then is generally to be preferred.

The M-estimation technique presented in this
paper is a natural extension of maximum likelihood
estimation. In fact, it is just a weighted
version in which the weights are determined
iteratively from the Mahalanobis distances. Thus,
these robust estimators are easy to understand and
interpret .

Lack of computer programs will make it
difficult to perform these analyses at the present
time. However, the senior author is currently
developing a multivariate data analysis package.
Some of these procedures should be published in
the SUGI (SAS User's Group International)
proceedings in 1981.

ACKNOWLEDGEMENTS

We wish to thank Gerald R. Hobbs, Harry V.
Wiant, and Jake C. Rice for reviewing early drafts
of this manuscript. This paper was published with
the approval of the Director of the West Virginia
University Agricultural and Forestry Experiment
Station as Scientific Paper No. 1665.

LITERATURE CITED

Bickel, P.J. 1973. On some analogues to linear
combinations of order statistics in the
linear model. The Annals of Statistics
1:597-616.

Hampel, F.R. 197^*. The influence curve and its
role in robust estimation. Journal of the
American Statistical Association 69:383-393.

Hettmansperger , T.P., and J.W. McKean. 1977. A
robust alternative based on ranks to least
squares in analyzing linear models.
Techometrics 19:275-284.

Hogg, R.V. 1979. Statistical robustness: one
view of its use in applications today. The
American Statistician 33:108-115.

Huber, P.J. 1964. Robust estimation of a
location parameter. The Annals of

Mathematical Statistics 35:73-101.

Huber. P.J. 1977a. Robust statistical
procedures. CBMS-NSF Regional Conference
Series in Applied Mathematics. 56p. J.W.
Arrowsmith Ltd., Bristol, England.

Huber, P.J. 1977b. Robust covariances. p.
165-191. In Gupta, S.S., and D.S. Moore,
editors. Statistical decision theory and
related topics II. Academic Press, New York,
N.y.

Maronna, R.A. 1976. Robust M-estimators of
multivariate Iqcation and scatter. The
Annals of Statistics 4:51-67.

Mosteller, F., and J.W. Tukey. 1977. Data
analysis and regression. 588p.
Addison-Wesley, Reading, Mass.

Randies, R.H., et al . 1978. Generalized linear
and quadratic discriminant functions using
robust estimates. Journal of the American
Statistical Association 73:554-568.

Tukey, J.W. 1958. Bias and confidence in
not-quite large samples. Abstract in the
Annals of Mathematical Statistics 29:614.

Whitmore, R.C. 1979. Temporal variation in the
selected habitats of a guild of grassland
sparrows. Wilson Bulletin 91:592-598.

216

Table 1. Measurements of four habitat variables for savannah and grass-
hopper sparrows.'

Savannah sparrow Grasshopper sparrow

Obs

BGC

LD

VH

TD

BGC

LD

VH

TD

1

24.3

1.02

0.98

897

19.8

6.95

1.21

899

2

36.4

1.45

0.74

449

38.7

1 .20

0.57

174

3

19. 1

1.02

1.00

579

4.5

3.55

0.87

364

4

42.5

0.26

0.89

197

55.0

0. 47

0.54

133

5

58.4

0. 18

0.56

113

33.0

2.87

0.60

418

6

15.3

1.97

1.04

432

0.5

2.95

0.73

714

7

48.8

0.27

0.52

145

33.0

0.51

0.71

222

8

50.0

0.47

0.64

212

20.5

2.41

0.79

743

9

18.0

3.31

0.83

830

22.3

0.34

0.81

188

10

44.4

2.67

0.88

381

10.5

2.00

0.83

454

1 1

20.0

3. 15

1.14

826

24. 9

1.18

0.77

404

12

14.8

1. 18

0.77

859

55.5

0.53

0.61

95

13

13.

3.10

0.92

604

66. 1

0.56

0.00

154

14

40.4

0.53

0.44

210

22.9

0.48

0.32

274

15

10.6

2.60

1.00

480

20.5

1.66

0.51

308

16

0.7

6.30

1.30

1208

24.3

1.20

0.45

271

17

11.3

4.46

0. 92

468

14.2

1.76

1.21

629

18

68.5

0.87

0.84

120

22,0

2.77

0.57

431

19

1.4

3.55

1.31

1090

58.5

0.41

0.56

127

20

13.7

2.42

1.07

810

14.2

2.31

0, 50

451

'Variables are BGC - bare ground cover (.%); LD - litter depth (cm); VH -
vertical diversity of grass (Shannon-Weaver index); and TD - total grass
density (hits/unit transect).

Table 2. Final weights for each four-variate observation using Huber's
robust procedure with cutoffs of 1.97 and 1.56.

Final Weights

Savannah

sparrow

Grasshopper

sparrow

Obs

H197

H156

H197

H156

1

0.786

0.537

0.319

0.271

2

1

1

1

1

3

1

0.736

0.495

0.335

4

0. 960

0.640 ■

1

0.987

5

1

0.986

1

0.887

6

0.924

0.648

0.871

0.608

7

1

1

1

1

8

1

1

0. 650

0.449

9

0.964

0.639

0.814

0.604

10

1

0. 900

1

0.984

11

1

1

1

1

12

0.757

0.540

1

0.888

13

1

1

0.479

0.328

14

1

1

0. 926

0.726

15

1

0.765

1

1

16

0.534

0.408

1

1

17

0.679

0. 471

0.651

0.487

18

0. 800

0.802

1

1

19

1

0.780

1

0.906

20

1

1

1

1

217

Table 3. Sample mean vectors, standard deviations, and correlations for
four variables measured on savannah sparrow habitat using maximum
likelihood and Huber's robust procedure with cutoffs of 1.97 and 1.56.

BGC

Sample mean vectors
LD VH

TD

ML

Huber (1.97)
Huber (1.56)

27. 60
28. 13
29.63

2.039
1.880

0.890
0.880
0.866

5^5.5
530.6
510.6

ML

Huber (1.97)
Huber (1.56)

19.50
19. 16
23. 14

Sample standard deviations

1.625
1.325
1.329

0.235
0.248
0.328

BGC

vs

LD

Sample correlations

334.4
301.2
309.7

BGC

vs

VH

BGC

vs

TD

LD
vs
VH

LD
vs
TD

VH
vs
TD

ML

Huber (1.97)
Huber (1.56)

-0.728
-0.715
-0. 421

-0.724
-0.500
0.032

-0.831
-0.757
-0.441

0.715
0.608
0.693

0.699
0.698
0.780

0.761
0.723
0.767

Table 4. Sample eigenvalues and eigenvectors based on correlation matrices
for four variables measured on savannah sparrow habitat using maximum
likelihood and Huber's robust procedure with cutoffs of 1.97 and 1.56.

Sample eigenvalues
Component

Estimator

1

2

3

4

ML

3.230

0.326

0.283

0.160

(Cum. Percent)

80.8

88.9

96.0

100.0

H197

3.007

0.516

0. 31 1

0. 166

(Cum. Percent)

75.2

88. 1

95.8

100. 0

H156

2.641

1. 028

0.234

0.097

(Cum. Percent)

66.0

91 .7

97.6

100. 0

Sample eigenvectors

Estimator

Component

BGC

LD

VH

TD

ML

1

0.509

-0.485

-0.495

-0.

510

2

-0. 360

-0.791

-0. 096

0.

485

3

-0.425

0.336

-0.838

0.

068

4

-0. 656

-0. 160

0.21 1

-0.

707

H197

1

0.496

-0.503

-0.468

-0.

531

2

-0.578

0.232

-0.779

-0.

073

3

-0.350

-0.812

-0.026

0.

466

4

-0. 546

-0. 183

0.416

-0.

704

HI 56

1

0.292

-0.565

-0.505

-0.

583

2

-0.857

0.022

-0.515

-0.

004

3

-0. 150

-0.809

0.210

0.

528

4

-0. 398

-0. 158

0.659

-0.

618

218

Table 5. Sample mean vectors, standard deviations, and correlations for
four variables measured on grasshopper sparrow habitat using maximum
likelihood and Huber's robust procedure with cutoffs of 1.97 and 1.56.

Sample mean vectors

BGC

LD

VH

TD

ML

28. 04

1.806

0.658

372. 0

Huber

(1

.97)

28.63

1.591

0.635

345.7

nuDer

1 1

. DO )

do- yy

1.569

0.630

551 - 0

Sample standard

deviations

ML

18.24

1.576

0.274

227.0

Huber

/ 1

V 1

1 0 '59

0.992

0.225

1 ( u . ^

Huber

(1

.56)

20.84

1.048

0.276

182.7

Sample correlations

BGC

BGC

BGC

LD

LD

VH

vs

vs

vs

vs

vs

vs

LD

VH

TD

VH

TD

TD

ML

-0.518

-0.542

-0.673

0.563

0.81 1

0.640

Huber

(1

.97)

-0.473

0.039

-0.537

0.266

0.779

0.478

Huber

(1

.56)

-0. 171

0.381

-0. 194

0.438

0.836

0.622

Table 6. Sample eigenvalues and eigenvectors based on the correlation
matrices for four variables measured on grasshopper sparrow habitat using
maximum likelihood and Huber's robust procedure with cutoffs of 1.97 and
1.56.

Sample eigenvalues
Component

Estimator

1

2

3

4

ML

2.883

0.502

0.457

0.158

(Cum. percent)

72.1

84.6

96.0

100.0

H197

2.349

1.048

0.442

0. 161

(Cum. percent)

58.7

84.9

96.0

100.0

H156

2.279

1.290

0.332

0.099

(Cum. percent)

57.0

89.2

97.5

100.0

Estimator Component

Sample

BGC

eigenvalues
LD VH

TD

ML

H197

H156

0.471
0.704
-0.477
-0.236
0.437
0.587
-0.641
-0.231
0,025
0.843
-0.479
0.245

-0.506
0.631
0. 151
0.568
-0.572
-0 . 082
-0.645
0.500
-0.589
-0.209
-0.632
-0.458

-0.473
-0.235
-0.847
0.065
-0.318
0.801
0.410
0.297
-0.502
0.474
0.602
-0.402

-0.547
0.225
0.182
-0.786
-0.617
0.078
-0.067
-0 . 780
-0.633
-0. 148
0.092
0.754

219

Table 7. Four-variate discriminant coefficient estimates and standard
errors for Fisher's linear discriminant model and Ruber's robust linear
discriminant models.

LDF

BGC

LD

VH

TD

0.0656
0,0431
1.52

-0.525
0.668
-0.79

5. 40
2.97
1.82

0.00427
0.00282
1.51

BGC

HLDF (1.97)
LD

VH

TD

i*

s^_

ti*

0.0249
0.0303
0.82

-0.336
0.655
-0.51

3.45
2.45
1.41

0.00338
0.00340
0.99

BGC

HLDF (1.56)
LD

VH

TD

-0.00402
0.0294
-0. 14

-0.683
0.682
-1.00

2.38
3.09
0.77

0.00308
0. 00369
0.83

Table 8. Discriminant predictions based on Fisher's linear discriminant
model and Huber's robust linear discriminant models.

Savannah sparrow

(1.97) (1.56)

Predictions

Grasshopper sparrow

(1.97)

(1.56)

Obs

LDF

HLDF

HLDF

LDF

HLDF

HLDF

1

3.22

2.47

2.51

1.07

1. 16

-0.96

2

0.59

0.28

0.22

-1.23

-1.09

-0.88

3

1.64

1.33

1.60

-2.27

-1.05

-1.04

1.35

0.50

0.58

-0. 11

-0.68

-0.64

5

0.29

-0.50

-0. 47

-1.27

-0.87

-1. 17

6

0.48

0.56

0.61

-1.48

-0.25

0. 13

7

-0. 47

-0.80

-0. 49

-0.28

-0.36

0. 10

8

0.44

-0.20

-0. 14

0.56

0.73

0.65

9

0.51

0.80

0.41

-0.49

-0. 34

0.39

10

0.94

0. 32

-0.53

-0.90

-0.22

0. 17

11

2.39

1.96

1.24

-0.06

0.04

0.38

12

1.22

1.33

1.82

0. 11

-0.58

-0.63

13

-0. 16

0.30

0.09

-2.26

-2.23

-1.97

14

-1.31

-1. 15

-0.62

-2.81

-1.77

-0.61

15

-0. 17

0.26

0.25

-2.41

-1.45

-0.85

16

1.96

2. 26

0.72

-2.40

-1.54

-0.80

17

-1.59

-0.67

-1.25

2.27

1.85

1.77

18

2. 13

0.51

-0.29

-2.05

-1. 17

-1.09

19

3.00

2.84

2.26

0.23

-0.53

-0.58

20

1.91

1.75

1.54

-2.61

-1.38

-0.85

Mis-

class-

ified

5

5

7

5

4

7

220

Table 9. Discriminant leave-one-out predictions based on Fisher's linear
discriminant and Huber's robust linear discriminant models.

