BOSTON PUBLIC LIBRARY

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RELIABILITY OF KILL AND ACTIVITY ESTIMATES
IN THE U.S. WATERFOWL HUNTER SURVEY

MO

UNITED STATES DEPARTMENT OF THE INTERIOR

FISH AND WILDLIFE SERVICE

Special Scientific Report - Wildlife No. 240

Library of Congress Cataloging in Publication Data

Couling, Leigh M.

Reliability of kill and activity estimates in the U.S. waterfowl hunter survey.

(Special scientific report — wildlife ; no. 240)

Supt. of Docs, no.: I 49.15/3:240

1. Waterfowl shooting— United States. 2. Hunting surveys — United States. I.
Sen, A. R. II. Martin, Elwood. III. Tide. IV. Series.
SK361.A256 no. 240 [SK331] 639. 9 79 '0973s 81-607087
[338.4'7799244'0973] AACR2

NOTE: Use of trade names does not imply U.S. Government endorsement of commercial products.

RELIABILITY OF KILL AND ACTIVITY ESTIMATES
IN THE U.S. WATERFOWL HUNTER SURVEY

By Leigh M. Couling
A. R. Sen
Elwood M. Martin

r.s.

MHH A W II Dl II I
SKKVK*K

UNITED STATES DEPARTMENT OF THE INTERIOR

FISH AND WILDLIFE SERVICE

Special Scientific Report — Wildlife No. 240
Washington, D.C. • 1982

Reliability of Kill and Activity Estimates
in the U.S. Waterfowl Hunter Survey

by

Leigh M. Couling1 and A. R. Sen

Canadian Wildlife Service

Migratory Birds Branch

Environment Canada

Ottawa, Ontario, Canada K1A 0E7

and

Elwood M. Martin

Office of Migratory Bird Management
Laurel, Maryland 20708

Abstract

A mail questionnaire survey of waterfowl hunters is conducted each year in the United States to
provide information on waterfowl kill and hunter activity. We carried out a study using data from the
1971-72 and 1972-73 hunting seasons to determine the effectiveness of the present U.S. sampling and
estimation techniques, and a number of modifications in both sampling and analysis is recommended.
We found that stratification of post offices based on the number of duck stamps sold did not give more
precise estimates of mean duck bag and days hunted per hunter than estimates obtained in the absence
of stratification. Estimation of error variance assuming simple random sampling of hunters instead of
cluster sampling of post offices should be avoided because it may lead to significant underestimates. Of
the cluster-type estimators examined, the ratio estimator is recommended for estimating means and
standard errors of duck bag and days hunted per hunter. Estimates of kill and hunter activity showed
wide departures from normality which led to inefficient estimates of the means and of their variances.
Log transformation resulted in approximate normality and in considerable increase in precision of the
estimates of error of kill. A possible nonresponse problem exists in the stratification scheme because
hunters from the larger post offices have lower apparent response rates than those from smaller post
offices. Some sampling problems are a result of and dictated by the type of sampling frame currently
available; however, it appears that, with the more advanced statistical techniques now available, some
improvements can be made in the sampling as well as in the analytical phase.

As part of its continuing efforts toward more efficient conformity to the sampling scheme (Martin and Carney

management and conservation of wildlife resources, the 1977). The present report will do so by (1) examining the

United States Fish and Wildlife Service (USFWS) efficiency of the current method of stratification and (2)

initiated a Mail Questionnaire Survey of U.S. waterfowl comparing the efficiency of other estimation techniques

hunters in 1952. Data obtained in this annual survey are with the one presently used. Survey estimates of kill and

used to estimate U.S. waterfowl hunting activity and activity have highly skewed distributions. The effect of

success based on a sample of hunters purchasing duck this on the estimates and their errors is examined and,

stamps. The reliability of these estimates has been given where appropriate, methods are recommended for

little consideration in terms of either efficiency or increasing the precision of these estimates. We also briefly

examine the important sources of non-sampling error: (1)
nonresponse bias due to differences between hunters who

do not report their activities and hunters who do, and (2)

■Present address: S&S Software Ltd., 444 MacLaren St., response bias due to improper or exaggerated reporting of

Ottawa, Ontario, Canada K2P 2C8. kill.

In this study we used hunter kill and activity data from
the 1971-72 and 1972-73 season USFWS surveys of 12 of
the largest States, including at least one State from each
of the four waterfowl flyways that span the continent.
The States included were Florida, Maryland, and New
York (Atlantic Flyway); Arkansas, Illinois, Louisiana,
Minnesota, Missouri, and Wisconsin (Mississippi
Flyway); Texas (Central Flyway); and California and
Washington (Pacific Flyway).

Design of the U.S. Survey

The U.S. survey sampling frame consists of a master list
of post offices that sell duck stamps (Martin and Carney
1977). Each State is stratified into several geographic
zones. In each zone, the post offices are further stratified
into three groups based on the number of duck stamps
sold the previous year. These are stratum 1 (less than 100
stamps), stratum 2 (100 to 1,000 stamps), and stratum 3
(more than 1 ,000 stamps) .

Post offices are selected at random within each
stratum. In strata 1 and 2 and in a sub-sample of branch
post offices in stratum 3, all duck stamp purchasers are
asked to participate in the survey; stratum 3 post offices
are usually made up of a number of branch post offices
and, to avoid unnecessarily large samples of hunters, a
sub-sample of branch offices is chosen. These branch of-
fices are identifiable individually in the sampling phase
but are combined under the main post office name in
other phases of the analysis because it is not feasible to
maintain the identity of each branch office. The post
offices are the primary sampling units, and the hunters
buying duck stamps are the elements or the ultimate units
of selection. Thus in strata 1 and 2, the U.S. survey design
is a stratified single-stage cluster sampling scheme, and in
stratum 3 there is a second stage consisting of branch post
offices. However, for our analysis, it was necessary to
treat the large post offices as single units and assume sin-
gle-stage cluster sampling for all strata.

All post offices selected in the survey are sent name and
address forms (survey contact cards) with instructions to
give one card to every duck stamp purchaser asking the
purchaser to complete the form and return it to the postal
clerk for mailing to the USFWS. The purchaser is thus
placed on the Service's mailing list. The purchaser retains
a portion of the contact card on which an explanation of
the survey and a hunting record form for his own use are
provided. Actual participation in the survey is variable
because some stamp buyers are unwilling to supply their
names and addresses and some postal clerks may neglect
to hand out cards. Hunters who fill out address cards can-
not be viewed as units of a second sampling stage because
the decision by hunters on whether to fill out contact
cards does not constitute a probability sample.

At the end of the hunting season, each hunter who
filled out a contact card is sent a questionnaire to com-
plete and return. Any hunter failing to do so within 3 to 4

weeks is sent a follow-up questionnaire. The data are used
to estimate hunting activity and success for each stratum
of each zone of each State. The results are then summed
to form zone, State, flyway, and national estimates
(Martin and Carney 1977).

Estimation of Stratum Means
Within a Zone

We will now obtain expressions for estimates of means
(and standard errors) for duck kill and number of days
hunted per active hunter for a post office stratum within
a zone in a State. Because the estimators of the two means
have the same mathematical form, we will simply use the
term "mean per hunter" in the following discussion.

