m
HARVARD UNIVERSITY
Library of the
Museum of
Comparative Zoology
Number 4
Kay 31, I938
A STATISTICAL STUDY OF THE RATTLESNAKES
s-nA-S
°K
/■ S^ By Laurence M. Klauber
' Curator of Reptiles and Amphibians, *%^ °% %\
San Diego Society of Natural Historic ^ '^ ^ ^^ '^■-^^
I
V* HEAD DIMENSIONS
Table of Contents
Introduction 2
Measurement Methods 3
Sexual Dimorphism 3
Nature of Head-to-Body Relationship 5
Comparison of Straight Line and Parabolic Regression . . 7
Example Colubrid Head Proportions 9
Dispersion in Head Length — Variation with Body Length . 11
Extent of Dispersion — Limited Age Groups 12
Character of Dispersion--Complete Age Series 15
Head Length as a Diagnostic Character 20
Example Taxonomic Problems 22
Validity of Head Length as a Diagnostic Character ... 24.
Short Cut Methods 25
Use of L/H Ratio 27
Dwarfed Forms 29
Effect of Ultimate Length on Species-Difference
Calculations 34
Effect of Ultimate Length on Dispersion 36
Sexual Dimorphism in L/H 36
Species Differences — General Discussion 39
Width and Depth 4-5
Summary and Conclusions 51
* Previous sections of this paper have appeared as follows:
I, Introduction; II, Sex Ratio in Rattlesnake Populations;
III, Birth Rate; Occasional Papers S.D.Soc.Nat .Hist . , No.l,
pp. 1-24, figs. 1-2, tables 1-4, Aug. 10, 1936: IV, The Growth
of the Rattlesnake; Occasional Papers S.D.Soc.Nat .Hist . ,
No. 3, pp. 1-56, figs. 3-6, tables 5-16, Dec. 15, 1937.
1,
A STATISTICAL STUDY OF THE RATTLESNAKES
By Laurence M. Klauber
V HEAD DIMENSIONS
Introduction
The head length of a snake is known to be a diag-
nostic character of some value and as such has often been
cited in species descriptions and in taxonomlc comparisons.
However, it is a quantity which changes with growth, during
the life of each individual, and therefore a basis of com-
parison other than mere linear measurement must be devised.
Usually this has taken the form of a proportion — the ratio
of the head length to the body length over-all; or its re-
ciprocal, the number of times the head length is contained
in the body length. It is the purpose of this study to de-
termine the value of this character In species diagnosis,
at least as it applies to the rattlesnakes. In such an in-
vestigation we must consider the accuracy (and therefore
the validity) of measurements. We must find how the head
size varies with the body size during growth — whetner this
relationship can be expressed as a simple ratio, as has
often been assumed to be the case. We must examine the
dispersion of this proportionality within a species to find
whether Intraspecies variation is so great as to limit or
Invalidate comparisons between species. The presence or
absence of sexual dimorphism must be ascertained.
In addition to the possible taxonomlc value of
head length studies, they afford a basis for the considera-
tion of variations in fang length and venom yield, which
are matters of some practical importance in the snake bite
problem. I expect, in a subsequent section of this series,
to report upon correlations between head size, fang length,
and venom secretion.
This study of the head length variations among
the rattlesnakes originally contemplated the preparation of
a table comparing the several species and subspecies statis-
tically, drawing attention to those which deviate conspicu-
ously from the rattlesnake mode. Examples were to be pre-
sented, showing how species differences in this character-
istic might be verified.
These simple objectives proved far more elusive
than was expected. The difficulty of dealing with a quan-
tity, which not only varies between individuals, as is to
be expected in any group of animals, but which falls to
remain constant with respect to any single individual dur-
ing his lifetime, became Increasingly apparent as the work
progressed.
In the taxonomlc study of a genus of snakes, head
length is only one of many characters which may be employed
as differential criteria. In most cases it will not be
found of critical Importance, although in a few instances
it may afford finality in determining differences and rela-
tionships. In view of these facts, the space which is here
given to the development of head length statistics is not to
be interpreted as an indication of the relative value which
the writer attaches to this character. But it appeared to
offer a useful example of how the proportion of a body part
may be handled statistically. In addition it seemed worth
while to investigate fully, as a typical example, the nature
of the variation of head length within a single species.
This is my reason for having worked out the complete trends
in the Platteville and Pierre series of Crotalus vlridls
viridis, as subsequently presented. But as a. survey of a
purely taxonomic criterion, this amount of detail is not
Justified.
Measurement Methods
The head length, as used in the present study, is
the distance between the front face of the rostral and the
posterior end of either mandible; this, it is found, can be
calipered with reasonable acc^lracy. This dimension, al-
though at a slight angle with the center-line of the body,
differs only in a minor degree from the distance between the
rostral and the mid-point of a line Joining the posterior
ends of the mandibles, which might be considered the true
length of the head.