Leave-One-Out Predictions

Savannah sparrow Grasshopper sparrow

(1.97) (1.56) (1.97) (1.56)
Obs LDF HLDF HLDF LDF HLDF HLDF

1

3.32

2.33

2.30

3.85

1.79

-0.37

2

0.55

0.24

0. 17

-1. 17

-1 .01

-0.78

3

1.53

1.26

1.48

-2. 13

-1.01

-0.82

4

1. 17

0.23

0.33

0.08

-0.45

-0.36

5

0.07

-0.79

-0.78

-1.17

-0.55

-0.91

6

0.25

0.32

0. 31

-1.32

0.01

0.29

7

-0.66

-1.08

-0.72

-0. 16

-0. 14

0.40

8

0.32

-0.35

-0.30

0.82

0.99

0.84

9

0.31

0.51

0. 11

-0. 14

0.15

0.97

10

0.76

-0.06

-0.89

-0.78

-0.01

0.49

11

2.33

2.05

1.31

0.01

0. 15

0.55

12

0.83

0.88

1.47

0.36

-0.34

-0.39

13

-0.26

0.20

-0.06

-1.96

-2.49

-1.69

14

-1.64

-1.47

-0.86

-2.78

-1.53

-0.45

15

-0.43

0.04

-0.08

-2.36

-1.41

-0.80

16

1.68

1.64

0. 10

-2.35

-1 .47

-0.76

17

-2.72

-2. 02

-2. 42

3.46

2.41

2.33

18

1.92

0. 18

-0.67

-1.98

-0.97

-0.83

19

3.01

3.13

2.29

0.52

-0.23

-0.29

20

1.84

1.86

1.69

-2.56

-1 .24

-0.67

Mis-

class-

ified

5

6

9

7

6

7

DISCUSSION

STEVEN PARREN: In addition to the elimination of
outliers, does robust DFA have an advantage over
transformations of habitat variables as we measure
(and perceive) these variables in the field?

E. JAMES HARNER: The use of robust estimates does
not preclude the use of transformations but it
often makes it unnecessary to transform. An
outlier can make it appear that a log (say)
transform is necessary. I prefer not to transform
unless absolutely necessary because of the
difficulty of interpretation.

STEVEN PARREN: Are homogenous covariance matrices
ever found in ecological studies?

E. JAMES HARNER: Yes, within reasonable
statistical variability. In most cases, however,
they are probably both biologically and
statistically different. We need more research on
the effect of not satisfying this assumption on
the linear discriminant function. I believe A. P.
Dempster (1969. Elements of continuous multi-
variate analysis. 388 p. Addi son-Wesley ,
Reading, Mass.) somewhat quantifies the
"difference" in covariance matrices.

STEVEN PARREN: What effects do transformations of
variables used in multivariate space (DFA) have on
the ecological interpretation of habitat
variables?

E. JAMES HARNER: If the object is strictly
classification then by using, for example, the
power family of transformations causes few
conceptual difficulties. However, if you desire
to "interpret" the coefficients I believe it does.
The robust approach can sometimes make it
unnecessary to transform.

LESLIE MARCUS: Standard errors of your jackknifed
coefficients are large, which support the idea
that it is difficult to interpret coefficients.
Please comment.

E. JAMES HARNER: The t-tests based on these
standard errors are not significant, but the
robust standard errors are on the whole somewhat
smaller than those of the standard analysis. We
did not use variable selection in determining this
model. In order to complete the model-building
process, it would be necessary to eliminate one or
more variables from the discriminant analysis to
remove the redundancies.

221

USE OF DISCRIMINANT ANALYSIS AND OTHER STATISTICAL
METHODS IN ANALYZING MICROHABITAT UTILIZATION

OF DUSKY-FOOTED WOODRATS'
Janet I. Cavallaro^, John W. Menke*. and William A. Williams*

Abstract. — In the past 10 years, discriminant analysis
has become an increasingly used statistical method in small
mammal studies. Although originally developed as a
classification procedure, it has been used by ecologists and
wildlife biologists for a variety of other purposes. No
generally accepted methods, however, have been developed for
interpreting the discriminant functions used for these
alternative purposes.

Using the dusky-footed woodrat (Neotoma fuscipes) as an
example, we present a method for interpreting discriminant
functions when the purpose is both to characterize micro-
habitats used by a species and to explain why individuals of
a species occur in some microhabitats and not in others.
Three types of discriminant functions were identified from
which hypotheses of different forms were developed. More
complete interpretation of two-group discriminant functions
can be made if a multiple regression is calculated for the
dependent variable, presence or absence of woodrats and if
partial correlations are calculated for the independent
variables. Since multiple regression coefficients are
unstable if multicollinearity exists among the independent
variables, the variance inflation factor was calculated and
ridge regression was performed so that the effect of
intercorrelated variables could thereby be examined and
unstable variables eliminated wherever necessary.

Key words: California chaparral; multicollinearity;
multiple regression; partial correlation; ridge regression;
variance inflation factor; wildlife habitat analysis.

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^Ph.D. Graduate Student, Department of
Forestry and Resource Management, University of
California, Berkeley, CA 94720.

'Associate Professor, Department of Agronomy
and Range Science, University of California,
Davis, CA 95616.

■•Professor, Department of Agronomy and Range
Science, University of California, Davis, CA
95616.

222

INTRODUCTION

Discriminant analysis was developed as a
classification procedure for assigning unknown
specimens to one of two or more groups. In small
mammal studies, however, discriminant analysis has
also been used for other purposes: 1) evidence of
niche separation, 2) hypothesis formulation about
the mechanism of coexistence of sympatric species,
and 3) characterization of microhabitats .

M'Closkey (1976) and Holbrook (1978) used
discriminant analysis to provide evidence of niche
separation of small mammal species by equating
structural variables in discriminant functions to
niche dimensions that separated the niches of
small mammal species. M'Closkey recommended
caution in equating statistical and biological
patterns of microhabitat use, but then went on to
suggest that

"the statistical patterns of separation
reflect ecological and evolutionary
adjustments to inter-specific

competition. Structural habitat

division by rodents is either the means
by which co-existence is achieved or is
correlated with other niche dimensions
(e.g., food) responsible for
coexistence. "

Unless hypotheses are developed from the
discriminant function to explain how species
partition the habitat according to structural
variables, however, little new insight is
achieved .

In other work, M'Closkey and Fieldwick (1975)
used discriminant analysis to develop hypotheses
to explain how two sympatric species could
coexist. They hypothesized that since the
discriminant function contained structural habitat
variables that were characteristic structural
features of optimal habitats of Microtus
pennsylvanicus or Peromyscus leucopus, the two
species were sympatric because animals of each
species could find microhabitats which
corresponded to their optimal habitat. They
observed that foliage height diversity and tree
basal area in Peromyscus microhabitats were
respectively 1.5 and 4 times greater than in
Microtus microhabitats, and the depth of the
grassmat in Microtus microhabitats was twice as
deep as in the Peromyscus microhabitats,

Dueser and Shugart (1978) used discriminant
analysis to characterize the microhabitats of
small mammal species and to explain why
individuals of a species occurred in some
microhabitats and not in others. By restricting
their analysis to significant univariate variables
they were able to conclude that species X occurred
where more of habitat variable A occurred, where
habitat variable B was larger, and so forth. They
used the simple correlation between the function
and each variable in the function to measure the
"relative contribution of the variable to the
power of the function to discriminate between

groups". Simple correlation coefficients,
however, are inappropirate whenever more than one
independent variable is involved since they only
show which univariate variables would separate the
groups similarly to the function. The relative
magnitude of the correlation coefficient
corresponds exactly to relative value of the
univariate F-ratios (Marascuilo and Levin').

Using live-trap data collected on
dusky-footed woodrats in chamise-wedgeleaf
ceanothus ( Adenostoma fasciculatum-Ceanothus
cuneatus) chaparral in northern California, we
present a method for interpreting discriminant
functions that includes: 1) determination of the
type of discriminant function to be used, 2)
assessment of multicollinearity with the variance
inflation factor and ridge regression, 3) use of
multiple regression and partial correlation
analysis, and 4) development of hypotheses to
explain why individuals of a species frequent some
microhabitats and not others.

FIELD METHODS

Woodrats were trapped on 0.5-ha and 1.0-ha
grids in 14- and 25-year old chamise-ceanothus
chaparral, respectively, for periods of 6 nights.
Fifty-two structural, botanical, and physical
habitat variables describing each microhabitat
were measured using point sampling and nearest
neighbor methods (Cavallaro 1978).

INTERPRETING DISCRIMINANT FUNCTIONS

Discriminant functions can serve as the basis
for characterizing habitats used by individuals of
a species. The characterization, however, is much
more useful if researchers hypothesize why a
species uses habitats with particular
characteristics. Partial correlation analysis can
assist in hypothesis development. Hereafter, we
present our approach to interpreting discriminant
functions in wildlife habitat studies using the
woodrat as an example.

Selection of Discriminant Function Type

We have identified three types of
discriminant functions that help both to
characterize the habitats used by a species and to
generate hypotheses explaining that habitat use.
The types are distinguished on the basis of
whether or not the function contains variables
which as univariate variables identify a
difference between where individuals of a species
do and do not occur. The three types of functions
are as follows: Type I contains only significant
univariate variables. Type II contains both
significant and nonsignificant univariate

'Textbook in preparation, Marascuilo, L.A.,
and J. Levin, Department of Education, University
of California, Berkeley.

223

Table 1. Variables included in two discriminant functions that separated trap sites used and not used by
dusky-footed woodrats on the 14- and 25 year-old plots; also shown are univariate statistics and variance
inflation factors.

Univariate Variance inflation factor

Discriminant

Variable function type F DF Type I DF Type II

IM-year-old plot

1.

Other shrub species density

I,

II

0. 175

0.

4

121.4

2.

Other shrub species live leaf density

I

0. 135

3

4

3.

Other shrub species live stem density

I,

II

0. 197

15

5

116.0

4.

Other shrub species dead stem density

I

0. 159

1 0

1

5.

Chamise dead stem density

I

0. 136

2.

8

6.

Chemise density

I ,

II

0. 099

2

o

8

2. 0

7.

Total vegetation cover

TT

U. UU'J

1.7

8.

Vertical canopy density 100-150 cm aboveground

II

0.021

3.0

9.

Vertical canopy density 150-200 cm aboveground

II

0.009

2.0

10.

Total stem (< 0.5 cm in diameter) density

II

0.022

2.3

1 1 .

Ceanothus live leaf density

II

0.007

1.4

12.

Yerba santa live leaf density

II

0.026

1.1

Type I discriminant function**, = 0.29

Type II discriminant function**, = 0.53

25-year-old plot

13.

Other ground cover

I,

II

0. 103

1.

1

1.2

14.

Live stem (< 0.5 cm in diameter) density

I,

II

«

0.070

1

1

1.2

15.

Live stem (1.0-2.5 cm in diameter) density

I,

II

»

0.064

16.

Chamise live leaf density

I,

II

*

0.068

1

1

7.1

17.

Vertical canopy density 0-2.5 cm aboveground

II

0.016

1.2

18.

Live leaf density

II

0.026

6.7

19.

Total stem (0.5-1.0 cm in diameter) density

II

0, 043

1.1

Type I discriminant function**, = 0.21
Type II discriminant function**, R^ = 0.40

»P < 0.05, **P < 0.01

variables, and Type III contains only
nonsignificant univariate variables. Hypotheses
based on Type I discriminant functions would state
that where woodrats occur variables x, y, and z,
respectively, are less, greater, and less than
where woodrats do not occur for some proposed
reason. In contrast, hypotheses based on Type III
discriminant functions would state that where
woodrats occur X = value D, Y = value E, and Z =
value F, and this particular combination of
variable values is important for some proposed
reason even though similar values of x, y, and z
can be found in microhabitats not used by
woodrats. Hypotheses based on Type II

discriminant functions would state that where
woodrats occur variable X is greater, Y = value H,
and Z = value I for a stated reason which must
explain why more of X is necessary and why
variables Y and Z have their particular values
even though similar values of Y and Z can be found
in microhabitats not used by woodrats.

It becomes apparent that hypotheses developed
from Type II or III discriminant functions are
much more difficult to justify because the
functions would indicate that animals are using
extremely specific microhabitats. Those
discriminant functions may often have
questionable ecological significance. For
comparison, however, we will include Type I and II
discriminant functions in our discussion.

If a satisfactory hypothesis can be developed
from a Type I discriminant function, selection of
variables is a simple matter. All significant
univariate variables are used. Selection of
variables to include in a Type II or Type III
discriminant function is more difficult. A useful
strategy is to use those variables that increment
the coefficient of determination, R^, by a
specified minimum amount (Marascuilo and Levin^);
we used a 2% increment for Type II and III
discriminant functions.

224

Discriminant Functions

For the 14-year-old plot, the Type I
discriminant function had a value of 29% and
contained all significant univariate variables
except one, other shrub species live leaf density
(2) (table 1). The Type II discriminant function
had a value of 53% and contained in addition to
all nonsignificant univariate variables listed in
table 1 three density variables, other shrub
species (1), other shrub species live stems (3),
and chamise (6). For the 25-year-old plot, the
Type I discriminant function had a R^ value of 21%
and contained all significant univariate
variables. The Type II discriminant function had
a R^ value of W% and contained all nonsignificant
univariate variables listed in table 1 as well as
all significant univariate variables.

The coefficient of determination, R^ , is
considered a useful statistic because a function
may be significant and yet very little variation
in the function may be associated with the
variables in the discriminant function. In such
cases, the function contributes little
information. In the field of educational
psychology, R^ values less than 10? are considered
too inconsequential to be reported.'

Knowing the importance of each variable in
the discriminant functions would be extremely
useful for developing hypotheses. Fortunately,
the two-group case of discriminant analysis can be
done as a multiple regression, with the dependent
variable being presence or absence of woodrats;
and thereafter, partial correlations can be
calculated to assess the importance of habitat
variables. Partial correlations are a measure of
association between each independent variable and
the dependent variable in the multiple regression.

Assessment of Multicollinearity
in Multiple Regression

In data sets containing measurements of
uncontrolled variables, as is usually the case in
wildlife habitat studies, multicollinearity often
exists among variables and - can cause ambiguous
results in ordinary multiple regression (Hoerl and
Kennard 1970). Resulting regression coefficients
can be inflated in absolute value or may even have
the wrong sign. The possibility that such
difficulties will occur increases as independent
variables diverge from orthogonality. Thus, it is
desirable to test for multicollinearity among
independent variables.