Let M, = number of active hunters (those hunting 1 or
more days during the season) who purchased
duck stamps from the ith post office of a stra-
tum,
yi( = number of ducks killed (or days hunted) by the
jth hunter who purchased a duck stamp from
the i,h post office,
Mi
y{ = E yy = number of ducks killed (or days
i = ' hunted) by all Mi hunters who pur-
chased duck stamps from the ith post office,
n = number of post offices randomly selected from
the population of N post offices in a stratum,
and

N N

y = E Vj /E Mj = population mean per hunter for a
1 = 1 ' = ' post office stratum within a zone.

A biased but consistent ratio estimator of the popula-
tion mean per hunter for a stratum ( y ) is

y =

n n

E yL/E M,

i = 1 i = l

(1)

The variance of the estimator is

V„(y)

N2d " £)

N

E

>

- M, y )*

N

- 1

,E (y, - Miy-)2
■ = 1

N - 1

(2)

Since the M;'s are known only for the post offices in

N M-

the sample, their average E — p must be estimated by

n ~_

E Mi . Hence Vcl( y ) is estimated by
i = I-jt

vci ( y) =

**-&

i = 1

n

-MiJF)2

n

- 1

(3)

(The problem of nonresponse in M( is examined in a later
section.)

Appraisal of Post Office Stratification

One reason for stratification of zones into three post of-
fice groups was to take advantage of any differences in
average hunting activity and success among post offices.
It was assumed that hunter characteristics would differ
most along a rural-urban axis; stamp sales volume would
indicate the rural-urban character. If no such difference
existed, this stratification would be inefficient because it
would not contribute to the sampling scheme in terms of
additional information about hunter characteristics.

We chose two variables, kill per active hunter and days
per active hunter, for this examination. One could as well
use kill and days per potential hunter, the procedure rou-
tinely followed by the USFWS. (A potential hunter is one
who bought a duck stamp with the intention of hunting.)
Our analysis indicated that the results differ little be-
tween the two approaches (because the vast majority of
duck stamp buyers go hunting). To determine if stratifi-
cation was effective, we carried out tests for significant
differences between stratum means for both duck kill and
days hunted for each of the 12 States (Steel and Torrie
1960).

Although they represent about 36% of all U.S. duck
stamp buyers, only about 320 post offices (2% of the sam-
pling frame) sell more than 1,000 duck stamps per season.
The entire United States is divided into 188 zones and, al-
though over 50% of the stratum 3 post offices are in-
cluded in the sample each year, the number sampled in
each zone is very small. These differences among strata
should be clear from Table 1, which shows the average
number of post offices per zone by post office stratum for
the States in this study, States which are all much more
populous than the average State. In some zones, the num-
ber of sample post offices in stratum 2 is also small.
Hence, for the 12 States under study, the data for strata 2
and 3 were combined for each zone. The zonal estimates
for stratum 1 and for combined strata 2 and 3 were next
summed to provide more reliable estimates at the State
level, and tests for differences between these totals were
then carried out for each State.

The non-normality of kill and days hunted will affect
many of the test procedures used for calculating signifi-
cant differences. Therefore, these normal tests will
usually tend to be less powerful than their non-para-
metric counterparts. However, at State and fly way levels
the means will generally be based on fairly large samples
to ensure normality and increase the power of test com-
parisons. Hence, estimates of mean kill and activity are

presented at State, flyway, and national levels.

The results of the t-tests for differences between stra-
tum 1 and strata 2 and 3 combined for each State (Table
2) show that kill per hunter differed significandy (P <
0.05) between strata in only two States each year; mean
days hunted differed significantly in three States during
1971-72 and in two States during 1972-73. Thus the
number of States to benefit from stratification by post of-
fice size was too small to warrant its continued use in the
analyses. For other characteristics of interest (e.g., kill
per hunter by species, age and sex ratio determination) it
may be advantageous to retain this stratification in sam-
pling.

Additional Testing of Zone
and State Estimates

Because no differences were found among post office
strata, estimates of duck kill per hunter and days hunted
per hunter were calculated with and without post office
stratification for comparison. Zonal estimates based on
post office stratification were obtained as weighted sums
of stratum estimates; the weights were proportions of
duck stamp sales in each stratum (the method of com-
bining stratum estimates currently being used) . Unstrati-
fied zone estimates were made by treating all data from a
zone as a single sample. State estimates for both methods
were obtained as weighted sums of zone estimates, and
the weights were proportions of duck stamp sales in each
zone. In practice, one would first choose the estimation
technique best suited to the State as a whole, apply the
method separately to each zone, and then form weighted
sums of these results as

ind

.'here

k £

= S , wzyz

z = 1

v( ys) = E w* v( yj

Z = 1

(4)

(5)

k = number of zones in the State,

yz = mean per hunter estimate for zone z,

ys = mean per hunter estimate for State s,

v( yj = estimated variance of yz,

v( yj) = estimated variance of ys, and

wz= Qz = the weight for zone z

k where Qz could be the

E Qz total number of duck
z = 1 i

stamp sales in zone z or

the number of post offices

or unity. When Qz = 1

for all zones, (4) and (5)

are of the "unweighted"

form.

Table 1. Average number of post offices available per zone by stratum and State.

1971-72: Average in stratum

1972-73: Average in stratum

State

1

2

3

1

2

3

Florida

31.6

8.7

0.3

28.9

9.7

1.1

Maryland

57.0

15.7

1.1

60.0

20.3

1.7

New York

130.1

18.1

0.6

122.4

35.3

0.7

Arkansas

60.7

19.3

2.3

51.3

29.3

3.3

Illinois

130.6

27.8

1.0

121.0

37.8

1.4

Louisiana

37.2

28.8

3.4

31.8

31.6

4.2

Minnesota

77.0

30.7

3.0

76.3

30.0

3.7

Missouri

76.8

12.2

1.0

68.4

19.6

1.4

Wisconsin

99.4

36.0

3.6

92.2

41.4

5.2

Texas

104.8

23.8

2.2

94.7

29.0

3.7

California

74.7

36.1

4.1

68.3

41.4

5.3

Washington

87.7

29.7

5.0

80.3

28.3

6.7

As expected with proportional sampling, none of the
States showed significant differences in means estimated
by either method (Table 3). The important finding is
that, in most instances, ignoring stratification did not re-
sult in decreased precision; on the contrary, there was an
apparent gain as indicated by the reduction in the stand-
ard errors which may be partly attributable to the small
number of post offices on which most of the stratum esti-
mates are based and partly to departures from propor-
tional allocation among strata.

From Cochran (1963:138-139) we estimate gains in ef-
ficiency attained when post office stratification is ig-
nored. The term "gain in efficiency" reflects the reduction
in the error of estimation obtained when one estimation
procedure is replaced by another. Table 4 presents values
at the State level obtained by combining zonal estimates
for each State. The percentage gains in efficiency of esti-
mates for most States are fairly large. By comparison, the
two negative gains are relatively small and might be at-

tributed to other causes such as sampling error. This find-
ing provides further evidence that the present stratifica-
tion by post office groups within zones is not efficient.