Head length, as above defined, is affected by pre-
servative somewhat more than body length. In a series of
specimens in which the body length was found to decrease in
alcohol by 2 per cent, the head length was decreased by an
average of 6.8 per cent. It is therefore desirable to mea-
sure specimens before they have set in the preservative,
although this is not as easy, because of the soft tissues,
as it is after the head has hardened. In any case where the
highest accuracy is required, it is necessary to handle all
comparative material in the same manner, that is, either
freshly killed, or after setting in preservative. In the
present investigation pre-set measurements were frequently
used, especially in the large homogeneous series from which
the character of variation has been determined.
Sexual Dimorphism
In coordinating the statistics of head length it
is advisable to investigate sexual dimorphism first, since,
if there be none, the subsequent problems will be simplified.
To investigate this question a determination was made (sep-
arately for each subspecies) of the average size of the
head for each size-class of body* in series of cinereous.
lucasensls, ruber, scutulatus , viridis, lutosus, and ore-
ganus. These comparisons were made only between size-
classes within the adult range, where sexual dimorphism, if
present, would naturally be manifest. It was found that
such variations as were apparent, as frequently showed one
sex to have a larger head as the other; the variations
seemed to be dictated entirely by chance. These series to-
talled 210 size-classes, involving 184.6 snakes. In these
210 classes it was found that the male heads averaged
* Body size-classes were taken by 10 mm. increments; for
example all the snakes of a subspecies from 740 to 7-49 mm.,
inclusive, were grouped in a single size-class.
HEAD LENGXH IN MM.
larger in 90, the females were larger in 108, and in 12
classes the sexes were equal. Thus we conclude, lAith re-
spect to these species and subspecies, that there is no
sexual dimorphism in head length at equal body lengths. A
more conclusive test of this equality, using viridis as an
example, and a probable deviation in the case of cerastes
are discussed on p. 36, subsequent to the development of
certain methods of analysis.
Nature of Head-to-Body Relationship
As is usual in considering the importance of diag-
nostic characters, there are two phases to be investigated:
first the consistency (or its opposite, dispersion) of the
character within a single species; secondly the differences
between species. The extent of the first determines the re-
lative importance of the second.
That there should be a definite correlation be-
tween head size and body length is obvious; the problem con-
cerns the nature of the correlation, whether linear or of
higher degree, and how closely individuals adhere to the
regression line.
As an initial visual survey, the head-length and
corresponding body-length coordinates of several series of
specimens, each series representing a different species or
subspecies, were set up on rectangular cross-section paper.
In each series the points are found to lie approximately on
a straight line. No definite and consistent curvilineal de-
parture is noted; for while some series appear to bow up-
ward slightly, an equal n\amber have a slight sag. Although
there is some scatter the adherence to the regression lines
is fairly close. Thus the relationship seems to be simple
and linear. However the regression lines do not, in any
species, pass through the origin, the intersection with the
line H = 0 being always on the negative side; the equation
therefore is of the form H = aL + b; and, because of the
presence of the constant positive term b, we know at once
that the head length does not bear a constant proportional-
ity to the body length throughout life.* Furthermore, b is
relatively of considerable magnitude, so that this devia-
tion from a constant ratio is marked — juvenile rattlers have
proportionally larger heads than adults. For example the
equations of the several viridis subspecies all approximate
H = 0.035L + 7.5, where H and L are expressed in millimeters.
Thus a juvenile 300 mm. long has a head length averaging 18
mm., while the head of a 1000 mm. adult measures about 42.5
mm. The juvenile head length is 6 per cent of the body
length; the adult only 4.25 per cent. This proportionately
larger juvenile head is a condition common to many animals.
For the better visualization of the relationships
above set forth, reference is made to Fig. 7 showing, on
rectangular coordinates, the head-body correlation in 100
C. ruber evenly distributed amongst juveniles, adolescents,
and adults, but otherwise chosen at random from a large
* For, if H = aL + b, then H/L = a + b/L; therefore the
ratio of the head to the body is not constant but changes
with L. Lower case algebraic quantities appearing in the
text outside of equations are underlined for clarity, as is
b above.
series. Attention is directed to the facts previously dis-
cussed: (l) a straight line fits the situation quite well;
(2) this line does not pass through the origin, that is,
the point where H = 0 when L = 0; (3) the adherence to the
regression line is fairly close, the scatter not being ex-
cessive.
Also Fig. 8 has been prepared from the same 100
specimens of C. ruber . This shows how L/H changes as the
snakes age, and how futile it is to make interspecies com-
parisons of L/H ratios, unless the material compared be
restricted to a narrow and comparable length-range.