Calculation of the variance inflation factor
(VIF) is the preferred method for determining
whether a multicollinearity problem exists,
because the VIF will detect relations that exist
among several variables that might not be evident

0.21

0.4 r

0.2

;S(k;

-0.2

0.3
K

0.6

'Personal communication with L.A. Marascuilo,
Professor of Statistics, Department of Education,
University of California, Berkeley.

Figure 1. Ridge trace for variables in the Type I
discriminant function for the 25-year-old plot.
Variables as defined in table 1.

from examination of simple correlations (Williams
et al. 1979). VIF equals 1/(1-R?) where R? is the
multiple correlation coefficient of one
independent variable with all other independent
variables. As R? approaches 0 (orthogonality) VIF
approaches 1; whereas, as R? approaches 1
(variables are highly intercorrelated ) VIF
approaches infinity. VIF values exceeding 10 are
considered likely to cause problems in estimating
regression coefficients (Chatterjee and Price
1977). "Tolerance" is the reciprocal of the VIF,
and the equivalent critical value is 0.1.

One or more variables in the two discriminant
functions for the 14-year-old plot has a VIF value
which exceeds 10, suggesting that for those
variables the regression coefficients probably are
not stable (table 1). In contrast, neither
function for the 25-year-old plot contained any
variables with a VIF greater than 10, thus all
those variables can be used in calculating a
multiple regression and partial correlations.

Ridge regression was developed to alleviate
problems from intercorrelated independent
variables by including a bias factor k when
regression coefficients are estimated (Hoerl and
Kennard 1970). The first step in ridge regression
involves making a ridge trace by adding increments
of k over the range of 0 to 1 and plotting the
respective standardized partial regression
coefficients 0(k) against k. As k increases the
coefficient of determination, R^, decreases.

The ridge trace for variables contained in

225

the Type I discriminant functions for the 25
year-old plot (fig. 1) shows that the regression
coefficients are stable as would be expected since
no variables in that discriminant function had a
VIF value greater than 10. As k increases, the
standardized regression coefficients 0(k) maintain
both their absolute value and their values
relative to each other quite well. In contrast,
in the ridge trace for variables contained in the
Type I discriminant function for the 14 year-old
plot (fig. 2) variables 2 and 4, other shrub
species live leaf density and other shrub species
dead stem density, were negative for k = 0 and
became positive for k > 0.13, indicating that
those variables are highly unstable. Variable 3,
other shrub species live stem density, was high
and positive at k = 0, but decreased rapidly as k
increased, suggesting that variable 3 is also
unstable .

The ridge trace shows either what k value to
use, if a multiple regression is desired for a
particular combination of variables (a value which
stabilizes the regression coefficients while not
greatly decreasing the value), or which

variables to eliminate because they are unstable.
The ridge trace shows visually what the VIF values
suggest .

Variables lacking stability were eliminated
from the Type I discriminant function for the 14-
year-old plot leaving other shrub species live
stem density (3) and chamise dead stem density
(5). Both had a VIF of 1.3 and stable regression
coefficients upon calculation (fig. 3).

0 30

-0,4

Figure 2. Ridge trace for variables in the Type I
discriminant function for the 14-year-old plot.
Variables are defined in table 1.

0 23
p2 0.22
0.21 -
0.20

0.3

0.6

0.4 r.

0.2

/3(K)

-0.2

0.3
K

0.6

Figure 3- Ridge trace for variables originally in

the Type I discriminant function for the
14-year-old plot which remained after assessing
intercorrelation among variables. Variables are
defined in table 1.

The ridge trace for variables in the Type II
discriminant function for the 14-year-old plot is
shown in figure 4. Variable 1, other shrub
species density, was strongly negative for k = 0
but became positive for k > 0.06. In contrast,
variable 3t other shrub species live stem density
was strongly positive for k = 0 but decreased
rapidly in value for k > 0. For k = 0 these two
variables had opposite signs, even though they had
a simple correlation of 0.99 and therefore
represent the same underlying variable. These
results demonstrate how multicollinear variables
can greatly affect the stability of regression
coefficents. After eliminating two variables' in
the Type II discriminant function, the VIF for all
variables was less than 3.0, and a new ridge trace
of the restricted variables show that the
variables were much more stable (fig. 5).

Microhabitat Use Hypotheses

Based on the Type I discriminant function for
the 14 year-old plot, microhabitats used by
woodrats can be characterized generally as having
greater density of other shrub species and lesser
density of chamise, and specifically as having
greater density of live leaves, live stems, and
dead stems of other shrub species and less density
of dead chamise stems, than microhabitats not
used. This combination of variables was
associated with 29% of variability in woodrat
occurrence (table 1 and 2). The reason woodrats

226

occur in these microhabitats can be hypothesized
from partial correlation coefficients. Although a
correlation does not prove cause and effect, it
does suggest variables to consider in trying to
establish a cause and effect relationship. A
positive relationship exists between woodrat
presence and amount of live stems of other shrub
species which included deerbrush (Ceanothus
integerrimus) , scrub oak (Quercus dumosa) , poison
oak (Toxicodendron radicans) , and toyon
(Heteromeles arbutifolia) with scrub oak being by
far most abundant. In contrast, no relationship
existed between woodrat presence and amount of
dead chamise stems. We, therefore, suggest that
woodrats occurred in particular microhabitats
because of presence of other shrub species, rather
than because of lesser amounts of chamise. This
hypothesis agrees with the observation that oak
species are the most important variable in
determining presence of dusky-footed woodrats
(Linsdale and Tevis 1951).

Based on the Type II discriminant function
for the 14-year-old plot, microhabitats used by
woodrats can be characterized generally as having
greater density of other shrub species and less
density of chamise, and specifically as having
greater density of live stems of other shrub
species, total vegetation cover of 80.9%, vertical
canopy density of 0.6 and 0.0, respectively, for
layers between 100-150 cm and between 150-200 cm
above the ground, total density of 1.5 for stems
less than 0.5 cm in diameter, and density of live
leaves of 0.1 and 0.0 respectively for ceanothus

0.50

0.40

0 30L-L

/3(K)

-0.2

Figure 5. Ridge trace for variables originally in

the Type II discriminant function for the It-
year-old plot which remained after assessing the
intercorrelation among the variables. Variables
are defined in table 1.

Figure 4. Ridge trace for variables in the Type

II discriminant function for the 14-year-old
plot. Variables are defined in table 1.

and yerba santa. In contrast, microhabitats where
woodrats did not occur can be characterized as
having less density of other shrub species,
particularly live stems of other shrub species,
greater density of chamise, total vegetation cover
of 78.7%, vertical canopy density of 0.5 and 0.1,
respectively, for layers between 100-150 cm and
between 150-200 cm above the ground, total density
of 1.7 for stems less than 0.5 cm in diameter, and
density of live leaves of 0.1 and 0.0,
respectively for ceanothus and yerba santa. We do
not think this function provides a good character-
ization of microhabitats used and not used by
woodrats because we can give no reason why wood-
rats would select such specific microhabitats.
Why, for example, should they distinguish between
80.9 and 78.7% total vegetation cover, and so
forth? Consequently, we suspect that the Type II
discriminant function has statistical significance
in this case but lacks ecological validity.

On the 25-year-old plot, where other shrub
species were essentially absent (they accounted
for only 1.5% of the shrub cover in contrast to 6%
on the 14-year-old plot) and where all other
species shrub cover was produced by deerbrush,
woodrats did not occur in microhabitats with
greater density of other shrub species. On the
25-year-old plot, they occurred instead where more
other ground cover was present, density of live
chamise leaves was greater, density of live stems
between 1.0 and 2.5 cm in diameter was greater,
and density of live stems less than 0.5 cm in

227

Table 2. Mean values (percent) for all variables in Type I and II discriminant
functions and partial correlations for variables in Type I and II multiple
regressions for the 14- and 25-year-old plots.

Variable

Microhabitat
Used Not Used

Partial
Correlation

14-year-old plot

Type I discriminant function

1 .

Other shrub species density

0.6

0,0

2.

Other shrub species live leaf density

0. 1

0.0

3.

Other shrub species live stem density

0.2

0.0

0.33»»

i».

Other shrub species dead stem density

0.3

0.0

5.

Chamise dead stem density

0.6

0.9

-0.20

6.

Chamise density

1.4

2.1

Type II discriminant function

1 .

Other shrub species denisty

0.6

0.0

3.

Other shrub species live stem density

0.2

0.0

0.48»»

6.

Chamise density

1.4

2.1

7.

Total vegetation cover

80.9

78.7

0.30*

8.

Vertical canopy density 100-150 cm aboveground

0.6

0.5

0.41»»

9.

Vertical canopy density 150-200 cm aboveground

0 . 0

0. 1

-0 . 28**

10.

Total stem (<0.5 cm in diameter) density

1.5

1.7

-0.53»»

1 1 .

Ceanothus live leaf density

0. 1

0. 1

-0 . 30*

12.

Yerba santa live leaf density

0.0

0.0

-0.33»

25-year-old plot

Type I discriminant function

13.

Other ground cover

2.1

0.3

0.28«»

Live stem (<0.5 cm diameter) density

0.2

0.5

-0.20

15.

Live stem (1.0-2.5 cm in diameter) density

0. 1

0.0

0. 11

16.

Chamise live leaf density

0.5

0.3

0.20

Type II discriminant function

13.

Other ground cover

2.1

0.3

0.32»»

14.

Live stem (<0.5 cm in diameter) density

0.2

0.5

-0.29»»

15.

Live stem (1.0-2.5 cm in diameter) density

0. 1

0.0

0.23»

16.

Chamise live leaf density

0.5

0.3

0.31»»

17.

Vertical canopy density 0-25 cm aboveground

0.2

0.3

-0.43»»

18.

Live leaf density

0.7

0.5

-0.25»

19.

Total stem (0.5-1.0 cm in diameter) density

0. 1

0.0

0.31*»

»P <

0.05, »»P < 0.01

diameter was less. This combination of variables
was associated with 21% of the variability in
woodrat occurrence (tables 1 and 2).

The only significant partial correlation
between woodrat presence and a habitat variable
was for amount of other ground cover. Other
ground cover consisted mainly of large downed
branches which might have attracted the woodrats.
Linsdale and Tevis (1951) stated: "Every pile of
dead wood within the rat habitat eventually

receives an accumulation of twigs deposited by the
rats . "

Since no significant partial correlation
existed between woodrat presence and any other
habitat variable and since we cannot explain why
other habitat variables in the Type I discriminant
function would be particularly important to
woodrats, we hypothesize that other significant
differences between where woodrats occurred were
the result of woodrat activity in their micro-

228

habitats rather than because woodrats select
mi crohabitats with those particular
characteristics. The lower density of live stems
less than 0.5 cm in diameter may have resulted
from woodrat utilization of the vegetation.
Woodrats cut the terminal branches of chamise and
wedgeleaf ceanothus and take them to their houses
where they feed on leaves and flowers (Linsdale
and Tevis 1951). This continual pruning of the
plants may cause improved vigor so that a greater
density of live chamise leaves and a greater
density of live stems between 1.0 cm and 2.5 cm in
diameter occur in microhabitats used by woodrats.
In this habitat dominated by mature chamise and
wedgeleaf ceanothus plants, it appears, then, that
woodrat selection of microhabitats may have been
in response to other ground cover, while other
significant differences between where woodrats did
and did not occur were the result of woodrat
activity.

The Type II discriminant function for the 25
year-old plot contained four significant and three
nonsignificant univariate variables. Again, we
could not explain why woodrats would select
microhabitats with such specific densities of live
leaves, stems between 0.5 cm and 1.0 cm in
diameter, or a vertical canopy density between 0
cm and 25 cm above ground, and therefore concluded
that this combination has little ecological
validity even though more of the variation in
woodrat presence was associated with that
combination of variables. The combination of
variables probably only had statistical
significance.

CONCLUSIONS

Discriminant analysis, multiple regression,
and partial correlation results can be used
together to characterize the microhabitats used by
individuals of a species and to generate
hypotheses explaining why animals occur in some
microhabitats and not in others. The specific
form of the hypothesis will vary depending on the
type of discriminant function.

If the habitat variables are highly
multicollinear , however, estimates of the
regression coefficients are unstable and
unreliable. In such cases VIFs can be calculated
or ridge regression can be used to identify the
unstable variables for elimination prior to
running the multiple regression and partial
correlations .

The multivariate approaches in wildlife
studies can suggest hypotheses to explain
observations made by the researcher. These
hypotheses should themselves then be tested so
that advances can be made in our understanding of
the habitat relationships of wildlife species.

ACKNOWLEDGMENTS
We thank L.A. Marascuilo for his assistance

in the statistical interpretation of our data and
we thank D.R. McCullough, W.E. Waters and D.E.
Capen for their helpful comments on an earlier
draft of our manuscript.

LITERATURE CITED

Cavallaro, J.I. 1978. Small mammal habitat in
different-aged chamise/wedgeleaf ceanothus
chaparral. M.S. Thesis. 112 p. University
of California, Berkeley.

Chatter jee, S., and B. Price. 1977. Regression
analysis by example. 312 p. John Wiley and
Sons, New York, N.Y.

Dueser, R.D., and H.H. Shugart, Jr. 1978.
Microhabitats in a forest-floor small mammal
fauna. Ecology 59:89-98.

Hoerl, A.E., and R.W. Kennard. 1970. Ridge
regression: applications to non-orthogonal
problems. Technometrics 12:69-82.

Holbrook, S.J. 1978. Habitat relationships and
coexistence of four sympatric species of
Peromyscus in Northwestern New Mexico.
Journal of Mammalogy 59:18-25.

Linsdale, J.M. , and L.P. Tevis, Jr. 1951. The
dusky-footed woodrat. 664 p. University of
California Press, Berkeley, Calif.

M'Closkey, R.T. 1976. Community structure in
sympatric rodents. Ecology 57:728-739.