There are reasons, however, to believe that some recog-
nition of the volume of post office duck stamp sales may
still be advisable. Mentioned previously is the two-stage
sampling often required for the larger post offices in-
volving the use of branch offices, which, under the pre-
sent accounting system, cannot be treated as separate post
offices. There is also the question of biases stemming from
lower response rates for larger post offices (discussed in a
later section). Finally, because post offices vary so much
in size, the number of post offices to be chosen in the sam-
ple selection is not a predetermined (constant) value as re-
quired by statistical theory; instead, drawing continues
until the supply of contact cards available for a stratum is
exhausted. Segregating these potential statistical
problems into a separate stratum of large post offices may
be an appropriate step in isolating and solving them.

Table 2. Observed probability level (P) for l-tests of differences in mean duck kill and days hunted between

stratum 1 and strata 2 and 3 combined.

Kill per

hunter

Days per

hunter

1971-72

1972-73

1971-72

1972-73

State

P

df

P

df

P

df

P

df

Florida

0.25"

53

0.00

50

0.46

53

0.01

50

Maryland

0.64

55

0.67

49

0.26

55

0.20

49

New York

0.28

112

0.24

106

0.17

112

0.14

106

Arkansas

0.08

34

0.94

28

0.67

34

0.95

28

Illinois

0.01

109

0.88

103

0.09

109

0.69

103

Louisiana

0.00

46

0.00

44

0.41

46

0.94

44

Minnesota

0.30

53

0.69

58

0.64

53

0.92

58

Missouri

0.07

55

0.83

49

0.00

55

0.00

49

Wisconsin

0.06

5.3

0.74

41

0.03

53

0.70

41

Texas

0.31

79

0.66

80

0.06

79

0.42

80

California

0.30

78

0.44

69

0.02

78

0.26

69

Washington

0.59

34

1)21

38

0.83

34

0.36

38

aThe observed probability levels are correct to two decimal places.

Table 3. Comparisons of State level kill-per-hunter and days-per-hunter estimates:
stratified by post office size vs. unstratified.

Kill per

hunter

Days per

hunter

Stratified

Unstratified

Stratified

Unstratified

Hunting season

Standard

Standard

Standard

Standard

and flyway

State

X

error

X

error

X

error

X

error

1971-72

Florida

7.65

0.83

7.30

0.73

6.20

0.41

6.17

0.35

Atlantic

Maryland

6.02

1.86

6.00

1.73

8.77

0.50

8.83

0.49

New York

3.91

0.27

3.89

0.26

7.14

0.26

7.09

0.25

Mississippi

Arkansas

9.19

3.14

8.73

2.92

6.70

0.90

6.49

0.84

Illinois

6.44

0.40

6.20

0.40

8.32

0.49

8.20

0.49

Louisiana

14.64

1.82

14.37

1.56

8.52

0.72

8.58

0.66

Minnesota

8.62

0.65

8.54

0.54

7.22

0.44

7.17

0.37

Missouri

7.68

0.53

7.52

0.51

8.36

0.45

8.15

0.45

Wisconsin

6.90

0.41

6.63

0.43

8.49

0.57

8.34

0.55

Central

Texas

8.09

0.80

8.29

0.74

6.89

0.38

6.84

0.36

Pacific

California

18.44

1.10

18.31

1.01

8.35

0.34

8.27

0.31

Washington

9.92

1.00

9.77

0.91

8.87

0.55

8.45

0.54

1972-73

Florida

9.62

0.55

9.35

0.52

6.84

0.39

6.72

0.36

Atlantic

Maryland

5.17

0.83

5.24

0.79

8.71

0.80

8.53

0.78

New York

4.03

0.32

4.12

0.31

7.04

0.26

7.15

0.25

Mississippi

Arkansas

10.35

1.55

10.28

1.55

8.67

0.92

8.79

0.89

Illinois

6.97

0.45

6.98

0.45

8.62

0.40

8.60

0.40

Louisiana

15.07

1.73

14.89

1.52

7.99

0.77

8.04

0.72

Minnesota

8.78

0.47

8.75

0.48

7.48

0.30

7.46

0.29

Missouri

6.13

0.52

6.29

0.49

6.84

0.37

6.84

0.36

Wisconsin

5.79

0.75

5.68

0.69

8.37

0.61

8.35

0.56

Central

Texas

10.70

2.74

11.04

2.58

5.96

0.54

6.05

0.51

Pacific

California

17.01

1.70

17.00

1.61

8.28

0.46

8.28

0.45

Washington

12.08

1.14

11.82

1.08

8.48

1.03

8.38

0.92

Although the present stratification by post office size
has apparently not led to better estimates, stratification
by geographic zone should be retained when making
State estimates. That is, estimates at the State level should
be obtained through some combination of the zonal esti-
mates, because there is much variation between zones in
kill per hunter and days per hunter. This significant zone-
to-zone variation indicates that stratification of each
State into zones was a good addition to the sampling
scheme (see Appendix).

Cluster Sampling Versus
Simple Random Sampling

It will now be assumed that the data on hunter kill and
activity were obtained through simple random selection
(SRS) of hunters instead of by cluster sampling as in the
current design. This assumption has been made in the
limited work done to date by the USFWS for estimating
standard errors of the survey estimates, although it is
recognized to be only marginally useful for providing a

crude approximation. The sample estimate of mean per
hunter ( y) is given by

y = E y( IT, M,
i = l i = 1

(6)

which has the same mathematical form as equation (1)
though it is based on a different selection scheme.
The variance of this estimate is

Vsr,(y);

(1 - E Mj/E Mj]

i = 1 i = 1

E M,

i = 1

M :

N

E
i = l j = l

e ' (Yii - y )2

N

E M( - 1

i = 1

and is estimated by

Vsrs(y> =

(1 --)

K NT

E M:

i = 1

n

E

i =

e (y.f - y)2

i = i

E M, - 1

i = 1

(7)

(8)

6

Table 4. Gains in the efficiency of State level estimates
achieved by ignoring stratification by post office size.