One mighl* suppose this type of relationship be-
tween head and body length to be the result of the short-
ening of the rattlesnake tail by the rattles — that if the
rattlesnake had a normal tail the increased body length
would produce a constant ratio between H and L. However
such is not the case. To produce a constant ratio between
H and L, a constant, b/a, must be added to L; which means
that regardless of the size of the snake we must assume
that the tail was foreshortened in a fixed amount by the
24
23
22
21
20
r
\I9
I
0
O
0
3
0
0
O
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0
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>-
oo8
0
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3 O
0
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0
0
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16
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17
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1 1 1 1 1 1
FIGURE e
16
/
FXAMPLF TREND OF l/h RATIO
/
o
100 Crotalus ruber
15
/
7
lA
/o
100 loo So 400" 500 600 700 800 900 1000
1_ENGXH OF BODY IN MM.
1100 1200 1300 1400
presence of the rattle. This is obviously not what has oc-
curred. Whatever shortening may have been produced through
the replacement of the usual tapered tail by the rattle, it
was in some degree proportional to the size of the snake,
as elementary comparative studies of tail lengths in Cro-
talus and Agkistrodon will show. Certainly if a large rat-
tler 1000 mm. long, with a tail 75 mm. long, lost 180 mm.
by shortening, it is impossible that a juvenile 300 mm. long
with a 22 mm. tail could have similarly lost 180 mm. Yet
something of this nature would be required to indicate de-
rivation from a constant relationship between head and body.
The accuracy of a straight-line relationship of
the form H = aL + b has been questioned by some with whom
the matter has been discussed. They point out that, at zero
body length, H has a value of b, an obvious impossibility.
This I grant, but it is not claimed that the equation covers
the prenatal as well as the postnatal stage. What form the
growth curve takes in the prenatal stage I do not know; I
merely suggest that the straight line closely represents the
relationship during life. Plotting a number of species on
log-log coordinates we find curves to result in almost every
series: thus the frequently used relative-growth equation
Y = cX^ does not represent the situation as well as the
simpler linear equation. Even two consecutive straight-line
sections on log-log coordinates, such as would indicate a
change in the rate of proportionate growth at some point in
the life cycle (adolescence for example) do not seem to be
applicable here.*
Comparison of Straight Line and Parabolic Regression
I do not mean to state that it is impossible to
find a parabolic curve to fit the head-body relationship
with a fair approximation. Taking, for example, the
Platteville series of 833 specimens of Crotalus viridis
vlridis, we find that the equation H = 0.279L>-'-''^4 fits
the situation moderately well, altnough the simpler and
more practicable equation H = 0.0355L + 7 fits consider-
ably better, over the complete range from birth to maximum
size.
To make certain of this matter of relative appro-
priateness of formulas, 19 representative species or terri-
torial series were plotted on log-log paper. In general
the resulting regression lines showed a distinctly greater
tendency to curvature than was the case with the same data
plotted on rectangular coSrdinates. Only in one or two
cases out of the 19, where uniform coordinates resulted in
regression lines perceptibly bowed upward in the center,
did the log-log paper tend to straighten the line; in most
cases logarithmic coordinates produced a curve having a
central sag, with considerably greater deviation from a
straight line than the quadrille ruling. However, for those
who prefer equations of the parabolic form, the following
table gives the constants for each species in the equation
H = cL^, which produce an approximate fit, although the
formulas will usually give juvenile and adult heads lower
than actuality, while the adolescents will be high.
* See Julian Huxley, Problems of Relative Growth; New York,
1932, p. 10.
Constants In the Equation H = cL^
Species or Subspecies
c
k
C.d. durissus
0.330
0.715
C . enyo
0.339
0.680
Cm. molossus
0.240
0.764
C. adamant e us
0.241
0.775
C. cinereous
0.300
0.722
C. lucasensis
0.339
0.708
C. ruber
0.275
0.741
C. scutulatus
0.196
0.634
C.v. vlrldis
Platteville series
0.279
0.724
Pierre series
0.343
0.692
C.v. lutosus
0.310
0.703
C.v. oreganus
San Diego County-
0.315
0.717
Cm. pyrrhus
0.226
0.762
C cerastes
0.159
0.817
Ch. horridus
0.374
0.684
CI. klauberl
0.230
0.748
Ct. pricei
0.196
0.778
S.m. streckeri
0.422
0.657
S.c. catenatus
0.663
0.584
It will be observed that there are considerable
differences in the constants of closely related species-
differences which do not occur to a similar extent in the
constants of the straight-line formulas, as will be subse-
quently shown. This is because of the major effect of the
Juveniles in determining a parabolic equation, and the con-
siderable changes in the constants with small changes in
the slope of the curve. In reality the statement of the
third figure in these constants gives an aspect of accuracy
which is not Justified by the facts.
One minor advantage in the use of an equation of
the form H = cL^ lies in its convertibility into an equa-
tion of L/H in terms of L, while the straight line form
H = aL + b is not convertible, as pointed out before. Thus
from H = cL^ we derive the equation L/H = c--L-'-~J^. For
example in the Platteville series, if H = 0.279L^- ''^'^, then
L/H = 3.58l0'276, But this is to be considered only an
approximation. In general there appears to be no advantage,
either of accuracy or workability in adhering to the more
involved parabolic formula, consequently the subsequent dis-
cussion is based on the more accurate straight-line rela-
tionship.