M'Closkey, R.T., and B. Fieldwick. 1975.
Ecological separation of sympatric rodents
( Peromyscus and Microtus ) . Journal of
Mammalogy 56: 119-129.

Williams, W.A., CO. Qualset, and S. Ceng. 1979.
Ridge regression for extracting soybean yield
factors. Crop Science 19:869-873.

DISCUSSION

LESLIE MARCUS: 1) The three types of discriminant
analyses, while simpler, remind me of original
suggestions in use of uncorrelated variables in
physical anthropology (Pearson coefficient of
racial likeness). However the type II variables
may represent a "gestalt" or common habitat factor
for the organism. One way of getting at that
might be a rotation of the multivariate
discriminant coefficient akin to oblique factor
rotations, of course only reasonable for K > 2
groups. 2) I would suggest a better choice for
your "types" would be the Mahalanobis distance d^
for each variable rather than a significance test.
The "type" test is both sample size dependent and
dependent on significance and level and, of
course, still requires a cutoff for D. 3) I would
prefer Mahalanobis D^ as a measure of difference
rather than R^ for overall discrimination, because
it gives a plottable metric in the canonical
variate of discriminant space. 4) It is suggested
that correlation between canonical variate and
original variate be looked at. It has been shown
that for two groups this is just a simple function
of the mean difference (Bergmann, R.E. 1970.
Interpretation and use of a generalized
discriminant function. p. 35-60. ln_ Bose, R.C.
et al., editors. Essays in probability and

229

statistics. University of North Carolina Press.)
In K groups, this is not so since vectors are not
orthogonal. Perhaps for full interpretation these
correlations should be given; they are almost
never reported.

JANET CAVALLARO: We find your comments on
discriminant analysis valuable and we think that
they may be very useful in further interpreting
discriminant functions. In response to your
fourth comment, in the two-group discriminant
case, the correlation between the canonical
variate and the original variables provides no new
information and cannot be used in interpreting the
original variables as you stated. In discriminant
analysis where the number of groups exceeds two,
the correlation may be useful; but it is useful
only in trying to interpret what the canonical
variate represents, not in interpreting the
importance of the original variables in separating
the groups. An original variable may be important
because it separates the groups or it may be
important because it in combination with the other
original variables creates a new variable, the
canonical variate, which separates the groups.

KEN MORRISON: In your first example, it was only
for Peromyscus maniculatus (the most numerous
species) that the discriminant function was
successful. Does this really mean that the
discriminant function failed?

JANET CAVALLARO: A discriminant function must be
evaluated both statistically and ecologically; and
although it may be significant statistically, we
contend that it may be ecologically meaningless.
In the case of the discriminant functions reported
by Holbrook ( 1978), we assume that at least some
of the discriminant functions must have been
statistically significant, but we question whether
or not they were ecologically meaningful. She
provided no ecological interpretation of the
functions except to infer that "species occupy
habitats with characteristic three-dimensional
structure". We found it surprising ecologically
that the microhabitats used by Peromyscus
maniculatus could be classified better than those
used by P^. truei , P^. boylii , P. dif icilis.

In our work in the chamise-ceanothus
chaparral, P. maniculatus and P_. truei occurred on
all study plots. We could separate those
microhabitats £. maniculatus used from those they
did not use on only one of those plots. In
contrast, on all the study plots, we could
separate the microhabitats P_. truei used from
those they did not use. Considering the
ubiquitous distribution of P_. maniculatus , these
results conform more with current understanding of
the habitat relations of these species. We should
mention, however, that as researchers we are
searching for new information so that the
discriminant functions Holbrook reported should
not necessarily be negated. We believe, however,
that the burden rests on her to hypothesize why
the microhabitat used by P. maniculatus could be
classifed more readily than those used by the
other three species.

GERALD SVENDSEN: Because many of your measured
variables are not independent from one another, do
you think that you would have gained any insight
by using a factor analysis to derive new composite
yet independent variables and then performing
multiple regression on these new variables?

JANET CAVALLARO: I think the question you are
asking is whether or not principal components
could have been used to create a set of
independent "habitat" variables for the multiple
regression analysis rather than using ridge
regression to identify redundant or highly
intercorrelated habitat variables. The answer is
definitely yes, but the use of principal
components can create problems in itself.
Primarily, each principal component must be
interpreted as to what it measures, and this
interpretation itself can be a difficult task. By
using the ridge regression to identify a set of
relatively independent habitat variables, that
problem is avoided.

In our approach to analyzing the data, we
used the discriminant analysis to characterize the
microhabitats used by a particular species and to
determine how much of the variance between where a
species did and did not occur could be accounted
for by the combination of variables in the
discriminant function. We were, therefore,
interested in the suite of characters which
described the microhabitats used by a species and
we were not concerned about whether or not the
variables were intercorrelated.

The variables in the discriminant function
which characterized the species' microhabitats,
however, may or may not be the variables to which
the animals respond. The subsequent multiple
regression and partial correlation analysis was
then used to hypothesize why the animals occurred
in particular microhabitats and not in others.
For this phase of the analysis, however,
independent habitat variables were required. We
preferred ridge regression over principal
components to obtain a set of independent habitat
variables because, as already mentioned, we were
able to use the habitat variables which were
actually measured rather than using statistical
habitat variables, principal components, which
themselves require interpretation.

PAUL GEISSLER: If I understand what you said, the
difference between the conclusions associated with
the types of discriminant functions is that
significant univariate variables can be inter-
preted individually while nonsignificant variables
cannot. It seems to me that whether or not
variables can be interpreted individually depends
on whether or not it is correlated with other
variables, and not whether or not a relationship
can be demonstrated with the response variable.
Also it seems to me that limiting the discriminant
functions to significant variables may cause one
to miss complex relationships which depend on more
than one variable.

JANET CAVALLARO: The different types of

230

discriminant functions do not determine whether or
not the individual variables in the discriminant
function can be interpreted; the different types
of discriminant functions determine how the
individual variables must be interpreted. The
discriminant function is a new "habitat" variable
which is a sum of weighted habitat variables, not
a response variable. If the discriminant function
contains only significant univariate variables
(Type I), the discriminant function can be
interpreted as separating microhabitats which have
more of variable X, less of variable Y, and more
of variable Z from microhabitats which have less
of variable X, more of variable Y, and less of
variable Z, If the discriminant function contains
insignificant variables as well (Type II), the
discriminant function can be interpreted as above
plus the fact that in microhabitats used by a
particular species, variable A has a certain
average value. The hypothesis generated from a
Type II discriminant function, therefore, may
include the idea that variable A, the
insignificant univariate variable, must be at a
certain threshold level before the animals of a
species will use a certain habitat. If that is
the case, however, a significant difference should
exist in variable A between where animals do and
do not occur if a broader range of habitats were
sampled. Thereafter, a Type I discriminant
function may well be developed to separate the
microhabitats used and not used.

In our study in the chamise-ceanothus
chaparral, ceanothus density did not differ
between where Peromyscus truei did and did not
occur on the 25-year-old plot, and yet ceanothus
density was a variable in the Type II discriminant
function. £. truei occurred where ceanothus
density equalled 1.7 in contrast to 1.5 where P_.
truei did not occur. It was not clear to us why
P. truei should select on the average
microhabitats with a ceanothus density of 1.7 in
contrast to 1.5 when those two values were
essentially no different. On the 8-year-old plot

ceanothus density was much less, however, and
P. truei occurred where the ceanothus density was
significantly greater, 1.2 in contrast to 0.2
where they did not occur. From these results on
the 8- and 25-year-old plots, we can comfortably
hypothesize that P^. truei requires a minimum
density of ceanothus before they will occupy the
chamise-ceanothus habitat. This is analagous to
the observations of McCabe and Blanchard (1950)
who found that P_. truei would not use pure stands
of Baccharis pilularis but that they would use the
habitat when shrubs of other species were also
present. The Type I discriminant function for
the 8-year-old plot enables us to develop the
above partial hypothesis much more easily than did
the Type II discriminant function for the 25-year-
old plot.

By restricting the discriminant function to
significant univariate variables, complex
relationships between animals and their habitats
potentially can be missed; but as we have
suggested, this will depend to a large extent on
the range of habitats sampled. If a broad enough
range of habitats is sampled, any complex
relationship should be detected through a Type I
discriminant function as well as through a Type II
or III discriminant function. In addition, we
need to consider whether or not a Type II or III
discriminant function will necessarily enable us
to better detect complex relationships between
habitat variables and a particular species. The
Type II or III discriminant function suggests that
animals are using very precisely defined habitats
and so for those discriminant functions to have
ecological meaning, we think the measured habitat
variables must be very close measures of variables
to which the animals are actually responding. We
question how often variables perceived by animals
when they make their habitat selection are
actually measured; and we therefore question
whether or not many of the Type II or III
discriminant functions are not just artifacts of
our sampling.

231

A DESCRIPTIVE MODEL OF SNOWSHOE HARE HABITAT
Kathryn A. Converse ^ and Bernard J. Morzuch^

Abstract. — The snowshoe hare (Lepus americanus) on the
southern edge of its range in Massachusetts provided an
opportunity for use of multivariate analysis to model
selection of common habitat components over an extensive area
during the critical winter months for comparison with
preferred habitat. Ten areas were chosen for variability in
their vegetation structure and composition within two
ecological zones of the Berkshire Mountains in western
Massachusetts. Track counts were used as indices of hare
distribution and regressed on measurements of ground cover;
shrub volume; shrub and sapling frequency, richness and
composition; tree composition, frequency and basal area; and
snow depth.

A linear regression model was used to analyze these
hypothesized relationships. The technique of principal
components is suggested as a method of dealing with
collinearity among independent variables. Tests were
performed to judge the appropriateness of pooling data from
different study areas. Seemingly unrelated regression was in
turn, used to analyze the possibility of mutual correlation
within the error structure.

Key words: Forest habitat; Massachusetts;

multicollinearity ; pooling cross section data; principal
components; regression analysis; seemingly unrelated
regression; Snowshoe hare.

INTRODUCTION

Massachusetts has over 5 million acres of
land with 3 million acres of forests classfied
into 40 types. In 1971-1972, 76% of these forests
contained trees greater than 12 m in height and at
present 45% of the state is harvestable

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980, Burlington, Vt .

^M.S. Candidate, Wildlife Biology, Department
of Forestry and Wildlife Management, University of
Massachusetts, Amherst, MA 01003.

'Assistant Professor, Department of Food and
Resource Economics, University of Massachusetts,
Amherst, MA 01003.

(MacConnell 1975). In addition to increasing
forestry operations, there are continuing losses
of wildlife habitat due to clearing for housing,
commercial and recreational developments, and
drainage and filling of wetlands. It is essential
that wildlife managers become concerned with the
effects of such practices and develop methods for
location and description of extensive areas of
wildlife habitat for game and nongame species.
One possible method that addresses these issues is
division of the state into ecological zones
(Sczerzenie 1980).

The snowshoe hare ( Lepus americanus ) is
diversely distributed across Massachusetts from
the oak-pitch pine barrens of Cape Cod to the
spruce-fir thickets of the Berkshire Mountains.
Some hare may have survived transplantation from

232

New Brunswick, a management plan in practice since
1919. It is assumed that survival of these hare
was a function of inherent selective mechanisms
for corresponding habitat components. Previous
studies have shown hare to use a variety of
conifer and hardwood stands of all ages, species,
and interspersions. In a marginal southern range
such as Massachusetts, it is difficult to specify
precisely the variables to be used in a habitat
model.

This paper discusses a study conducted May
1978 to March 1980 where the physical features of
a variety of forest types within two ecological
zones in western Massachusetts were sampled
intensively. A linear regression model was used
to analyze the hypothesized relationships between
hare activity and selection of common habitat
components during the critical winter months.

STUDY AREAS

Known hare habitat was determined through
correspondence and interviews with district
wildlife managers, hare hunters, and local
residents. Ten research areas were chosen along a
90-km circuit through eight towns located in
Berkshire, Hampshire, and Franklin counties in
western Massachusetts. Five areas were chosen
within each of two ecological zones — transitional
(zone 1) and central Berkshires (zone
2) — designated by physiographic, vegetative, and
demographic variation (Sczerzenie 1980) (fig. 1).
Suitability of all areas was determined by field
investigation of size, permission of private land
owners, year-round road access, distance from
residential or recreational disturbance,
vegetation structure and composition variability,
presence of browse and/or fecal pellets, and
ability to census all areas on snowshoes within 24
hours.

The most recent land use inventory
(MacConnell 1975) indicated that the zone 1 study

Figure 1. Location of the ten research areas
within the eight Massachusetts towns and two
ecological zones.

areas fell within predominately hardwood forests,
while zone 2 study areas were predominantely
softwood forests (table 1). Forest maturity was
confirmed in both zones by the 12.5-18.6 m height
class and by high density, 81-100% crown closure.
When the study areas were viewed separately,
forest types of documented importance to the hare,
e.g., spruce, became more significant.
Discontinuity within habitat types was not usually
reflected on land use maps, but the resulting
edges provided juxtaposition of food and cover.

METHODS

Habitat Measurements and Hare Activity

Transect lines (total 400 m) were located
randomly on each area with permanent stations
every 25 m and plot centers each 50 m. Vegetation
was measured September 1978 through July 1979 on
all transects at each 50 m plot center.
Frequency, composition, and basal area of trees 10
cm dbh and above were measured using a 10 factor

2

prism plot. Circular plots, 40.5 m , were used to
determine frequency, composition and richness of
shrubs 30.5 cm high to 7.5 cm dbh, saplings 7.6 cm
to 9.9 cm dbh, and shrub volume for mountain
laurel (Kalmla latifolia) and yew (Taxus
canadensis) . Percent ground cover less than 30.5
cm high was visually estimated on four 2 m x 2 m
quadrats and then averaged.