State

Percent gain in

efficiency3

Kill per

hunter

Days per

hunter

Flyway

1971-72

1972-73

1971-72

1972-73

Atlantic

Florida

29.3

11.9

37.2

17.4

Maryland

15.6

10.4

4.1

5.2

New York

7.8

6.6

8.2

8.2

Mississippi

Arkansas

15.6

0.0

14.8

6.9

Illinois

0.0

0.0

0.0

0.0

Louisiana

36.1

29.5

19.0

14.4

Minnesota

44.9

-4.1

41.4

7.0

Missouri

8.0

12.6

0.0

5.6

Wisconsin

-9.1

18.1

7.4

18.7

Central

Texas

16.9

12.8

11.4

12.1

Pacific

California

18.6

11.5

20.3

4.5

Washington

20.8

11.4

3.7

25.3

aGain in efficiency is defined as the relative gain in information
where information is the reciprocal of the variance. The gain for
estimator 2 (ignoring stratification of a zone by post office size)
over estimator 1 (with stratified sampling) is given by

1 1

v- v- = vi - v2 _ 21 _ i

V„ V,

1

v7

Since the data were obtained through cluster sampling
of post offices and not of individuals chosen at random,
the above estimate of variance (herein called SRS esti-
mate) will be biased unless the variation among post of-
fices in terms of duck kill or days hunted per hunter is the
same as the variation among hunters within post offices,
i.e., there is no intracluster correlation (defined below).
This relationship is given by the following approximate
equation (Cochran 1963:210).

vd (y) = §£r~r [vsrs(y) U + (» - i)*i] 0)

M(N - 1)

where M = L M;/n and p is the intracluster correlation

i = 1

between pairs of hunters buying stamps at the same post
office in the cluster sample and is defined as

E(yiry)(yiu-y)
Efoj-y)2

N

where the numerator is averaged over £ M,(M; - l)/2

i = 1

N

distinct pairs and the denominator over all £ \f values

of y;i. fl i = 1

If p is greater than zero, then Vcl( y) will be larger than
Vsrs( y). Because Vc,( y) is an estimate which is appropri-
ate for the cluster sampling scheme, use of Vsrs( y) instead
of Vcl( y) would lead to a biased estimate and should be
avoided. Similarly, if p is less than zero, Vsrs( y ) will over-

estimate the true variance. However, if p = 0, then

— i —
Vsrs(y) and Vc!(y) are approximately equal (where N is

sufficiently large) so that either one can be safely used to
estimate the variance. Consequently, the choice of an
estimator should be based either on a prior knowledge of
the amount of intracluster correlation present in the pop-
ulation or on the amount indicated by data obtained
through a sampling of the population. To be reliable,
estimates of intraclass correlation should be based on
large samples at State, flyway, and national levels.

Results at the State level (Tables 5 and 6) illustrate the
relationship in equation (9). For obtaining estimates, the
best weighting method (wz in expressions [4] and [5]) is by
duck stamp sales in all post offices (both sampled and un-
sampled). The results given in Tables 5 and 6 were ob-
tained by using the "unweighted" form because (a) we are
comparing two methods, and the weights, if common to
both, should have little effect on the comparisons, and (b)
some of the tests used become very complex if carried out
in other than the unweighted form. This accounts for the
slight differences in estimates of kill and days per hunter
contained in Tables 5 and 6 as compared with those in
Table 3.

Tables 5 and 6 show that for the majority of the States
the intracluster correlation coefficient is significandy
greater than zero, and in most instances the SRS estimate
of standard error is much less than the cluster estimate, an
indication that it is underestimating the true error. The
percentage underestimate in Vsrs( y) compared with
Vcl( y ) is shown in the last columns of the tables. Hence,
as suggested by equation (9), estimation must be carried
out by the cluster approach.

It must be emphasized that since Vsrs( y ) assumes a de-
sign other than that which was implemented, its use
rather than that of Vd( y) is incorrect and should be
avoided. Only when the sample intracluster correlation
coefficient is zero or close to zero may one utilize Vsrs( y).

In comparing Vd(y) with Vsrs(y), N (the number of
post offices in the stratum) becomes very important only
when it is very small. A small value of N would tend to in-
crease the value of Vd(y) with respect to Vsrs(y) which
again suggests that Vsrs ( y ) will underestimate the true
variance (or overestimate it depending on whether p is
positive or negative).

The highly significant and positive intracluster correla-
tion in most of the States for both mean duck kill and days
hunted indicates that variation between post offices was
greater than variation among hunters within post offices.
This result suggests the use of a sampling scheme which
selects a large number of post offices but subsamples a
small percentage of the hunters from each of these post of-
fices. However, since this is not operationally feasible, a
somewhat less efficient procedure would be to obtain a
representative sample of post offices by selecting a syste-
matic sample of post offices by location within each zone
of each State, and selecting all hunters from the sampled
post offices.

Table 5. Estimated duck kill per hunter, standard error, intracluster correlation, and underestimate in variance ob-
tained at the State, "flyway, "and "national" levels by using individual hunters (SRS) rather than post offices (cluster)
as sampling units.

Hunting season
and flyway

State

Kill

per

hunter

Standard error

Cluster

SRS

Intracluster
correlation

Underestimate"
in error

(percent)

1971-72
Atlantic

Mississippi

Central
Pacific

Entire Season

1972-73
Atlantic

Mississippi

Central
Pacific

Entire Season

Florida
Maryland
New York
Combined

Arkansas
Illinois
Louisiana
Minnesota
Missouri
Wisconsin
Combined

Texas

California
Washington
Combined

Florida
Maryland
New York
Combined

Arkansas
Illinois
Louisiana
Minnesota
Missouri
Wisconsin
Combined

Texas

California
Washington
Combined

7.15

0.52

5.71

0.38

4.22

0.33

5.69

0.26

9.47

2.17

6.49

0.37

13.48

0.94

8.66

0.32

7.74

0.46

7.20

0.48

8.79

0.31

7.67

0.36

17.16

0.86

10.39

0.67

15.13

0.63

8.85

'0.19

8.80

0.48

4.88

0.29

4.52

0.36

6.35

0.25

10.96

0.82

7.13

0.43

15.55

0.67

8.80

0.34

6.26

0.33

5.82

0.36

8.94

0.19

10.30

0.55

16.70

0.73

12.27

0.80

15.37

0.56

9.39

0.15

0.34
0.30
0.18

0.17

0.64
0.23
0.23
0.21
0.36
0.18
0.12

0.32

0.61
0.51
0.45

0.10

0.45
0.28
0.21
0.21

0.77
0.27
0.27
0.25
0.29
0.20
0.13

0.46

0.68
0.71
0.51

0.12

0.0451"b
0.2396"
0.0457"
0.0838"

0.1323"
0.0315"
0.0500"
0.0263"
0.0093
0.0557"
0.0547"

0.0325"

0.0071
0.0089
0.0072

0.0170"

-0.0139
0.0116
0.0384"
0.0062

0.0713"

0.0330"

0.0227"

0.0201"

0.0024

0.0404"

0.0255"

0.0104

0.0204"
0.0417"
0.0246'

0.0186"

57.2
37.7
70.2
57.2

91.3
61.3
94.0
56.9

85.9
85.0

21.0

72.3

66.0

11.8
60.6
83.8
45.9

69.1

53.2

13.2
23.4
17.1

36.0

[Vc, (y) - Vsrs(y)]

"Underestimate

x 100.

Vci(y)

^Indicates significant correlation with P < 0.05.

Estimates for Groups of States

These estimates were obtained in the same manner as
was done at the State level. In the following, the terms
"flyway" and "national" are used to indicate simulated
totals at these levels because only 12 States are repre-
sented. For "flyway" level estimates, all zones of all States
represented in each flyway (ignoring State boundaries)
were combined in "unweighted" form by using equations
(4) and (5). To obtain "national" estimates for each hunt-
ing season, zone means were combined ignoring both
State and flyway boundaries (Tables 5 and 6).