Of course there is no theoretical reason why rat-
tler H on L regression lines should conform either to the
straight line H = aL + b or the simple parabola H = cL^.
They might equally well follow some curve of hifeher degree
lying between the two.* But as I have said, plotting many
* To determine the nature of the curves followed by those
species which (probably due to the exigencies of sampling)
diverge from straight lines, the two showing the most con-
spicuous deviations were selected for analysis. These are
cinereous which sags slightly, and c. catenatus which bows
upward in the center. By the usual methods of curve fit-
ting both are found to adhere to second degree parabolas
very closely. As the curves are quite flat, the L"^ compo-
8
series results in deviations on both sides of a straight
line, in about equal numbers; so that I am convinced that
the straight line gives the nearest approach to a universal
fit. With respect to one species the straight line is shovii-n
analytically to give the best fit (p. 16).
Example Colubrid Head Proportions
Before leaving the matter of the suitability of
certain types of equations to represent this relationship,
I might mention some results of a similar investigation of
several colubrine snakes. Having determined that the rat-
tlers rather unexpectedly followed a straight line regres-
sion in the head length relationship, it was natural to
check the situation in other genera to find if this were a
common or universal relationship amongst the snakes. Pre-
liminary results indicate that it is not. Of six species
tested, two seem to follow straight lines. Arizona elegans
occidentalis, based on 64. specimens, adheres closely to the
equation H = 0.022L + 6.5j while 82 specimens of Lampropel-
tls getulus californiae give H = 0.024 + 6.3. But 4. other
colubrids selected at random all show distinct sags when
plotted on quadrille ruled paper and essentially straight
lines when plotted on semi-log coordinates. Hence their
equations are of the form H = mii^, where m and n are con-
stants. In using this equation it is best to express L in
meters, rather than in millimeters, since to do otherwise
makes n unwieldy in form. The results secured in the cases
tested were as follows:
Niomber of
Species Specimens
Trimorphodon vandenburghi 14-
Phyllorhynchus d. perkinsi 70
Masticophis f. frenatus 36
Masticophis lateralis 23
For example the head length relationship in Tri-
morphodon vendenburghi may be written H = 7.35(4.52)^, H
being the head length in millimeters and L the length over-
all in meters.
Whether there is anything significant in the fact
that some species appear to follow a straight line, while
an exponential equation best suits the others, I am not pre-
pared to say. This offers an interesting field of research.
Possibly some intermediate form of curve, falling between
the two, may be more universally applicable. In any case
it is interesting to note that in none of these colubrids
does the head length have a constant ratio to body length.
Most colubrids are considerably more difficult to measure
with accuracy than such broad-headed snakes as the rattlers.
nents are small. The straight lines of best fit are also
presented for comparison:
C. cinereous Parabola H = 0.000004321^ + 0.0306L +9.5
Straight line H = 0.0351L + 8.7
S.c.catenatus Parabola H = -0.00000$04L2 + 0.0344L +8.6
Straight line H = 0.0302L +9.3
It will be noted that the direction of the deviation from
the straight line is indicated by the sign of the L^ term.
m
n
7.35
4.52
6.25
5.60
12.8
3.28
11.2
3.70
H
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rH
EH
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1 1 1
OOO
O mvO
790- 819
820- 849
850- 879
880- 909
910- 939
940- 969
970- 999
1000-1029
1030-1059
1060-1089
H
O
EH
SJ3q.9inTiiTn UT Ht).3u9T Xpog
Having found an exponential equation to fit the
head-body relationship In certain colubrlds a test was made
on several rattlers (the three species showing the greatest
sag when plotted on rectangular coordinates) to see if, in
their cases, an equation of this type might not be more
suitable than a straight line. This was found not to be the
case; even in these extreme Instances adherence to a straight
line was closer than to an exponential formula of the type
followed by these colubrlne snakes.