An understory density screen 1 m wide by 2 m
high (Telfer 1974, Oldenmeyer 1975) gridded into
5% blocks was placed 15 m from and at right angles
to the transect line (Nudds 1977). The screen was
photographed and the lower (0-1 m) and upper (1-2
m) percent obscured by vegetation recorded as
maximum density in July and minimum in March.
Habitat variables used in the analyses are
presented below:

RICH = number of species in the shrub

and sapling plot,
EVER = evergreen trees >10 cm dbh/0.01

ha,

HARD = hardwood trees >10 cm dbh/0.01
ha, 2

BA = basal area m /ha,

SHRUB = shrub volume 100 m /ha,

EDBHO = evergreen stems >30.5 cm high,
up to 2.5 cm dbh/0.01 ha,

HDBHO = hardwood stems >30.5 cm high, up
to 2.5 cm dbh/0.01 ha,

EDBH1 = evergreen saplings 2.6-9.9 cm
dbh/0.01 ha,

HDBH1 = hardwood saplings 2.6-9.9 cm
dbh/0,01 ha,

MINL = lower percent minimum under-
story,

MINU = upper percent minimum under-
story,

MAXL = lower percent maximum under-
story,

GCl = percent ground cover in annuals.

233

GC2 = percent ground cover in

perennials,
GC3 = percent bare ground (leaves,

duff and rock) .

Hare tracks intersecting the transect line
were totalled and snow depth measured for each 50
m section after each fresh snowfall (Hartmann
I960, Brocke 1975). Problems of high winds
drifting fresh snow, and the change from snow to
sleet along the altitudinal gradient eliminated
many observation days. If any sections of a
transect line were not trackable the count for the
whole line was not used.

Track counts were used as indices of hare
cover preference because they can be measured
easily by one person, provided for analysis by
habitat, and are good for extensive studies where
only 'relative population levels are needed.
Tracks provide an index to the average number of
hare using an area part of the time or as part of
their home range (Adams 1959, Hartman I960).

Statistical Model

Ordinary Least Squares

Multivariate ordinary least squares (OLS)
regression techniques were used to determine the
relationships between hare activity as a dependent
variable and weather and habitat measures as
independent variables.

The hypothesized relationship was assumed to
be linear and of the form:

'O * hh^

. + 6, X. , + e .
k ik 1

(1)

.th

where Y. = the i

n observations on the dependent

value of the entire set of
•vati
variable Y;

X. .

the i value of the j
independent variable X., where j
ranges from 1 to k;

th

Table 1. Dominant species of forest layers and corresponding forest classification system
(MacConnell 1975) for zones 1 and 2.

Research
areas

Forest
types

Tree
>10 cm dbh

Saplings
7.5-10 cm dbh

Shrubs
up to 7.5 cm dbh

Zone 1

0

HS3A

SH3A

Bi

H

Be

Wh

Be

M

L

Wh

Be

3

S3A

SH3A

H

M

S

H

H

S

M

H3A

SH2A

0

H

Wh

M

Be

L

Be

Wh

8

SH2A

HS3A

H

M

Wp

M

BI

M

L H

9

HS3A

M

H

Bi

Be

H

L

Wo

H M

Zone 2

1

SH2A

HS3A SH3A

H

M

S

M

H

L

Y

M

2

SH2A

SH3A

Wp

S

M

M

Wp

Sb

M

S

5

S3A

SH3A HS3A

S

M

S

M

S

BI

6

HS3A

HS2A H2A

M

Bi

H

Bi

S

M

M

S

7

S3A

SH2A SH3A

S

S

F

S

81

M F

' Type:

H = hardwood and S =

softwood ,

80% stands;

mixed

HS

= hardwood

and

SH

- softwood

predominating

Height: 2= 6.4-12.4 m; 3= 12.5-18.6 m
Crown closure: A= 81-100?; B= 30-80?

H=hemlock, M=maple, F=fir, S=spruce, Bi=birch, O=oak, Be=beech, Wp=white pine,
Wh=witch hazel, L=laurel, Bl=blueberr y, Wo=witch hobble, Sb=steeplebush , Y=yew.

234

= the j^*^ unknown parameter to be
estimated ;

B = a constant term;
o

e. = the i^*^ value of the set of n
random disturbances about the mean
of Y.

Equation (1) can be conveniently expressed in
matrix notation as

Y = xe_ + £ (2)

where Y = an n x 1 observation vector;

X = an n X k matrix of independent
variables ;

6_ = a k x 1 parameter vector on X;

£ = an n X 1 vector of error terms.

Assumptions of the model are that X is of rank k
and that each is normally and identically

distributed, independent of the other disturbance
terms, and has mean zero and constant variance a^.

Violation of any of these assumptions will
render OLS estimation inadequate. Specifically,
difficulty with respect to the rank condition of
the X matrix, i.e., multicollinearity , requires
the implementation of some "treatment." Likewise
problems with respect to the error term, e.g.,
heteroscedasticity or autocorrelation, require
some manipulative corrective measure to the data
before OLS can be applied.

Pooling Data

A very real problem when dealing with
cross-section data and model specification in this
case is the matter of organizing the data between
the two zones for estimation purposes. That is,
the model for each zone relating hare activity to
weather and habitat measures can be expressed
generally as in equation (2), and the question
revolves around the manner of implementing the
entire set of data from both zones to get the most
efficient parameter estimates.

0

X,

(5)

A most important feature of this model is that
arranging the data in this fashion does not
restrict the parameters of the independent
variables in zone 1 to be equal to those in zone
2, i.e., there are 2k parameters to be estimated
in the unrestricted model represented by equation
(5).

The inclination of many researchers in such a
situation is to "stack" equations (3) and (4) as:

-1

X,

(6)

Notice that estimation of this model imposes
severe resrictions upon the two parameter vectors
of equations (3) and (4), i.e., B.-]=^2"— *

Specifically, the implication is that any
particular parameter Q. in zone 1 is equal to the

corresponding Bj in zone 2. It is suggesed that,

rather than estimating only the stacked model of
equation (6) and naively imposing severe
restrictions on the parameter space, alternative
model specifications ought to be tested against
each other and one judged more appropriate than
another on the basis of some statistical
criterion. One such criterion is the classical
F-test (Fisher 1970).

Briefly, the idea is to calculate the
residual sum of squares of the unrestricted model
(RSSy) represented by equation (5) and make

Let equation (2) be rewritten to represent
zone 1 as

comparisons with the residual sum of squares of

the restricted model (RSS„) in equation (6). The

n

Y.
—1

X,B

1^1 ^ ^1

(3)

format for making these comparisons is;

where equation (3) is identical to equation (2)
except that the subscript 1 relates to zone 1
information. Correspondingly, the model

representing zone 2 is

h - ^^2 ^ ^2-

(4)

Estimating the model for zone 1 and zone 2
individually by OLS amounts to the following
formulation:

F =

(RSSp-RSSy) / (DFj^-DFy)
RSS^j / DFy

(7)

which is distributed as central F (centrality

parameter equal to zero) with (DF -DF ) and DF

n U U

degrees of freedom for the numerator and
denominator, respectively. Notice, furthermore,
that DFj^ and DF^ are the degrees of freedom

235

associated with the restricted and unrestricted
models, respectively.

Under the null hypothesis that 6.^=6.2=6.. i.e.,

that the restrictions suggested by equation (6)
hold, versus the alternate hypothesis that
^1 ^ A2' i-®*' that the restrictions do not hold,

equation (7) has particular appeal. Simply
stated, if RSS^ and RSSy are "close" to each

other, the implication is that the restrictions
are not in conflict with the sample data. This is
reflected in a small calculated F-statistic
relative to a critical F such that the null
hypothesis cannot be rejected. If, on the other
hand, RSS^ diverges greatly from RSS^j, the

calculated F-statistic will exceed the critical F
and the null hypothesis will be rejected, implying
that the restrictions are in conflict with the
sample data.

Seemingly Unrelated Regression

Once either the stacked or unstacked model
has been selected as the appropriate
specification, an important issue revolves around
whether or not the OLS procedure used for the
estimation of the selected model leads to the most
efficient parameter estimates. That is, even
though 1 ) the system has been purged of any
autocorrelation or heterscedasticity relating to
the error structure and 2) possible restrictions
relating to the parameters have been tested,
additional information may still be gleaned from
the estimated error vectors e^ and e_^. In fact,

up to now, the assumptions relating to the error
terms of either model (5) or model (5) have been
very restrictive in the sense that they do not
allow for mutual correlation between e^^ and e^2'

Testing whether or not the error vectors of the
selected model (5) or (6) above are actually
unrelated or only seemingly unrelated involves a
generalized least squares procedure (Zellner
1962). Basically, testing for mutual correlation
involves comparing the estimate of the variance of
the selected model (5) or (6) above (call this

^2

estimate o 0^3) with the estimate of the variance

of that same model cast in a generalized least
squares or seemingly unrelated regression

framework (call this estimate S

The

hypothesis to be tested is one of homogeneity of

v2

variance with the null hypotheis being 3

OLS

^2 ^2

o gypj and the alternate hypotheis being a >^

o <,,,„. The appropriate test statistic is

/v2 ^2

^ " ° OLS^° SUR

^2 ^2
If a is much larger than o the null

hypothesis of no mutual correlation must be
rejected .

The appeal of the seemingly unrelated
regression approach is that the off-diagonal
elements of the variance-covariance matrix of the
error term are not restricted to zero (as in the
ordinary least squares case) due to possible
mutual correlation. Through the generalized least
squares algorithm, the degree of mutual
correlation in the error structure is analyzed.
If there exists little or no mutual correlation,
the zero restrictions on the off-diagonals hold,

/v2 ^2
and o approaches a q^^- On the other hand,

the existence of mutual correlation in the error
vectors implies the admissibility of additional
information about the system which in turn implies

^2 ^2

that a < a and allows for more efficient

parameter estimates.

Principal Components

Collinearity among the independent variables
can be detected by observing rather high pairwise
correlations between explanatory variables and
also by performing auxiliary regressions of
individual independent variables on the remaining
set. This latter test is suggested by Farrar and
Glauber (1967).

The way to deal with multicollinearity is to
introduce additional sample information to
hopefully increase the selective variation among
the independent variables. Such information is
normally added by way of restrictions on the
parameters suggested by theory. These
restrictions may take the form of exact linear
restrictions (Goldberger 1964: 256-8), inequality
restrictions (Judge and Takayama 1966), and
stochastic restrictions (Theil and Goldberger
1961).

In the absence of any theoretical basis for
admitting restrictions on the parameters, an
alternative is to place restrictions on the
independent variables themselves. This can be
accomplished by transforming the original
variables into artificial constructs that are
orthogonal to each other and then retaining
certain of these constructs in a regression model
on the basis of their contribution to variability
in the original data, while eliminating — placing
zero restrictions on--those constructs that
contribute little or nothing to the variability in
the original data. This is precisely the focus of
principal components.

In principal components regression, a
transformed variable is determined to be important

236

and included or unimportant and excluded in the
regression model depending upon the size of the
characteristic root (eigenvalue) associated with
its corresponding characteristic vector
(eigenvector) (Massy 1965), the statistical
significance of its regression coefficient
( Mittelhammer and Baritelle 1977), or some
combination of eigenvalue size and correlation
with the dependent variable (Johnson et al . 1973).

A complication results with respect to the
optimal number of components to delete. The
tendency traditionally has been to delete
components asociated with small eigenvalues, e.g.,
less than one. The limitation of this approach is
that components with small eigenvalues may be
correlated very highly with the dependent
variable. Thus, a structural norm which
simultaneously considers the amount of variability
accounted for by a particular component and its
correlation with the dependent variable has
greater appeal. A particular norm which accounts
for these two measures is the F-test described in
equation (7). Components can be sequentially
deleted until a new restriction, i.e., the
deletion of an additional component, causes no
improvement with respect to the fit of the
equation. Finally, once the "optimal" number of
components has been deleted, the principal
component estimators can easily be transformed to
estimators on the original independent variables
(Johnson et al . 1973).

RESULTS AND DISCUSSION

OLS regression models were estimated
individually for zone 1 (n=384) and zone 2 (n=480)
in the spirit of equation (5) and with the zones
stacked (n=864) as suggested by equation (6).
Preliminary to performing these regressions, any
problems with respect to heteroscedasticity among
the error variances of the individual areas were
corrected. The model specifications were tested
using equation (7) to determine if the parameter
restrictions held. The F-values were significant
using the central F-test. This separation was
consistent with development of these ecological
zones and indicates that there were major habitat
differences .

Seemingly unrelated regression (SUR) was done
on the unstacked data and the possibility of
mutual correlation explored. The disturbances in
the two equations were found to be not mutually
correlated, and the SUR parameters estimators were
the same as the OLS estimators. The above tests
showed that both the set of explanatory variables
and disturbances between zones were not
correlated, and there was no justification for
combining the data.

Multicollinearity among the explanatory
variables was indicated by significant
relationships in the simple correlations and
auxilliary regressions. Almost perfect

collinearity existed between each of the three
ground cover measurements (GC1, GC2, AND GC3) and

the remaining variables. This was expected, as
ground cover was a function of the amount of light
reaching the- forest floor which, in turn, was
determined by canopy closure and understory.
Basal area (BA), a measure of canopy closure, was
also collinear with the rest of the variables.
Furthermore, collinearity among the independent
variables was confirmed in that the OLS
regressions for each zone had high standard errors
resulting in non-significant t-statistics.