For the majority of the "flyways," the intracluster cor-

relation coefficient is significantly larger than zero and in
most of these instances the SRS approach would under-
estimate the variance by about 72% in the 1971-72 sea-
son and 36% in the 1972-73 season for kill per hunter; for
days per hunter the figures were respectively 56 and 60 % .
It is evident that at flyway and national levels, just as at
the State level, estimation of variance or error should be
made by the cluster technique rather than by SRS. This
implies that the zonal estimates must all be calculated by
a cluster approach such as equations (1) and (3). State,
flyway, and national results are then automatically of the
cluster type.

The choice of estimation method can influence subse-

8

Table 6. Estimated days hunted per hunter, standard error, intracluster correlation, and underestimate in variance ob-
tained at State, "fly way," and "national" levels by using individual hunters (SRS) rather than post offices (cluster)
as sampling units.

Hunting season

Days
per

Standard

error

Intracluster

Underestimate3

i r% Art*r\r

111 < 1 I < 'I

and flyway

State

hunter

Cluster

SRS

correlation

(percent)

1971-72

Florida

5.95

0.29

0.17

0.0541"b

65.6

Atlantic

Maryland

8.00

0.29

0.24

0.0018

—

New York

7.29

0.18

0.15

0.021 r

30.6

Combined

6.86

0.15

0.10

0.0223"

55.6

Mississippi

Arkansas

7.22

0.59

0.26

0.1316"

80.6

Illinois

8.38

0.31

0.18

0.0481'

66.3

Louisiana

8.50

0.30

0.18

0.0604"

64.0

Minnesota

7.47

0.22

0.16

0.0188"

47.1

Missouri

8.70

0.39

0.28

0.0451'

48.4

Wisconsin

8.31

0.35

0.13

0.0667"

86.2

Combined

8.11

0.14

0.08

0.0567"

67.3

Central

Texas

6.40

0.16

0.14

0.0222'

23.4

Pacific

California

8.50

0.37

0.27

0.0107

_

Washington

8.82

0.42

0.33

0.0167

—

Combined

8.60

0.29

0.21

0.0125"

47.6

Entire Season

7.69

0.09

0.06

0.0170"

55.6

1972-73

Florida

6.67

0.22

0.23

0.0028

_

Atlantic

Maryland

7.89

0.29

0.29

0.0082

—

New York

7.48

0.26

0.19

0.0279'

46.6

Combined

7.22

0.15

0.13

0.0182'

24.9

Mississippi

Arkansas

9.05

0.64

0.40

0.1052'

60.9

Illinois

8.80

0.35

0.20

0.0458"

67.3

Louisiana

8.38

0.25

0.20

0.0487"

36.0

Minnesota

7.73

0.15

0.16

0.0081

—

Missouri

6.90

0.35

0.23

-0.0027

—

Wisconsin

8.27

0.20

0.17

0.0343'

27.7

Combined

8.10

0.12

0.09

0.0379"

92.4

Central

Texas

6.18

0.23

0.19

0.0110

-

Pacific

California

8.40

0.27

0.31

0.0056

—

Washington

8.70

0.48

0.35

0.0487"

46.8

Combined

8.49

0.23

0.24

0.0194*

-8.9

Entire Season

7.72

0.11

0.07

0.0264"

59.5

[Vc, (y) - Vsrs(y)]

aUnderestimate =

■ x 100.

Vc,(y)
b'Indicates significant correlation with P < 0.05.

quent statistical analysis. The results given in Table 5 for
the State of Illinois can serve as an example. The hypothe-
sis can be advanced that in Illinois there was no difference
between the 1971-72 and 1972-73 hunting seasons in
terms of kill per hunter. The estimates of mean duck kill
for the two seasons are 6.49 and 7.13, respectively. If the
null hypothesis is tested at the 10% significance level by
the usual <-test, it is accepted when standard error is cal-
culated by the cluster approach but rejected when stand-
ard error is calculated by the SRS approach. Because the
intracluster correlation coefficients for both seasons are
significantly greater than zero, the incorrect choice of the
SRS estimator would have led to an unreliable con-
clusion.

Efficiency of Estimators Based
on Cluster Sampling

We have seen that the ratio estimator (equation [1])
based on a simple random sample of post offices (Table 3,
ignoring stratification) was most suited for use with data
obtained through the U.S. Hunter Survey. Two other
estimators will be considered (Cochran 1963:250-252).
These are the mean of the unit means (MUM) and the
estimate based on probability proportional to an estimate
of size (PPES). All three estimators were used to estimate
mean duck kill and days hunted per hunter, and coeffi-
cients of variation for all 12 States and both hunting sea-
sons (Tables 7 and 8). As before, State estimates are

Table 7. Estimates of kill per hunter (KPH) and coefficients of variation (C.V.)
by using three "cluster" methods.

Hunting season

State

Ratio

PPES

MUM

and flyway

KPH

100 C.V.

KPH

100 C.V.

KPH

100 C.V.

1971-72

Florida

7.30

7.67

9.03

14.62

7.20

10.56

Atlantic

Maryland

6.00

6.00

10.97

15.86

6.98

17.91

New York

3.89

5.91

3.95

7.59

3.68

6.25

Mississippi

Arkansas

8.73

12.83

10.61

18.85

19.92

25.65

Illinois

6.20

5.48

7.65

9.41

5.34

7.68

Louisiana

14.37

6.12

24.74

15.36

14.53

8.88

Minnesota

8.54

3.16

15.70

16.81

8.30

4.70

Missouri

7.52

6.38

9.62

15.59

6.19

7.92

Wisconsin

6.63

6.64

6.35

8.82

5.72

9.26

Central

Texas

8.29

6.39

10.28

18.29

7.77

11.97

Pacific

California

18.31

3.93

42.58

16.13

18.02

4.72

Washington

9.77

4.71

17.26

16.63

10.01

8.69

1972-73

Florida

9.35

6.84

7.14

13.45

7.70

7.53

Atlantic

Maryland

5.24

5.92

6.94

19.74

5.12

11.91

New York

4.12

5.10

5.45

7.71

4.57

7.44

Mississippi

Arkansas

10.28

6.42

15.35

14.14

10.03

12.06

Illinois

6.98

5.44

13.32

7.06

6.31

6.18

Louisiana

14.89

3.69

15.30

16.14

15.16

10.55

Minnesota

8.75

3.09

16.76

17.18

8.50

4.94

Missouri

6.29

4.45

9.27

17.37

5.90

7.80

Wisconsin

5.68

4.93

8.02

17.58

6.00

8.00

Central

Texas

11.04

7.79

25.33

28.94

18.89

29.54

Pacific

California

17.00

4.65

29.24

19.32

16.50

7.15

Washington

11.82

8.63

16.01

16.55

11.14

10.68

weighted sums of zone estimates; the weight for a zone is
the proportion of the State's total duck stamp sales that
occurred in that zone. Of the three, the estimator most
often of highest precision was the ratio estimator, as indi-
cated by the much smaller coefficients of variation gener-
ally obtained with this method.

Effect of Non-Normality on the Estimates

Mean ducks shot and days hunted per hunter within
each State based on post offices as sampling units were
used to estimate departures from normality by skewness

(gi = m3/m23/2) and kurtosis (g2 = —^ - 3)

m2

where

m2 = E (Yi - P)2/n; m3 = E (yt - y)3/n;
i i

m4 = L (y, - y)4/n;

i

and n is the number of sample post offices in a State.
Sample post offices with less than five duck stamp buyers
responding were omitted from this study.