Dispersion in Head Length — Variation with Body Length
We now retTirn to the rattlers and to a more de-
tailed study of the straight line of best fit, and the
closeness of adherence of individual specimens to this
regression line. The coefficient of correlation was deter-
mined in three large homogeneous series, the Cape San Lucas
series of Crotalus lucasensis and the Platteville and Pierre
series of Crotalus vlrldis vlrldis mentioned in previous
sections of this paper The correlation table of the latter
group is set forth as an example In Table 17. The results
follow:
Lucasensis Vlrldis
Platteville Pierre
Number of specimens 247 833 715
Coefficient of correlation (r) 0.831 0.989 0.988
Standard error of estimate, mm. U.ll 1.03 1.10
Despite the relatively high coefficient of corre-
lation, thus determined, I do not consider the standard
error of estimate to be particularly Indicative of the val-
ue of head length as a diagnostic character: first because
the value of r will depend too much on the distribution (in
length) of the available specimens;* and, secondly, because
the method gives an equal value to all deviations in terms
of absolute measurement. By this I mean that a deviation
in head length of 3 mm. from the regression line at a body
length of 300 mm. has the same weight in determining the
value of the standard error of estimate as a deviation of
3 mm. at a body length of 1000 mm. But (using as an ex-
ample the equation H = 0.035L + 7.5) In the one case the
deviation from normal is 16.7 per cent, whereas in the other
it is only 7.1 per cent. I see no reason to assume that if
a snake differs 3 mm. in head length from the normal of his
species as a Juvenile, he will continue to differ by this
absolute amount as an adult, when 3 mm. will have become a
relatively smaller deviation. Rather it would be more logi-
cal to assume a constant proportional deviation; the snake
deviating 3 mm. in head length, or 16.7 per cent, at a body
length of 300 mm. will also probably deviate about 16.7 per
cent, or 7.1 mm., when he has grown to a body length of
1000 mm.
This assumption might be tested if we had two
* As an example observe the following variations in the
value of r: Platteville series — entire series 0.989, juve-
niles 0.823, adult males 0.943, adult females 0.906; Pierre
series — entire series 0.983, Juveniles 0.881, adult males,
0.94.3, adult females 0.926. The limited age groups always
have a lower correlation than the entire series, since the
scatteap is proportionately Increased.
11
large series of specimens available, all of one, for example,
being exactly 300 mm. long, the other all 1000 mm. These are
not at hand; if they were we could at once determine the co-
efficients of variation which we are seeking and thus obviate
the assumptions which we are trying to substantiate. But we
do have available some fairly numerous concentrations within
several Jiivenile and adult groups. Using these we may test
certain homogeneous Juvenile groups falling within a 10 mm.
body-length range, and adult groups within a 30 mm. range,
and determine their respective coefficients of variation. It
appears fair to have the ranges cover approximately the same
percentages of the length over-all. Such a calculation is
equivalent to assuming that the heads are without change in
size while the bod^ makes this small growth. This assiomption
will lead to a slightly higher value of the coefficient of
variation than the true figure. The results of such a deter-
mination follow:
Length
Coefficient
Number
of
range
of
variation
specimens
in mm.
in
per cent
Juveniles
San Patricio
cinereous
AO
310-319
2.74
Platteville
viridis
37
270-279
3.35
Pierre
virldis
27
320-329
2.31
San Diego Co.
oreganus
27
330-339
2.76
Adults
Cape
lucasensls 27
1110-1139
4.08
Platteville
viridis
65
760-789
3.76
Pierre
viridis
57
800-829
3.16
It will be observed that there is no decrease in
the coefficient of variation — if anything an increase is
shown in the adults over the Juveniles.* The same conclusion
is evident in all species, from a visual Inspection of the
scatter about the regression lines. Thus it is indicated
that during ontogeny any deviation in head length from the
normal is more likely to be maintained at a constant percent-
age than at a constant absolute value. The subsequent calcu-
lations are continued on this assumption, and we abandon the
usual method of computing the coefficient of correlation and
the standard error of estimate as being of no particular val-
ue in the present instance. It is our purpose to devise a
statistic which gives a clearer and more useful picture of
the dispersion.
Extent of Dispersion — Limited Age Groups
It has been stated that the value of the head
* Some might presume that the increase in the coefficient
of variation of the adults as compared to the Juveniles is
the result of taking a wider length-range (30 mm.) in the
former group, as compared to 10 mm. in the latter. This is
not so. For example, taking the Platteville adults having
the range 760 to 789 mm. and dividing these into three groups
having 10 mm. increments, we find the several separate coef-
ficients of variation to be 3.97, 3.68 and 3.31 per cent.
The average, 3.65, is not greatly different from the figxire
3.76, arrived at by taking the 30 mm. zone as a single unit.
12
length as a diagnostic character depends on the relative
dispersion within a homogeneous group, as compared with the
difference between the means of two groups representing sep-
arate species or subspecies. In using the assumption of
constant percentage deviation, as previously tested, we first
determine the regression line for the group by the method of
least squares* or other suitable means. We then use this
equation to transpose each measured head size to the equiva-
lent head size which that individual would probably have at
some arbitrarily selected standard length.^ If enough speci-
mens are available it is always advisable to employ only
those within a relatively narrow length-range. By so doing
we eliminate, or minimize, the increased variance which might
result from any unsuspected deviation of the regression curve
from a straight line, or from an inaccurate slope in the re-
gression line; for with a limited length-range the specimen
is translated only a short distance along the line.