Principal components regression (PCR) was
done on each zone. Equation (7) was used as the
deletion criterium with one component deleted from
zone 1 and four from zone 2. Because information
was removed from the models, the t-tests were
recognized only as indicators of more precise
estimators and suggest variables that may be most
important habitat descriptors (Freund 197'+).
Final PCR model coefficients and significance
tests appear in tables 2 and 3. The fact that
zone 1 was an area of transition between the
Connecticut River Valley lowlands and upland
Berkshire Mountains indicated that it would be
difficult to find common parameters within this
zone's heterogeniety .

The final model for zone 1 had eleven
significant effects while zone 2 had eight. Three
variates of major importance were retained with
the same sign in both models. Species richness
(RICH), indicating open areas, and basal area
(BA) , or canopy closure, had negative effects on
hare abundance while hardwood shrub frequency
(HDBHO) had a positive effect. Four additional
variable effects were significant in both zones
but carried opposite signs: number of evergreen
(EVER) and hardwood (HARD) trees, number of
evergreen saplings (EDBH1), and upper percent
winter understory (MINU). Sample data were
carefully examined for the possibility that the
signs changed in response to a particular
sensitivity in habitat measurements. The number
of hardwood trees appears to have a sample
frequency that could explain the change in signs
(fig. 2), but because zone 1 already had more
hardwood trees, the only interpretation was that
increasing numbers of hardwood trees in zone 1 is
related to increased hare activity, while in zone
2 the inverse held. The histogram in figure 3
demonstrates the relative frequencies for
evergreen trees was similar in both zones, as they
were for EDBH1 and MINU, so the same effects were
expected but not found.

One hypothesis for the different effects
between zones is that the low hare track count of
298 for zone 1, compared with 2226 tracks for zone
2, had low information content for selection of
variable effects and the quality of the parameter
estimates were questionable. When EVER and HARD
were plotted on a three dimensional graph (fig. '+)
the same frequency distributions were seen but in
addition the low track count values of zone 1 were
evident.

There are possible explanations for the
different variable effects between zones based on

237

Table 2, Principal components regression model of
the relationship between hare activity and
selection of habitat in zone 1.

Table 3. Principal components regression model of
the relationship between hare activity and
habitat component selection in zone 2.

Variable'
Names

Estimated
coefficients

T- ratio^

Variable'
Names

Estimated
coef f icients

T- ratio^

RICH

-0. 1 15

-1.933»

RICH

-0. 148

-2.337**

SNOW

0.010

2.292»»

SNOW

-0.013

-1.523

EVER

0. 310

7.238»» —

EVER

-0. 120

-2.240»»

HARD

0.088

2.347«»

HARD

-0.315

-3.732»»

BA

-0. 064

_4.796»«

BA

-0.040

-2.816**

SHRUB

-0.001

-0. 321

SHRUB

0.008

0.512

EDBHO

-0.009

-1 .745*

EDBHO

-0.006

-0.968

EDBHl

-0.028

-2.471**

EDBHl

0.066

3. 166**

HDBHO

0.003

3.189**

HDBHO

0.010

2.761**

HDBH1

-0. 002

-0.419

HDBH1

0.096

5.430»»

MINL

0.014

3.056»»

MINL

-0,012

-1 .074

MINU

-0. 016

-2. 382**

MINU

0. 051

2. 747**

MA XL

-0.012

-2.472**

MA XL

0. 007

0.690

GC1

0.001

0.101

GC1

-0.019

-1.294

GC2

-0.006

-0.679

GC2

-0.020

-1.233

GC3

-0.001

-0, 103

GC3

0.016

1.698*

Intercept

1.459

Intercept

3.181

' see text for

variable description

' see text for

variable description

2 »» P<0.01, » P<0.05

2 »• P<0.01, * P<0.05

field observation. For example, median snow
depths were the same for both zones while the
maximum depth was 41 cm deeper on zone 2. Heavier
snows in zone 2 had a greater immediate effect on
the habitat. Accumulation of snow on tree
branches and crowns bent them down within reach
for browsing, and often broke them off entirely
providing tips, and buds for food on top of the
snow. Shrubs and saplings became completely
covered with snow in some storms and hare burrowed
under them and used these areas for food and
shelter. Snowfalls in zone 1 were lighter and
melted faster with less accumulation, due to more
mature trees and denser canopy intercepting the
snow, preventing most tree damage. This
relationship between snow and vegetation in the
two zones explained the hares' selection of
different components in the sapling (EDBHl and
HDBH1) and density (MINL, MINU and MAXL) classes.

Additional hypotheses could be suggested

pertaining to differences between the models for
each zone, e.g., response to different species of
vegetation, interspecific competition between hare
and white-tailed deer (Odocoileus virginianus) ,
and the predator-food complex, but the important
point has already surfaced. If the parameter
estimates of these models had been restricted by
pooling the two data sets, none of these
differences between zones and ultimately hare
habitat would have been qualified. Opposite
effects for the same variable between zones would
have eliminated detection of some significant
habitat descriptors and given poor quality
estimates for others. This problem has severe
limitations when these data are being used as a
basis for designing and implementing management
plans. Too often in wildlife research, extensive
habitat management plans are based on local
intensive studies. Wildlife managers cannot
afford to have unrealized negative effects of
habitat manipulation surface, by either short-term

238

/

15

HARDWOOD TREES /0.01 hectare

Figure 2. Comparison of frequency distributions
of hardwood trees > 10 cm dbh for both zones.

Figure 4. Three dimensional plot of the
relationship between hare track counts and
number of evergreen and hardwood trees > 10 cm
dbh for both zones.

losses of the wildlife in that habitat or long
term destruction that will discourage potential
immigrating populations.

15-

EVERGREEN TREES/a01 hectare

Figure 3. Comparison of frequency distributions
between evergreen trees > 10 cm dbh for both
zones .

More specifically, the two zones in this
study should be managed differently assuming the
parameter estimates are valid. Zone 2 model
estimates agreed with our observations of hare
activity over the past 2 years, and were
consistent with earlier studies in good hare
range. The predominately hardwood habitat in zone
1 has only been studied quantitatively recently,
and not in areas with the same species composition
as this study. Sheldon (1957) stated that northen
hardwood-laurel, and hemlock-laurel habitats in
zone 1 were the most widespread and important in
Massachusetts yet in this study they have the
lowest populations. The zone 1 model should be
tested with more data to see if the parameter
effects hold or will move closer to the other zone
model .

ACKNOWLEDGMENTS

This study was jointly supported by the
Massachusetts Cooperative Wildlife Research Unit
of the U.S. Fish and Wildlife Service,
Massachusetts Divi^sion of Fisheries and Wildlife,
Wildlife Management Institute, and the University
of Massachusetts.

LITERATURE CITED

Adams, L. 1959. An analysis of a population of

snowshoe hares in north western Montana.

Ecological Monographs 29:141-170.
Brocke, R.H. 1975. Preliminary guidelines for

managing snowshoe hare in the Adirondacks.

Transactions Northeast Section of the

Wildlife Society 32:46-66.

239

Farrar, D.E., and R.R. Glauber. 1967.

Multicollinearity in regression analysis:

the problem revisited. Review of Economics

and Statistics 49:92-107.
Fisher, P.M. 1970. Tests of equality between

sets of coefficients in two linear

regressions: an expository note.

Econometrica 38:361-366.
Freund, R.J. 197^. On the misuse of significance

tests. American Journal of Agricultural

Statistics 56:192.
Goldberger, A.S. 1964. Econometric Theory. 399 p.

John Wiley and Sons, New York, N.Y.
Hartman, F.H. I960. Census techniques for

snowshoe hares. M.S. Thesis. 46 p. Michigan

State University, East Lansing, Mich.
Johnson, S.R., S.C. Reimer, and T.P. Rothrock.

1973. Principal components and the problem

of multicollinearity. Metroeconomica

25:306-317.

Judge, G.G., and T. Takayama. 1966. Inequality
restrictions in regression analysis. Journal
of the American Statistical Association 61:
166-181.

MacConnell, W.P. 1975. Classification manual:
remote sensing 20 years of change in
Massachusetts 1952-1972. 23 p. Massachusetts
Agricultural Experiment Station Resource
Bulletin No. 631.

Massey, W.F. 1965. Principal components
regression in exploratory statistical
research. American Statistical Association
Journal 60:234-256.

Mittelhammer, R.C, and J.L. Baritelle. 1977. On
two strategies for choosing principal
components in regression analysis. American
Journal of Agricultural Economics 59:336-343.

Nudds, T.D. 1977. Quantifying the vegetative
structure of wildlife cover. Wildlife
Society Bulletin 5:113-117.

Oldemeyer, J.L. 1975. Characteristics of paper
birch saplings browse by moose and snowshoe
hares, p. 53-60. l£ Eleventh North American
Moose Conference and Workshop. Winnipeg,
Manitoba .

Sczerzenie, P.J. 1980. Ecological zoning of

Massachusetts for wildlife management.

Transactions of the Northeast Section of The

Wildlife Society 37:104-112.
Sheldon, W. 1957. Snowshoe 'hare in

Massachusetts. p. 12-15. Iji Massachusetts

Wildlife, January-February.
Telfer, E.S. 1974. Vertical distribution of

cervid and snowshoe hare browsing. Journal

of Wildlife Management 38:944-946.
Theil, H.. and A.S. Goldberger. 1961. On pure

and mixed statistical estimation in

economics. International Economic Review

2:65-78.

Zellner, A. 1962. An efficient method of
estimating seemingly unrelated regressions
and tests for aggregation bias. American
Journal of the Statistical Association 57:
350-367.

DISCUSSION

GEORGE BURGOYNE, JR.: Would you care to comment
on the justification for treating a large number
of observations collected at five sites in two
regions as completely independent observations.
The fact that only ten sites were used would be
expected to yield more homogenous results than if
the same number of transects were really located
at random. It is difficult to believe that two 50
meter transects adjacently located are as
independent as two 50 meter transects independ-
ently, randomly located.

KATHRYN CONVERSE: Vegetation frequency,

composition, and structure was heterogenous along
the ten transect lines. Any problems of the
dependence with repeated observations was
corrected by orthogonalizing variables using
Principle Components, Certainly adjacent 50 m
transects are less independent than random 50 m
transects, but they did allow for repeated
detailed observations of changing hare activity
and habitat use with varying weather conditions as
the winter progressed. Intensive vegetation
measurements and winter tracking over 960 randomly
located transects would have been logistically
impossible for one person.

KEN MORRISON: Being familiar with animal
tracking, I have a gut feeling that the error in
measurement increases with increasing activity
levels, suggesting that a transformation, perhaps
log, would be appropriate? Is it possible that
this could have been the case in your study?

KATHRYN CONVERSE: Indeed, a log transformation
may be appropriate. We failed to test among
competing functional specifications for our model.
We merely assumed linear relationships.

KEN MORRISON: The number of crossings will be a
function of time since the last snowfall. When
you were sampling, is it possible that you
consistently sampled one area before the other,
thereby creating a biased difference between the
two?

KATHRYN CONVERSE: Two sets of five study areas
that were a mixture from both zones were tracked
in alternating order after each snowfall. I think
this would eliminate any bias between zones.

DOUGLAS INKLEY: Do you think that snow depth may
have had an effect on hare mobility, and therefore
different regional track counts may have been due
to both population differences and the effect of
snow depth on mobility?

KATHRYN CONVERSE: Regional hare activity was
definitely influenced by snow depth, consistency,
and accumulation. Inclusion of these snow
conditions as variables in this model helped to
explain changing food and cover availability by
habitat types throughout the winter months.

240

MARflN RAPHAEL: How do you derive the individual
parameter estimators, i.e., how do you get unique
estimators using principal components?

BERNARD MORZUCH: The process is very mechanical
and straightforward (See Morzuch, B.J. 1981.
Principal components and the problem of multi-
collinear ity . Journal of the Northeastern
Agriculture Economics Council. In press.).

LESLIE MARCUS: The "seemingly unrelated
regression" application to the two zone hare data
seems inappropriate. Since the individual plots
between zones ^"'^ ''21^ ^""^

"contemporaneous elements" (Zellner, A. 1963.
Estimators for seemingly unrelated regression
equations: some exact finite sample results.
American Statistical Association Journal
57:350-367.), there is no reason to relate them.
The value of C^C^ calculated to estimate a^^

depends on the permutation of the and

observations, and over permutations E(C^C2) = 0.

Since it might happen that i depending on

sampling design etc., this is an unnecessary
application of the interesting idea. Note if the
sampling sites had been sampled in the same order
in the two zones or at similar times of the day,
this might be a useful test. The way Ms. Converse
explained the data collection it does not seem
appropriate.

BERNARD MORZUCH: As background to answering this
question, two entirely different zones (within

which five areas were identifed, with eight plots
per area) were delineated on the basis of
vegetation and elevation. There was no attempt to
match a particular plot in zone 1 against any one
plot in zone 2. Alternatively, performing all
possible permutations to find one ordering that
provided the highest degree of mutual correlation
was not the central theme of this analysis.

Relative to the question, who is to say that
ordering the cross sections in zones 1 and 2 in a
particular fashion to make certain that we have a
matching of contemporaneous elements is the one
correct ordering so as to make use of Zellner' s
framework. Given the heterogeneous nature of
plots between zones and the lack of theory to
suggest a proper matching, alternative orderings,
i.e., random or otherwise, are appropriate.

The purpose of the study was to establish a
functional relationship between hare distribution
and habitat variables. The resulting estimators
of the pooled model have, at least, nice
asymptotic properties. Therefore, any additional
testing of the restrictive assumptions of the OLS
model that might lead to greater efficiencies in
the estimators would be most welcome. Such is the
purpose of testing for mutual correlation in a
seemingly unrelated (SUR) framework. Applying SUR
in this fashion is, most certainly, appropriate
and not an unnecessary application of an
interesting idea. Given our ordering of
contemporaneous elements, we were unable to
conclude the presence of mutual correlation. This
is not to say that another ordering would lead to
the same conclusions on efficiency grounds.
However, consistency and asymptotic unbiasedness
of the estimators would remain unaffected under
any ordering.