The estimates of mean kill, days hunted, skewness, and
kurtosis are presented in Tables 9 and 10 (tests of skewness

and kurtosis follow Snedecor and Cochran [1967:86-
87]). It is evident that kill per hunter was positively and
highly skewed in almost all States, which had the effect of
increasing the variance of the mean kill and decreasing its
precision. Kurtosis was highly pronounced in about 50%
of the States, which also reduced the precision of the er-
rors of the means. Both skewness and kurtosis were some-
what lower for mean days hunted than for mean kill.

The question arises as to how large n (the sample size
for post offices) must be within a State for the normal
approximation to be accurate enough for estimating
mean kill and activity. In a personal communication with
the second author, W. G. Cochran, whose work
(1963:41) assumes only marked skewness in a population,
advised us of more recent unpublished work by
G. Bartsch for populations in which the deviation from
normality involves both marked skewness and kurtosis.
Using this information, we have

n > 25G,2 + 1.64G2 (10)

where G, and G2 are Fisher's measures of skewness and
kurtosis in the population and are estimated by g, and g2,
so that a 95% confidence probability statement will not
be wrong more than 6 % of the time.

Of the 12 States considered in the study, the sample size
for estimating kill and activity in 6 (New York, Arkansas,
Louisiana, Minnesota, Texas, and Washington), was less

10

Table 8

Estimates of days hunted per hunter (DPH) and coefficients of variation (C.V.)
by using three "cluster" methods.

Hunting season

Ratio

PPES

MUM

and flyway

State

DPH

100 C.V.

DPH

100 C.V.

DPH

100 C.V.

1971-72

Florida

6.17

4.70

7.56

12.96

5.84

6.34

Atlantic

Maryland

8.83

3.51

15.80

14.24

8.85

5.65

New York

7.09

2.26

7.57

6.74

6.73

3.42

Mississippi

Arkansas

6.49

12.33

7.88

13.20

10.06

13.22

Illinois

8.20

3.41

11.24

8.10

7.71

5.32

Louisiana

8.58

3.73

15.40

13.12

9.15

6.88

Minnesota

7.17

2.65

14.33

16.33

7.07

3.11

Missouri

8.15

4.42

11.82

16.33

7.08

5.93

Wisconsin

8.34

5.28

10.40

11.15

8.48

6.72

Central

Texas

6.84

3.22

7.54

11.67

6.65

6.77

Pacific

California

8.27

2.66

16.26

12.85

7.74

3.10

Washington

8.45

4.26

15.37

15.68

8.91

7.63

1972-73

Florida

6.72

3.87

5.98

12.37

6.11

5.73

Atlantic

Maryland

8.53

3.99

10.03

13.16

8.72

8.03

New York

7.15

2.80

9.45

5.93

7.45

11.79

Mississippi

Arkansas

8.79

6.03

14.80

12.84

9.41

7.86

Illinois

8.60

3.95

15.45

32.36

7.94

4.66

Louisiana

8.04

3.11

10.22

18.20

9.00

9.44

Minnesota

7.46

2.41

15.37

16.07

7.44

2.96

Missouri

6.84

3.80

11.77

15.72

6.79

5.30

Wisconsin

8.35

4.79

12.35

15.95

8.49

6.12

Central

Texas

6.05

8.43

11.14

13.64

7.69

11.96

Pacific

California

8.28

2.78

13.13

13.10

7.92

4.29

Washington

8.38

5.61

12.13

12.61

8.94

7.83

than that needed for application of normal approxima-
tion. Because these 12 States had much larger samples
than most in the U.S. survey, kill and activity would be
estimated in almost all others with even less confidence
unless sample sizes were increased considerably, which is
not generally feasible.

However, when the data were transformed by setting
Xj = ln(Vj + 1) where y> = yj /Mh the mean kill or activity
for the ith post office in the sample, the distributions were
approximately normal; only 2 of the 12 States showed sig-
nificant skewness and kurtosis, and this, too, was not con-
sistent over the years. For detecting real differences be-
tween States when the distributions were highly skewed,
estimates of means on the transformed scale were more
precise, as evidenced by the confidence intervals pre-
sented in Table 1 1 . There was no significant difference on
the original scale in mean kill for 1971 between
Minnesota and Missouri or Louisiana and California;
however, real differences became evident after adjust-
ment for non-normality. Similarly, for 1972 the adjusted
mean kills (x) were significantly different between
Arkansas and Wisconsin and between Arkansas and Illi-
nois, although no differences were shown in means calcu-
lated directly (y). Further improvement in sensitivity of
tests can be made by pooling estimates of error on the

transformed scale because the transformed data are ex-
pected to have a more constant variance.

We will now transform the x back into the original
variates (i.e., obtain antilogarithms) and see if any gain
in efficiency has been achieved. This is important because
mean kill (or days hunted) can be readily interpreted and
is, therefore, more useful to management than its
logarithm. The efficiency of y with respect to mean m
when the means of the logarithms are transformed back is
approximately estimated by

(11)

(s* + 2 )

es - 1

where m is approximately equal to e(x + T' - 1 and s2 is
an unbiased estimate of the variance of x (Finney 1941);
in large samples and for small values of a2, the efficiency
of y as given by (11) can be almost 100% . The efficiency
of the direct sample mean kill ranged from 97 to 100%
(Table 12); for days hunted, the efficiency was 100% in
almost all instances.

The efficiency of direct estimates of population var-
iance with respect to estimates based on the transformed
data (Finney 1941:159) is also included in Table 12. This
shows that direct estimates of the variance of the mean

11

Table 9. Estimates of mean duck kill (K/H), skewness (gt), and kurtosis (g2) at the State level.

1971-72

1972-73

State

K/H

gi

g2

K/H

gi

g2

Florida

Maryland

New York

Arkansas

Illinois

Louisiana

Minnesota

Missouri

Wisconsin

Texas

California

Washington

7.21 (38)»

0.82*b

5.67 (43)

1.25"

4.01 (98)

2.07"

11.36 (28)

3.30"

5.82 (76)

1.28"

13.78 (39)

1.75"

8.63 (50)

3.49"

7.06(40)

0.82*

5.86 (50)

1.18"

7.62 (59)

1.47"

18.65 (61)

0.85"

10.41 (29)

1.38"

0.07

1.90*

3.99*'

12.74"
1.72"
3.83**

17.52"
0.30
1.28*
2.73"
1.07*
2.07*

7.43 (30)

1.05"

5.15 (33)

0.48

4.37 (89)

1.91"

11.62 (18)

3.01"

6.95 (72)

1.32"

15.20 (34)

1.68**

9.14 (50)

0.61*

5.88 (33)

0.80*

5.96 (39)

0.79*

12.57 (47)

3.45**

15.97 (51)

0.76*

11.66 (31)

2.06"

0.94
-0.32

5.90"

8.96"

3.34**

3.30"

0.16

0.87

0.23
12.34"

0.40

4.48"

"Figures in parentheses are numbers of post offices on which estimates are based.
b*Indicates significance with P < 0.05; "indicates significance with P < 0.01.