Thus we secure an array of equivalent measurements
from which, not only the dispersion constants can be computed,
but the nature of the dispersion may be ascertained. For ex-
ample, 185 Juvenile specimens of C.v. or eg anus from San Diego
County, varying in length from 260 to 398 mm. were measured
and the head dimensions were reduced (by the application of
the equation of H on L determined from these individuals) to
a standard body length of 330 mm., and the statistics of dis-
persion were then determined. In a similar manner other
groups were investigated, including the adult series of viri-
dis from Pierre, adult lucasensls, and the San Patricio Juve-
nile cinereous. The results are set forth in Table 18.
In these limited geographical groups the head size
is seen to be quite consistent. The extremes are not to be
taken too seriously; they often represent errors in measure-
ment, injured specimens, or distortion in preservation. It
is important to note that half the specimens (as indicated
by the semi-interquartile range in per cent) are likely to
fall within ±3i per cent of the mean. We find again, in
comparing the Juvenile and adult groups, that deviations (as
indicated by the coefficients of variation) are similar on a
percentage basis, rather than on an absolute basis; thus we
visualize the dispersion surface as widening with increased
body length. If we assume, as is indicated by graphical
studies of dispersions about the H on L lines of best fit,
that the dispersions in other species are similar in degree
to the four developed in Table 18, the head size may be of
real value in diagnosis, particularly if a moderate number
♦ The method of least squares, being based on absolute, not
proportionate deviations, may introduce a slight error, but
the correlation is so high that it may be neglected. (See
p. 20) If one prefers the parabolic approximation, H = cL*^,
in its straight line form logH = k logL + logc, in deriving
the standard error of estimate, deviations proportionate to
size are automatically assumed.
i) The deviation varies in proportion to L + b/a, not in
proportion to L alone. Thus if h represents the deviation
at length L and we desire to find the equivalent deviation
ho at standard length Lq we have hg = h(Lo + b/a)/(L + b/a).
As Lo + b/a is a constant, it is seen that deviations vary
with L + b/a. For an example computation of the reduction
of head length to a standard see p. 22.
13
Table 18
DISPERSION CONSTANTS OF HEAD LENGTH REDUCED TO STANDARD BODY SIZE
Juvenile
San Diego
oreganus
Juvenile
San Patricio
cinereous
Adult
Pierre
viridis
Adult
lucasensis
Number of specimens
185
139
154
180
Length over -all:
Length range, mm.
260-393
27-4-3U
800-899
850-1139
Assumed standard
body length, mm.
330
310
850
1000
Statistics of head
length reduced to
standard body length:
Mean head length, mm.
20.382
19.468
36.603
46.372
Probable error of
the mean, mm.
±0.039
±0.027
±0.074
±0.089
Standard devia-
tion, mm.
0.787
0.476
1.356
1.765
Coefficient of va-
riation, per cent
3.87
2.U
3.70
3.81
Extreme range, mm.
18.2-22.1
18.-4-20.7
31.1-39.2
40.3-51.7
Interquartile
range, mm.
19.9-20.9
19.2-19.8
35.7-37.5
45.2-47.6
Semi-interquartile
range, per cent
±2.61
±1.65
±2.50
±2.57
Variation of L/H
within the inter-
quartile range
16.6-15.8
16.1-15.7
23.8-22.7
22.1-21.0
of specimens be available. A character having a coefficient
of variation usually below 5 per cent does not compare un-
favorably with others used in taxonomy, as shown in Table 14 .
While discussing the extent of the variation in
head length at any fixed body length, it should be pointed
out that at least a part of this variation may not be a mere
random divergence in the proportional parts of head and body;
on the contrary, it may result from an attempt upon the part
of nature to produce a tiniformity of L/H at corresponding
ages. For, as is subsequently shown (p. 36), morphological
homogeneity at corresponding ages, within a population of
snakes growing to different ultimate lengths (a condition
common to all populations), will in itself produce a scatter
about the regression line of H on L Just as we find here.
In other words, this scatter may be attributed to a certain
type of uniformity rather than nonuniformity.
Character of Dispersion — Complete Age Series
The data contained in Table IS were based on snakes
of limited body-length ranges, to be assured of a straight
regression line. As a final survey of the nature and con-
sistency of the head-body relationship we proceed to investi-
gate our two largest available series (the Platteville and
Pierre series of C.v. vlridis) from birth to ultimate adult
size, using in each case the proportionate method of reduc-
ing head size to the equivalent size at a standard body
length. In other words every specimen in the series is re-
duced to a single standard body length; the resulting head
lengths are entirely comparable, and may be gathered in a
statistical array for computation of the character and de-
gree of dispersion. Standard tests for linearity of regres-
sion may be made. Finally, by the chi-square test, the re-
semblance of this cross section of the dispersion surface
to a normal curve may be investigated.
The Platteville series of 833 specimens was
checked analytically; but in order that the nature of the
dispersion might be visualized from juvenile to adult, a
graphic as well as an analytic study was made of the 715
specimens of the Pierre series of vlridis. The regression
line having been determined from a correlation table, the
area immediately adjacent thereto was divided into a central
zone with eight equal zones (i.e., with equal intercepts on
the H-lines) on either side. The zones widen proportionate
to L + b/a, so that all points having the same per cent de-
viation from the regression line (rather than the same abso-
lute deviation) fall within the same zone. The total points
within each zone were counted and the results tabulated.