241

A DISCUSSION OF ROBUST PROCEDURES IN
MULTIVARIATE ANALYSIS'
Lyman L. McDonald^

Abstract. — Robust procedures in multivariate analysis
are briefly reviewed. Conjectures are given for applications
of existing methods in other areas of multivariate analysis.

Key words: Discriminant analysis; multicollinearity ;
outliers; principal components; robust procedures.

Upon review of titles of papers to be
presented at this meeting, my first reaction was
to discuss a combination of three subjects: 1)
the danger of "overf itting" multivariate data with
descriptive procedures such as principal
components, factor analysis and canonical
correlation analysis; 2) the ultra-conservative
nature of inference procedures in multivariate
analysis of variance (MANOVA) and in multivariate
regression analysis (i.e., simultaneous regression
of several dependent variables on a collection of
independent variables); and 3) the need for
"robust" analysis procedures in multivariate
studies .

The first of my concerns is addressed by Karr
and Martin (1981). Their observations reinforce
my experiences and I endorse their suggestion that
"One possibility might be a table of expected
amount of variation accounted for by random number
matrices that can be used as a test relative to
the amount of variation accounted for in real
data." Actually tables of upper percentage points
in the null distribution of variance accounted for
by random number matrices would be more
appropriate for a formal test of hypothesis, but
the idea is the same.

The second of my concerns is not of much
interest in the papers presented. MANOVA is
mentioned rarely and insofar as I am aware, the

'Paper presented at The use of multivariate
statistics in studies of wildlife habitat: a
workshop, April 23-25, 1980. Burlington Vt.

^Professor, Departments of Statistics and
Zoology, University of Wyoming, Laramie, WY
82071.

corresponding simultaneous inference procedures
are not strongly recommended.

I emphasize, then, the third topic: a need
for robust procedures. This need becomes obvious
when reading several of the papers. For example,
Williams (1981) correctly pointed out that an
important aspect of canonical variates in
discriminant analysis is the ecological meaning of
the coefficients. However he noted that "Most
people who have worked with discriminant analysis
have probably seen cases in which positively
correlated variates have canonical coefficients
with different signs," and quoting from Weiner and
Dunn (1966), "For complex structures, magnitudes
and even signs of coefficients are dependent on
what additional variables are included in the
model." In a study of discriminant analysis by
use of multiple regression methods, Cavallaro et
al. (1981) stated that "Multicollinearity can
cause the regression coefficients to be inflated
in absolute value or even to have the wrong sign."
Smith (1981) in his paper on canonical correlation
analysis noted that "Most often coefficients are
exceedingly difficult to interpret, if not
meaningless (Cassie and Michael 1968). . .."
Also, ". . .Cassie (1969) found that adding and
subtracting variables at random 'capriciously'
changed the values of the corresponding canonical
coefficients." Harner and Whitmore (1981) stated
that "Outliers in multivariate data can have
pronounced effects on the interpretations and
conclusions of statistical analyses."

With the exception of Harner and Whitmore
(1981) and Cavallaro et al . (1981), little
information has been given in these proceedings to
help the reader understand and solve these
problems when they arise in a particular

242

application. Instead, the reader is warned that
problems may exist and in some cases analysis
procedures for detection of the need for more
robust methods have been given (e.g., data
splitting). However, alternative methods have not
always been given if unstable results occur in the
analysis of a particular data set.

There appear to be two basic problems and
hence at least two approaches to "robust"
multivariate analysis procedures: 1) elimination
(or reduction) of multicollinearity before
inversion of a variance-covariance type matrix — or
equivalently , improved estimation of the inverse
of a variance-covariance matrix; and 2) robust
estimation of parameters by some procedure for
outright trimming of "outliers" or for reduction
of the influence of outliers.

Consider first the problem of reducing
effects of multicollinearity on multivariate
procedures. Most multivariate methods, except for
descriptive procedures such as principal
components and factor analysis, involve the
inverse of a matrix. Si, of sums of squares and
cross-products. Correlation matrices, variance-
covariance matrices, etc., are all special cases.
Letting , i = 1 , . . . , p , denote the characteristic

roots of a (p x p) nonsingular matrix S, one can
write

diagonal and the regression coefficients are

,-1

= Zd/X. ) a. a!
i = 1 ^ -'-^

(1)

where denotes the corresponding normalized

characteristic vectors. If some of the variables
used to compute S are strongly intercorrelated
(i.e., there is multicollinearity) then some of
the characteristic roots will be "small" and the
matrix S is said to be ill-conditioned. Clearly
from Eq.(1) division by small values is involved
and the inverse of S will be unstable in such
cases. Consider the obvious effect in regression

analysis [g = S~^X'Y], discriminant functions

[D(X) = (X^-X2) 'S~^X] , Mahalanobis' distance

[D^ = (Ii-l2^ '-"^ ^?1"?2^-' ^""^ canonical variables
and correlations [characteristic roots and vectors

-1 -1 -1 -1

of -1^^2^22—21 —22—21—11—12"' "'^^'"^ standard

notation is used. The regression coefficients,
discriminant coefficients, . . ., will be
impossible to interpret if the problem is severe.

£» = (diag (X^,

,X^))

-1

X'Y

Increased stability can be expected with
usually only a minor increase in bias. Sczerzenie
(1981), in his paper on analysis of deer harvest-
land use relationships, correctly noted that the
procedure will often reduce the variance of the
estimators. Converse and Morzuch (1981) noted
that the procedure will decrease the effect of
multicollinearity among independent variables.
For further details on this procedure, the
interested reader is referred to Sczerzenie (1981)
and Converse and Morzuch (1981) and their cited
references .

Use of principal components to reduce the
effect of multicollinearity in multivariate
procedures other than regression analysis has been
proposed but not studied (insofar as I am aware).
The approach is to transform all data to the first
few principal components before starting the
analysis. Variables are then uncorrelated and
only diagonal matrices need to be inverted. For
example. Fisher's linear discriminant function

D(X) = [(X^ - X^) 'S
would be replaced by

]X

,-1,

D*(X) = [(X^ - X2)'A')(ASA') A]X where
■^xp^pxl '"^P'"S2S'^ts the transformation of all data

to the "first" q principal components, q < p.
Hopefully, the loadings in D*(X) will be more
stable and easier to interpret than in the
original function without sacrificing the
discriminating ability. Similar adjustments can
be made in canonical variate analysis and other
multivariate procedures.

The second approach to solving the problem of
multicollinearity in regression analysis is by
ridge regression or biased-regression, first
developed by Hoerl and Kennard (1970). Cavallaro
et al . (1981) gave an excellent application of
this procedure in their use of multiple regression
techniques for discriminant analysis. The basic
idea is to allow a certain bias in the regression
coefficient by adding a constant k to the diagonal
of the coefficient matrix of the normal equations
before inverting, i.e.,

g = (S+kI)~^X'Y.

Regression analysis has, of course, received
the most attention under two general approaches.
First, regression on the first few principal
components (say q with q < p) eliminating those
corresponding to small characteristic roots. The
results are obvious in that the reduced
coefficient matrix of the normal equations is

This operation will add the constant
characteristic roots of S and hence

k to all

(S+kl)

-1

Z [1/(x.+k)]a.a.'
i = 1 ^ ^

243

Ridge regression can be viewed as a special
application of a much broader problem, namely
improved estimation of the inverse of a
variance-covariance matirx, e.g., Efron and Morris
(1976). In general, small characteristic roots of
S are underestimates of the corresponding
population values and large roots are
overestimates. Efron and Morris sought to improve

S ^ by shrinking all roots toward a common value.

The ridge-adjusted estimate, (S+1<I_)~\ works by
increasing all roots by a constant amount, k.
Thus the ill-conditioning effect of
underestimating small roots is countered without
having much influence on the larger roots. I will

refer to estimates of based on adjusted roots
of S as simply ridge-adjusted estimators.

1 am aware of two investigations into the use

of ridge-adjusted estimates of £ ^ in linear
discriminant analysis, i.e.,

D«(X) = [(X^ - X2)'(S + kI)"'']X.

The first by DiPillo ( 1976) showed that when the
variance-covariance matrices are ill-conditioned,
improvements can be made in the probability of
correct classification. In an independent
investigation, Smidt and McDonald (1976) found no
improvement in classification rate but obtained
increased stability in estimates of coefficients
of discriminant functions.

I am not aware of research of this type into
the study of canonical correlation analysis and
other procedures but would conjecture that similar
improvements can be made by using ridge-adjusted
estimators of the inverse of variance-covariance
matrices.

The second area of development of robust
multivariate analysis procedures is in the
reduction of the effect of outliers on resulting
analyses. It is well known that outliers in
multidimensional studies are very difficult to
detect, and if undetected can have a pronounced
effect on the analysis. Harner aod Whitmore
(1981) developed a method for building robust
discriminant models. In general, they found that
in the presence of outliers robust models
performed much better than Fisher's discriminant
model. With such an improvement in discriminant
analysis it is almost a certainty that other
multivariate procedures will perform better if the
influence of outliers is reduced. Guarding
against the adverse effects of outliers should be
a goal in every data analysis. The interested
reader is referred to cited references in Harner
and Whitmore 's excellent paper for further work in
this area.

LITERATURE CITED

Cassie, R.M. 1969. Multivariate analysis in
ecology. Proceedings of the New Zealand
Ecological Society 16:53-57.

Cassie, R.M., and A.D. Michael. 1968. Fauna and
sediments of an interidal mudflat: a
multivariate analysis. Journal of

Experimental Marine Biology and Ecology
2: 1-23.

Cavallaro, J. I., J.W. Menke, and W.A. Williams.
1981. Use of discriminant analysis and other
statistical methods in analyzing microhabitat
utilization of dusky-footed woodrats.
Proceedings of this workshop.

Converse, K.A., and B.J. Morzuch. 1981. A
descriptive model of snowshoe hare habitat.
Proceedings of this workshop.

DiPillo, P.J. 1976. The application of bias to
discriminant analysis. Communications in
Statistics A, Theory and Methods 5:83'4-84iJ.

Efron, B., and C. Morris. 1976. Multivariate
empirical Bayes and estimation of covariance
matrices. The Annals of Statistics 4:22-32.

Harner, E.J., and R.C. Whitmore. 1981. Robust
principal component and discriminant analysis
of two grassland bird species habitat.
Proceedings of this workshop.

Hoerl, A.E., and R.W. Kennard. 1970. Ridge
regression: biased estimation for

non-orthogonal problems. Technometrics
12; 55-67.

Karr, J.R., and T.E. Martin. 1981. Random

numbers and principal components: further

searches for the unicorn? Proceedings of

this workshop.
Sczerzenie, P.J. 1981. Principal components

analysis of deer harvest — land use

relationships in Massachusetts. Proceedings

of this workshop.
Smidt, R.K., and L.L. McDonald. 1976. Ridge

discriminant analysis. Technical report No.

108, Department of Statistics, University of

Wyoming, Laramie.
Smith, K.G. 1981. Canonical correlation analysis

and its use in wildlife habitat studies.

Proceedings of this workshop.
Weiner, J.M., and O.J. Dunn. 1966. Elimination

of variates in linear discrimination

problems. Biometrics 22:268-275.
Williams, B.K. 1981. Discriminant analysis in

wildlife research: theory and applications.

Proceedings of this workshop.

244

WORKSHOP PARTICIPANTS

Robert W. Aho

Michigan Department of Natural Resources
8562 East Stoll Road, Route 1
East Lansing, MI 48823

Bruce Ambuel

Department of Wildlife Ecology
University of VJisconsin
Madison, WI 53705

Robert Anthony

Department of Fisheries and Wildlife
Oregon State University
Corvallis, OR 97330

Barbara A. Bajusz
Pesticide Research Laboratory
The Pennsylvania State University
University Park, PA 16802

Pierre Beland

Canadian Wildlife Service

2700 Blv. Laurier

Ste-Foy, Quebec, Canada G1V 4H5

John A. Bissonette

Oklahoma Cooperative Wildlife Research Unit
Oklahoma State University
Stillwater, OK 74074

Jo Black

School of Forest Resources and Conservation
University of Florida
Gainesville, FL 32611

Debbie Boles
Zoology Department
University of Vermont
Burlington, VT 05405

Andre Bourget

Canadian Wildlife Service

2700 Blv. Laurier

Ste-Foy, Quebec, Canada G1V 4H5

Allen Boynton

North Carolina Wildlife Resources Commission
Rt. 5. Box 17
Burnsville, NC 28714

List does not include authors of papers in these
proceedings .

Richard Bramwell
Wildlife Resources
Box 400

MacDonald College

Ste. Anne-de-Bel levue , Quebec, Canada H9X ICO
Steve Brown

West Virginia Department of Natural Resources
Box 67

Elkins, WV 26241

Francine Buckley
Buckley Associates
372 South Street
Carlisle, MA 01741

Paul A. Buckley
National Park Service
15 State Street
Boston, MA 02129

George E. Burgoyne, Jr.