Table 10. Estimates of mean days hunted (D/H), skewness (gj), and kurtosis (g2) at the State level.

1971-72 1972-73

State

Florida

Maryland

New York

Arkansas

Illinois

Louisiana

Minnesota

Missouri

Wisconsin

Texas

California

Washington

D/H

gl

g2

D/H

gi

gst

6.54 (38)a

-0.32

0.69

6.47 (30)

0.21

-1.19

8.72 (43)

1.17"b

2.04"

7.81 (33)

1.24"

1.15*

6.97 (98)

1.33**

2.92"

7.44 (89)

0.68**

0.16

8.74 (28)

1.12"

1.60*

9.05 (18)

1.38"

1.90*

7.90 (76)

0.67*

-0.09

8.47 (72)

0.22

-0.06

8.78 (39)

1.36**

3.33"

8.56 (34)

0.79*

0.02

7.53 (50)

3.15"

14.81"

7.66 (50)

0.51

0.16

7.67 (40)

0.54

0.33

6.60 (33)

1.05"

0.60

7.53 (50)

1.26"

1.58*

8.17 (39)

0.43

0.49

6.54 (59)

0.77**

2.49"

6.87 (47)

1.78"

4.66*

8.15 (61)

0.56*

-0.02

7.97 (51)

0.35

0.76

9.17 (29)

1.49"

2.02*

8.66 (31)

1.44**

1.78

aFigures in parentheses are numbers of post offices on which estimates are based.
b*Indicates significance with P < 0.05; "indicates significance with P < 0.01.

Table 11. Comparisons of 95% confidence interval (C.I.) estimates of mean kill calculated from original units

n n

(y = £ yj/n) and from transformed data (x = L x/n where Xj = ln[yj + I]) transformed back to original units.
i = 1 i = 1

95% C.I. fory

State

1971

1972

95% C.I. forx
(Converted to original units)

1971

1972

Florida

Maryland

New York

Arkansas

Illinois

Louisiana

Minnesota

Missouri

Wisconsin

Texas

California

Washington

5.71-8.70

4.66-6.68

3.41-4.62

7.45-15.27

4.99-6.65

10.80-16.70
7.65-9.61
5.81-8.31
4.93-6.79
6.34-8.90

16.34-20.95
8.72-12.10

6.09-8.77

4.26-6.04

3.82^.92

6.79-16.45

5.95-7.94

12.13-18.26
8.18-10.10
4.94-6.83
4.97-6.94
9.58-15.57

13.70-18.23
9.27-14.05

5.75-8.50

5.16-6.89

4.01-4.85

7.46-12.81

5.31-6.75

10.49-15.03
8.50-9.97
6.05-8.41
5.42-7.03
6.36-8.58

15.33-20.09
9.30-12.18

6.62-9.02

4.81-6.55

4.44-5.37

8.00-13.74

6.05-7.85

12.06-16.95

8.67-10.59

5.47-7.39

5.47-7.32

10.18-13.46

13.07-17.46

9.78-13.46

12

Table 12. Efficiency of direct estimates of sample mean kill and days hunted and their variances relative to estimates
obtained after logarithmic transformation.

State

Efficiency of direct mean (%)

Kill

Days 1
1971

lunted

1971

1972

1972

98

99

100

100

99

99

99

100

99

99

100

100

97

100

100

99

99

99

100

100

98

99

100

100

100

100

100

100

99

100

99

100

99

99

100

100

98

99

100

100

99

99

100

100

100

99

100

100

Efficiency of direct variance (%)

Kill

Days 1
1971

lunted

1971

1972

1972

51

68

81

84

62

67

69

77

63

67

78

82

39

53

71

64

60

55

78

77

54

61

74

76

82

76

87

84

59

69

66

87

64

66

82

80

54

62

81

79

59

60

81

70

73

67

83

83

Florida

Maryland

New York

Arkansas

Illinois

Louisiana

Minnesota

Missouri

Wisconsin

Texas

California

Washington

kill for a State would generally be inefficient compared
with estimates of variance obtained by transforming back
the variance of the logarithms; for days hunted, however,
direct estimates of the variance of the sample mean were
generally of high efficiency.

We suggest, therefore, that direct estimates of mean
and variance always be adopted except for estimating
standard error of mean kill, which should be obtained by
applying logarithmic transformation procedures. There is
need for examination of data from other States and for a
number of years to obtain confirmatory evidence.

Table 13. Response rates (%) at the post office stratum
level in the 1971-72 and 1972-73 Hunter Question-
naire Surveys for the combined 12-State sample.

1971-72

1972-73

Address

Question-

Address

Question-

Stratum

cards

naires

Total

cards

naires

Total

1

44.7

67.6

30.2

35.7

67.0

23.9

2

38.4

66.8

25.6

34.0

66.2

22.5

3

21.2

72.4

15.3

22.4

71.2

15.9

Combined

33.9

68.2

23.1

30.1

67.8

20.4

Nonresponse

We will use the term nonresponse to refer to failure to
measure some of the units in the selected sample. Nonre-
sponse at a selected post office or branch office will,
therefore, be defined as the ratio of the number of people
who fail to return completed questionnaires to the num-
ber of potential hunters at the selected outlet (about 1 %
of stamp purchasers have no intention of hunting but are
stamp collectors or wish to support conservation by buy-
ing a stamp and thus do not qualify as potential hunters).
Although nonresponse does not necessarily affect estima-
tion procedures, it can introduce a serious bias.

Nonresponse in the U.S. mail survey occurs in three
stages: (1) the postal clerk may fail to give a card to the
hunter; (2) the hunter may choose not to fill out an
address card; and (3) after having sent in an address card
the hunter may not complete and return the question-
naire. Nonresponse at the first stage may be due to an in-
sufficient supply of cards or the postal clerk failing to
hand a card to every duck stamp purchaser. Nonresponse
at the second and third stages is dependent on the hunter
and may add seriously to the error of the estimates. The
hunter's refusal to supply his name and address probably

holds more potential for nonresponse bias than his failure
to return a questionnaire (Table 13).

The number of contact cards sent to sample post offices
is based on an estimate of expected stamp sales during the
current season. Table 13 shows that questionnaire re-
sponse rates are very similar among strata but address
card response rates decrease with increases in post office
size. The very low response rate for stratum 3 is due to a
relatively high nonresponse at the address card level as
compared with strata 1 and 2. The percentage of re-
spondents is far too low for reliable estimates if the
hunters not responding have activity characteristics
which differ from those of respondents and, therefore,
may cause a substantial nonresponse bias.