Thus, there was secured an average cross-section of the dis-
persion surface as Intercepted by a plane parallel to the
H-axis and perpendicular to the L-axis.
A study of the graphical presentation of the re-
sults derived from these 715 specimens, which is not here
reproduced, as a very large sheet would be required for the
purpose, gives impressive visual verification of the assump-
tion that dispersion is not uniform in terms of absolute
measurement, but rather is proportionate to L + b/a, with a
slight increase in variability in the last adult stages.
In the analytic study of the two series, correla-
15
tlon tables (see Table 17 for example) were first prepared
in order to determine the constants of the regression equa-
tions. These were then used, in the manner previously out-
lined, to reduce all head lengths to a standard body length
of 900 mm. This length was selected as representing a
fairly large adult, and is employed in order that the varia-
tions in head size might be visualized. The variations at
any other body size may be easily determined, using the re-
gression equations and the values of the coefficient of
variation. These, in fact, constitute the important result
of the computation. The statistics follow:
Number of specimens
Regression constants
(in the equation H
Platteville Pierre
series series
833 715
aL + b) a 0.03553 0.03350
b 6.968 8.363
Mean length of head, mm.
(at standard body length
of 900 mm.) 38.94 ± 0.0317 38.51 ± 0.0347
Interquartile range, mm. 38.02 to 39.86 37.58 to 39.44
Standard deviation, mm. 1.359 1.376
Coefficient of variation, per cent 3.49 3.57
Again we note, in this low coefficient of varia-
tion, a rather consistent character which will Justify an
investigation of species differences. Of course it may be
said that a thorough examination of dispersion has been made
in only one species and it may not be true that all species
are equally consistent. Such large series, especially with
all individuals carefully preserved and measured under uni-
form conditions, are not available in other species; and the
method of computation is a rather laborious one. However,
41 other species, subspecies, and geographical groups were
plotted and there was evidence of a comparable narrowness of
dispersion.
The first use to which the correlation tables of
these two large series are put is an analytic study of lin-
earity. We calculate the eta coefficient of determination
and, comparing this with the Pearsonian coefficient of de-
termination, find them practically equal. Using the method
of Fisher we find that the regression is not significantly
curvilinear, F being below the 5 per cent level in both
cases.* The conclusion may therefore be definitely drawn
that a straight line substantially fits the H-L relationship
in this species. As I have stated elsewhere, a visual sur-
vey of the regressions of the other species Justifies the
opinion that this relationship is general among the rattle-
snakes.
* Fisher, Statistical Methods for Research Workers, 4th Ed.,
1932, p. 237; Yule and Kendall, An Introduction to the Theory
of Statistics, 1937, p. 455; Davies and Yoder, Business Sta-
tistics, 1937, p. 379.
16
The coefficient of variation given in the tabula-
tion above Is the average dispersion about the regression
line on a percentage basis. An important question has to do
with the trend in this dispersion — are Juveniles more, or
less, variable than adults, and is the difference conspicu-
ous? This has been touched upon before, but we now have
available complete sets of data from birth to matxirity in
two homogeneous series. Dividing our material into 100 mm.
body-length zones and calculating the coefficient of varia-
tion of each zone separately, we have the following table:
Cpefficient of
Specimens
in Zone
Variation in
Per Cent
Length
Platteville
Pierre
Platteville
Pierre
Zone, mm.
series
series
13
series
series
200- 299
129
3.32
2.35
300- 399
100
146
2.88
2.72
^00- 499
11
21
2.72
4.16
500- 599
120
69
2.91
4.28
600- 699
186
111
3.88
2.90
700- 799
211
114
3.30
3.75
800- 899
63
154
4.05
3.73
900- 999
13
73
2.40
3.19
1000-1099
srage 833
715
-
2.96
Total or Ave
3.36
3.37
It will be noted that the weighted average of the zonal co-
efficients of variation is in each instance somewhat below
the general average for the series as a whole, as set forth
in the previous tabulation.
Because of the fluctuations between zones, the
trend of the coefficient of variation is not clearly in evi-
dence. To determine this trend the straight line of best
fit, for these coordinates of the body length (L) and the
coefficient of variation of H (Vg) , was determined for each
series by the method of least squares with the following re-
sults:
Platteville Series Vtj = 0.000818L + 2.886
Pierre Series Vg = 0.000981L + 2.717
In order that the meaning of these equations may
be more clearly visualized, we set forth the values of Vg at
various body lengths, as derived from the equations:
Platteville Series
Pierre Series
L = 250 mm.
3.09
2.96
Values of V^
L = 500 mm.
3.30
3.21
L = 900 mm.