Michigan Department of Natural Resources

Box 30028

Lansing, MI 48910

M. Gail Butler

Canadian Wildlife Service

Environment Canada

Ottawa, Ontario, Canada K1A 0E7

John Gary

Department of Wildlife Ecology
University of Wisconsin
226 Russell Laboratory
Madison, WI 53706

Gilles Chapdelaine

Canadian Wildlife Service

2700 Blv. Laurier

Ste-Foy, Quebec, Canada G1V 4H5

Carl J. Christianson

University of Maryland

Appalachian Environmental Laboratory

57 First Street

Frostburg, MD 21532

Sukwoo Chang

National Marine Fisheries Service
Highlands, NJ 07732

245

Robert Clark
Wildlife Resources
Box 400

MacDonald College

Ste. Anne-de-Bel levue , Quebec, Canada H9X ICO
Wayne L. Cornelius

Southeastern Cooperative Fish and Game Statistics
Project

North Carolina State University
Box 5457

Raleigh. NC 27650

Brian Dennis

213 Mueller Laboratory

The Pennsylvania State University

University Park, PA 16802

Tim G. Dilworth
Department of Biology
University of New Brutvswick
Fredericton, New Brunswick, Canada

Susan M. Doehlert
Department of Wildlife
Division of Forestry
Morgantown, WV 26506

Jean Doucet
Wildlife Resources
Box 400

MacDonald College

Ste. Anne-de-Bel levue , Quebec, Canada, H9X ICO

Tom Dowhan
Zoology Department
University of Vermont
Burlington, VT 05405

Tom Dwyer

United States Fish and Wildlife Service
Migratory Bird and Habitat Research Laboratory
Laurel, MD 2081 1

Chuck Evans
USDA Forest Service
2081 East Sierra
Fresno, CA 93710

John T. Finn

Department of Forestry and Wildlife
University of Massachusetts
Amherst, MA 01003

Sidney S. Frissell
School of Forestry
University of Montana
Missoula, MT 59812

J. Edward Gates

Appalachian Environmental Laboratory
Frostburg State College
Frostburg, MD 21532

Jean Gauthier

Canadian Wildlife Service

2700 Blv. Laurier

Ste-Foy, Quebec, Canada GIV 4H5

Paul H. Geissler

U.S. Fish and Wildlife Service

Migratory Bird and Habitat Research Laboratory

Laurel, MD 20811

Arnie Gotfryd
Department of Zoology
University of Toronto
Toronto, Ontario, Canada M5S 1A1

W.E. Grant
Nagle Hall

Texas A&M University
College Station, TX 77843

Bart Guetti

New York State Department of Environmental Conser-
vation
South Wolf Road
Albany, NY 12233

Kevin J. Gutzwiller
The Pennsylvania State University
209B Ferguson Building '
University Park, PA 16802

Jonathan Haufler '
Department of Fisheries and Wildlife^
Michigan State University \
East Lansing, MI 48824 \

Larry Haugh \
Statistics Program \
University of Vermont \
Burlington, VT 05405

Cliff Hawkes

USDA Forest Service

240 West Prospect

Fort Collins, CO 80521

William M. Mealy
USDA Forest Service
180 Canfield Street
Morgantown, WV 26505

Doug Heimbuch
Fernow Hall
Cornell University
Ithaca, NY 14853

Jonas Hedberg
Hyltemasa

S-284 00 Perstorp, Sweden

Barbara T. Hill
USDA Forest Service
Box 638

Laconia, NH 03246
Douglas Inkley

U.S. Fish and Wildlife Service

Migratory Bird and Habitat Research Laboratory

Laurel, MD 20811

Rich Kahl

Missouri Cooperative Wildlife Research Unit
University of Missouri
Columbia, MO 6521 1

246

Ron Kalinoski
Biology Department
Syracuse University
Syracuse, NY 13210

Marie Kautz

New York State Department of Environmental Conser-
vation

Wildlife Resource Center
Delmar, NY 12054

Jeff Keller
Fernow Hall
Cornell University
Ithaca, NY 14853

C. William Kilpatrick
Zoology Department
University of Vermont
Burlington, VT 05405

M. Kingsley

Canadian Wildlife Service
5320-122 Street
Edmonton, Alberta, Canada

Kevin R. Kinsley

132 Land and Water Resources

The Pennsylvania State University

University Park, PA 16802

J. Thomas Kitchings

Oak Ridge National Laboratory

Building 1505, Box X

Oak Ridge, TN 37830

George LaBar

Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Pierre Laporte

Canadian Wildlife Service

2700 Blv. Laurier

Ste-Foy, Quebec, Canada G1V 4H5

Terry A, Larson

Department of Biological Sciences
Illinois State University
Normal, IL 61761

Richard Lindeborg
USDA Forest Service
240 West Prospect
Fort Collins, CO 80521

David A. MacKenzie
Department of Zoology
University of Toronto
Toronto, Ontario, Canada M5S 1A1

Linda K. Mann

Oak Ridge National Laboratory
Building 1505, Box X
Oak Ridge, TN 37830

Leslie F. Marcus

American Museum of Natural History
79th at Central Park West
New York, NY 10024

Don McCarty

Texas Parks and Wildlife Department
4200 Smith School Road
Austin, TX 78744

Donald A. McCrimmon, Jr.

National Audubon Research Department

Cornell University

159 Sapsucker Woods Road

Ithaca, NY 14850

Eileen Miller
Box 181, RR 2
Dover, NH 03820

Linda Morris

3OID Forest Resources Laboratory
The Pennsylvania State University
University Park, PA 16802

Guy Morrison

Canadian Wildlife Service
Ontario Region Headquarters
1725 Woodward Drive
Ottawa, Ontario, Canada KIG 3Z7

Ken Morrison

Department of Chemical Engineering
University of Sherbrooke
Sherbrooke, Quebec, Canada

Ken Myers

Zoology Department
University of Guelph
Guelph, Ontario, Canada

John L. Oldemeyer

Denver Wildlife Research Center

1300 Blue Spruce Drive

Fort Collins, CO 80524

Richard J. Olson

Oak Ridge National Laboratory

Building 1505, Box X

Oak Ridge, TN 37830

John W. Ozard

New York State Department of Environmenal Conser-
vation

Wildlife Resources Center
Delmar, NY 12054

Stephen G, Parren
Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Barbara W. Patty
National Audubon Society
115 Indian Mound Trail
Tavernier, FL 33070

John Paul

Department of Entomology

University of Delaware
Newark, DE 19711

247

John H. Porter

Department of Environmental Sciences
Clark Hall

University of Virginia
Charlottesville, VA 22903

Dale Rabe

Department of Fish and Wildlife
Michigan State University
East Lansing, MI 18823

Richard L. Raesly

University of Maryland

Appalachian Environmental Laboratory

1 1 Bowery Street

Frostburg, MD 21532

Gerald Rasmussen

New York State Department of Environmental Conser-
vation

Wildlife Resources Center
Delmar, NY 12054

William Roberts
Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Ken Ross

Canadiatj Wildlife Service
Ontario Region Headquarters
1725 Woodward Drive
Ottawa. Ontario, Canada K1G 3Z7

Douglas E. Runde
Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Hans Schreuder
USDA Forest Service
240 West Prospect
Fort Collins, CO 80521

Mark Scott

Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Steven L. Sheriff

Missouri Department of Conservation
1110 College
Columbia, MO 65201

Jeffrey Short

United States Air Force

AFESC/DEVN

Tyndall Air Force Base, FL 32403
Nova Si Ivy

Wildlife and Fisheries Sciences Department
Texas A&M University
College Station, TX 77843

James E. Skaley
Fernow Hall
Cornell University
Ithaca, NY 14853

Charles R. Smith
Laboratory of Ornithology
Cornell University
159 Sapsucker Woods Road
Ithaca, NY 14850

C.E. John Smith
Biometrics Division
Canadian Wildlife Service
Department of the Environment
Ottawa, Ontario, Canada K1A 0E7

Graham Smith
University of Idaho
2712 Pleasanton
Boise, ID 83702

Ronald Smith

Arizona Game and Fish Department
2222 W. Greenway Road
Phoenix, AZ 85023

Alan J. Steiner

Department of Forestry and Wildlife Management
204 Holdsworth Hall
University of Massachusetts
Amherst, MA 01003

Craig Stihler

West Virginia Department of Natural Resources
Box 67

Elkins, WV 26241
Gerald L. Storm

Pennsylvania Cooperative Wildlife Research Unit
The Pennsylvania State University
University Park, PA 16802

Scott Sutcliffe
Loon Preservation Committee
Audubon Society of New Hampshire
RFD 2

Meredith, NH 03253

Gerald E. Svendsen
Zoology Department
Ohio University
Athens, OH 45701

Diane L. Tessaglia
Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Frank R. Thompson
Wildlife Biology Program
University of Vermont
Burlington, VT 05405

Ian Thompson

Canadian Wildlife Service

1725 Woodward Drive

Ottawa, Ontario, Canada K1G 3Z7

248

Nancy Tilghman
USDA Forest Service
Hilton House

University of Massachusetts
Amherst, MA 01003

Alan Tipton

Department of Fisheries and Wildlife Science
Virginia Polytechnic Institute and State

University
Blacksburg, VA 2U060

Kiraberly Titus
University of Maryland
Appalachian Environmental Laboratory
Frostburg, MD 21532

James Traynor

New York State Department of Environmental Conser-
vation

Wildlife Resources Center
Delmar, NY 12054

Walter M. Tzilkowski
205 Forest Resources Laboratory
The Pennsylvania State University
University Park, PA 16802

Beatrice VanHorne
Biology Department
University of New Mexico
Albuquerque, NM 87131

James S. Wakeley
205 Forest Resources Laboratory
The Pennsylvania State University
University Park, PA 16802

Dan Welsh

Canadian Wildlife Service

1725 Woodward Drive

Ottawa, Ontario, Canada K1G 3Z7

Glenn White

6653 Tiffin Avenue

San Diego, CA 921 1U

Jim Woehr

State University of New York at Plattsburgh
Plattsburgh, NY 12901

249

it U. S. GOVERNMENT PRINTING OFFICE 1981 - 781-065 A66 Reg.

1

^ n ^ u

(0 Q> o o

-rH I. o u.

t- O
10 u.
>

*> Q • O

^ CO "--^

E (0

a- *J

^ . CM w

°"

^ 1^ OS 0)

b H S

9. ^ °
o

c -

O ID C

-H -H +J -H

*J > C .-H

<IJ I. 3 <H

4^ 01 O O

n CO E o

c *>

o

n L.

I. <u

0) >

> tH

3 V

CO

C iH
(0 OJ

T3

0) Q.
*i O

•a m

1

»

V

XI

c

a

s:

nd

£

ra

ID

•H

>

n

(0

le

o.

c

E

10

iH

X

a

a>

e

m

n

ns

on

de

o

■etati

lu

at

o

o

in

li

a.

a.

t-

pics

. ap

inte

To

w

T3

nd

2 *^

4-> E o

9> E n 1.

CL (. 4J a

(0 0) (0 o>

a> *J CO

V 4J >> 4J

ON CO

cj CO

> CO 11

o *i

■ ^ CC 4)

"5 o-:

4) £

Q. 41

P -p CO n c

uj ra 4) cj

«- CO
C

CO D

4) Q.

•H £

"O n

4) ^

£ 1

o o

o1i

4) th n

O CO c

•rH -H

> C rH

I. 3 rH

4) O O

CO £ U

O 4) "

, CO E ^
>.

4> ^

■P CO

■H -H CO
I- t-

o. (0 o

•* >: -p

0) Jr .rf rH

C CO

o - o

E n t.

k t. *> o.

4) CO O.

b> 4J CO

4) -P >> 4^

*> n I.

CO 4> O O

■r< (. O U.

1. O <S

CO u.
>

■H <: c

.P Q • O

rH CO °- -H

3 O -P

e C7^ CO
3- -P

V, . CM CO

CO J3

>>*3

o o

O b.

.rf m c CO

rH CO
3 =>

E

0\ CO
CM CO

O O -H 4)

C P

° "Si-

> -rH CO f4;
-3 00 O.

1) 4J 05

4>

4)

" »rH =

m S T>

• 4> c

<- -rH " ra

S ^ M

T3 OT rH p)

■o -o

C ,H 3
(0 <y <-H

^ -rH

= >- °

•rl «> O rH

C

bJ CO 4}

CJ o

_ -rH

5 -P 4)

■T to o

i -rH -rH

to CO s: cj

o -p =°

O CO ^
4) 4) 4

ra

S ra c

u o

V E

a (.

CO 4>

a>

.2 ^

Zh
Q.

a <«
co-g

■H

= m
>>

4; rH

•P to
(0 c

•H (0
L.

(0 o

> -p

4> -H ,
3 to

3 4>

4,

-H -P

r^ ^ °= I"

t- -H
O -o

. -rH 4) O rH

C U. O

u n 4> o

o o

2 -P 4) ra

T; m o ro c

r; -rH -rt p -rH

a P > C rH

" to J. 3 rH

-P 4J O O

• to CO 2: O

a. 41

E

to . •

"go

c 3 3 -P

■o -o

C rH
to QJ

4) CX

^ °
-H £

■O W

4) ^

to

o o «; >
^ ?

a c

4> 41

o to

-ri

I • p m

o g J

, 00 E M

I t3^ >,

■- 4) rH

•P to

rH to C

•rH -rH (0
I i- t-

' O. CO o

<" c HJ ra

a > .P to

Rocky
Mountains

Great
Plains

U.S. Department of Agriculture
Forest Service

Rocky Mountain Forest and
Range Experiment Station

The Rocky Mountain Station is one of eight
regional experiment stations, plus the Forest
Products Laboratory and the Washington Office
Staff, that make up the Forest Service research
organization.

RESEARCH FOCUS

Research programs at the Rocky Mountain
Station are coordinated with area universities and
with other institutions. Many studies are
conducted on a cooperative basis to accelerate
solutions to problems involving range, water,
wildlife and fish habitat, human and community
development, timber, recreation, protection, and
multiresource evaluation.

RESEARCH LOCATIONS

Research Work Units of the Rocky Mountain
Station are operated in cooperation with
universities in the following cities:

Albuquerque, New Mexico

Bottineau, North Dakota

Flagstaff, Arizona

Fort Collins, Colorado*

Laramie, Wyoming

Lincoln, Nebraska

Lubbock, Texas

Rapid City, South Dakota

Tempe, Arizona

•Station Headquarters: 240 W. Prospect St., Fort Collins, CO 80526