Our estimates of expected duck stamp sales, and there-
fore of response rates, are poorest for stratum 3, but
whether the low response is inherent in the stratification
scheme or arises from characteristics of the post offices or
of the hunters in stratum 3 is not known. Possibly the
large offices and branches of stratum 3 have responsibil-
ities for stamp sales divided among more clerks or the
clerks have greater workloads, and these contribute to
their large address card nonresponse. Whether this
nonresponse is real or an artifact of stratification, it is es-
sential that we determine if it introduces nonresponse

13

biases in these kill and activity estimates. In the other two
strata, nonrespondents may represent a different class of
hunters from those who respond (Atwood 1956; Sen 1972;
Filion 1974). If there is such a bias, elimination of
stratification by post office size will not reduce or control
it but simply mask it in the overall sample. In the absence
of more information on nonresponse, the present stratifi-
cation system cannot be utilized to improve the efficiency
of our estimates.

Our testing for differences in hunter characteristics
among strata was handicapped by the relatively small
samples of post offices from stratum 3 and sometimes
stratum 2; therefore, we combined them. This weakness
of the analysis, and therefore its results, is brought into
question again by this finding of substantial differences in
hunter response among strata and the increased suspicion
that other characteristics may also differ. Only further
study can answer these questions and provide appropriate
corrective measures where needed. Investigations into
methods of inducing increased response both at the ad-
dress card level and in questionnaire returns appear hope-
ful. In the United States, a reminder letter sent to sample
post offices several weeks before the hunting season opens
increases address card response materially and, in the
Canadian survey, the questionnaire response rate has
been substantially increased by sending a reminder card
immediately following the first mailing of the question-
naire. Much more effort must be put into these and re-
lated studies if the reliability of our harvest surveys is to
meet the increasingly high standards expected of them.

Intracluster correlation coefficients are almost always
positive, which indicates that for the sample data there is
more variation in kill and activity among post office
clusters than among hunters within post offices. This sug-
gests that a sampling scheme in which fewer hunters are
selected, each from a large number of post offices, would
be more efficient than one in which the same sample is
confined to a few post offices. Because such sampling is
not feasible, a systematic sample of post offices by loca-
tion within zones (i.e., increased emphasis on broader
geographic distribution of the sample) with sampling of
all hunters in each post office appears to be a more effec-
tive approach.

Both kill and days hunted per hunter within a State
showed marked skewness from normality and yielded in-
efficient estimates of means and their errors. Logarithmic
transformation of the data resulted in approximate nor-
mality and in considerable increase in the precision of the
estimated error of mean kill.

Nonresponse was highly pronounced at the "address
card" level, especially for large post offices, and might
lead to sizable bias in estimates of kill and activity. There
is need for investigation into this problem in the interest
of overall efficiency and precision.

We extend our thanks to G. E. J. Smith and H. Boyd of
the Canadian Wildlife Service and to P. H. Geissler of the
U.S. Fish and Wildlife Service for their useful comments
and suggestions.

References

Summary

The present U.S. method of substratifying geographic
zones into post office groups based on the number of duck
stamps sold is ineffective in terms of estimation of the pre-
cision of hunter success and activity figures. Elimination
of post office stratification would not decrease the effi-
ciency of the estimates but would decrease the amount of
work required for implementation. However, other
problems involving post office size and other considera-
tions such as sampling for the closely related duck wing
survey might require retaining such stratification. These
aspects need further evaluation.

Estimates of error for both ducks killed and days
hunted should be consistent with the selection scheme
which is essentially one of cluster sampling of post offices
and not of individual hunters buying duck stamps at post
offices. Estimation by methods other than those based on
cluster sampling is not recommended because it can intro-
duce bias.

Atwood, E. L. 1956. Validity of mail survey data on bagged wa-
terfowl. J. Wildl. Manage. 20(1):1-16.

Cochran, W. G. 1963. Sampling techniques. 2nd Edition. J.
Wiley & Sons, Inc., New York. 413 pp.

Filion, F. L. 1974. Estimating bias due to nonresponse in harvest
surveys. Can. Wildl. Serv. Biom. Sect. Manuscr. Rep. 5. 57
pp.

Finney, D. J. 1941. On the distribution of a variate whose loga-
rithm is normally distributed. J. R. Stat. Soc., Ser. B., 7:155-
161.

Martin, E. M., and S. M. Carney. 1977. Population ecology of
the mallard: IV. A review of duck hunting regulations, activ-
ity, and success, with special reference to the mallard. U.S.
Fish Wildl. Serv., Resour. Publ. 130. 137 pp.

Sen, A. R. 1972. Some nonsampling errors in the Canadian wa-
terfowl mail survey. J. Wildl. Manage. 36(3): 95 1-954.

Snedecor, G. W., and W. G. Cochran. 1967. Statistical meth-
ods. 6th Edition. Iowa State University Press, Ames. 593 pp.

Steel, R. G. D., and J. H. Torrie. 1960. Principles and proce-
dures of statistics. McGraw-Hill Co., New York. 481 pp.

Wright, V. L. 1978. Causes and effects of biases on waterfowl
harvest estimates. J. Wildl. Manage. 42(2): 25 1-262.

14

APPENDIX

Stratification of States into geographic zones is an ef-
fective procedure (improves accuracy of estimation) if
there are significant differences between the zones of a
State in terms of hunter characteristics. To test for zone

differences within each State, we performed analysis of
variance (ANOVA) and the results are presented in the
following table.

Observed probability levels (P) of F-vahies calculated to test for zone differences ("between" vs. "within" zone)

for each State.

Hunting season

x duck kill

x days hunted

Hunting season
and flyway

x duck kill

it days hunted

and flyway

State

P

df

P

df

State

P

df

P

df

1971-72

Florida

0.14"

6, 48

0.01

6. 48

1972-73

Florida

0.07

6,

45

0.01

6, 45

Atlantic

Maryland

0.47

2, 55

0.00

2, 55

Atlantic

Maryland

0.00

2,

49

0.00

2, 49

New York

0.00

6, 110

0.00

6, 110

New York

0.00

6,

102

0.00

6, 102

Mississippi

Arkansas

0.26

2, 33

0.00

2, 33

Mississippi

Arkansas

0.14

2,

27

0.27

2, 27

Illinois

0.00

4, 107

0.00

4, 107

Illinois

0.15

4,

98

0.01

4, 98

Louisiana

0.01

4, 43

0.12

4, 43

Louisiana

0.01

4,

41

0.00

4. 41

Minnesota

0.21

6, 48

0.00

6, 48

Minnesota

0.01

6,

53

0.00

6, 53

Missouri

0.00

4, 52

0.00

4, 52

Missouri

0.13

4,

46

0.05

4, 46

Wisconsin

0.00

4, 50

0.01

4, 50

Wisconsin

0.06

4,

38

0.13

4. 38

Central

Texas

0.05

5, 79

0.00

5, 79

Central

Texas

0.85

5,

76

0.60

5. 76

Pacific

California

0.00

6, 73

0.00

6, 73

Pacific

California

0.01

6,

64

0.00

6, 64

Washington

0.16

2, 34

0.07

2, 34

Washington

0.60

2,

37

0.47

2, 37

aThe observed probability levels are correct to two decimal places.

As the Nation's principal conservation agency, the Department of the
Interior has responsibility for most of our nationally owned public lands
and natural resources. This includes fostering the wisest use of our land
and water resources, protecting our fish and wildlife, preserving the
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our people. The Department also has a major responsibility for American
Indian reservation communities and for people who live in island
territories under U.S. administration.

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