3T&2
3.60
Thus we see (as is evident from the constants of the equa-
tions) that the trend of the coefficient of variation is to
increase slightly as the snakes increase in length. In
other words the dispersion of H about the regression line
becomes somewhat greater with age and we have demonstrated
that, in this species at least, young rattlers are less var-
iable than adults — not only less variable on a basis of ab-
solute measurements, which might well be assumed, but less
even on a basis of percentage deviation from the mean head
size at any body length.
It appears to me that an analysis of the trend in
the coefficient of variation in a problem of this type gives
17
a useful plctvire of the nature of change in a body propor-
tionality during ontogeny. This differs from the method of
Harris* where the trend in the proportionality Itself is
determined. The latter, it seems to me, is evident from
the regression equation, for where the correlation is linear
a negative value of Harris' r^z merely proves that b in the
equation X = aY + b is positive. In the present instance
we have:
Platteville
series
0.989
-0.9U
Pierre
series
0.988
-0.931
Pearson coefficient of correlation r^^^
Harris coefficient of correlation t-^^
Since Ti is negative we know that H is a decreasing propor-
tion of E, as L increases, which we knew originally from the
positive value of b. Thus the Harris method does not appear
pertinent in the present case.
So much for the directional trend of dispersion;
we return once more to its character, that is, the shape of
the dispersion surface. Taking our complete arrays of equi-
valent head lengths at a standard body length, we have the
following statements of dispersion in terms of percentage
of the mean head length:
Percentage
of Mean
Class :
Limits
Class Center
Below
89.5
_
89.5-
90.5
90
90.5-
91.5
91
91.5-
92.5
92
92.5-
93.5
93
93.5-
94.5
94
94.5-
95.5
95
95.5-
96.5
96
%.5-
97.5
97
97.5-
98.5
98
98.5-
99.5
99
99.5-
100.5
100
100.5-
101.5
101
101.5-
102.5
102
102.5-
103.5
103
103.5-
104.5
104
104.5-
105.5
105
105.5-
106.5
106
106.5-
107.5
107
107.5-
108.5
108
108.5-
109.5
109
109.5-
110.5
110
Above
110.5
—
Number of
Specimens
Platteville
Pierre
1
3
1
1
3
3
10
4
14
10
21
22
38
45
55
48
79
59
87
80
97
76
103
89
84
85
84
65
52
48
36
25
24
17
19
19
12
-4
7
5
1
1
2
1
3
5
833
715
These distributions were compared with the distri-
bution under the normal curve, by means of the chi-square
test, with the following results:
Series
Platteville viridis
Pierre viridis
Value of P
0.893
0.226
* J. Arthur Harris, Biometrica, Vol.6, p. 436, 1909; Gene-
tics, Vol. 3, p. 328, 1918; Davenport and Ekas, Statistical
Methods in Biology, Medicine and Psychology, p. 95, 1936;
Treloar, An Outline of Biometric Analysis, p. 69, 1936.
18
FIGURE d
DISPERSION OF HEAD-LENGTH DEVIATIONS
A Q/^l \T DC/^ D C C C I^K.1 1 IKIC
ABOU 1 KLCjKLooIUN LIINL
Plafteville Series of C.v.viridis
ORMAL CURVE
ESULT FROM MEASUREMENT
o R
(
»
0
>
<
/
o\c
y
I
/
\
/
\
j
'
A
/
\
I
/
<
\
/
\
/
,
A
I
<
/
\
— 8-t
^
\
V
)
8
8 9
0 9
2 9
4 9
6 S
8 l(
30 IC
)2 l(
)4 l(
)6 11
D8 1
10 1
2
PER CENT OF MEAN
Thus it appears that the distributions are probably-
normal. In order that the character of the adherence to the
normal curve may be visualized, the situation in the Platte-
ville series is given in Fig. 9.
It will be observed, both from the table on p. 18
and from Fig. 9, that these distributions have a slightly-
negative bias; the means of the distributions instead of
being 100 per cent are found to be 99. 6 7^ per cent in the
Platteville and 99.550 in the Pierre series. This probably
results from the original calculation of the regression lines
having been made on a basis of absolute, rather than propor-
tional, deviations. However, this is the only way in which
b/a can be initially determined, and until this constant is
known the method of translation to a standard body-size can-
not be used. Based on the weighted trends in the deviation
from 100 per cent in each length-zone, a correction can be
made in the regression equation which will bring the mean to
100 per cent. In the Platteville series this correction is
found to change the equation from H = 0.03553L + 6.968 to
H = 0.03552L + 6.891; the corresponding correction in the
case of the Pierre series is from H = 0.03350L + 8.363 to
H = 0.03336L + 8.329. The differences are of no practical
importance and are only mentioned to indicate the source and
correction of the deviation from 100 per cent. Since the
zonal coefficients of variation were computed using the ac-
tual means, they are not affected by this slight error in
the regression line.
Thus by a complete consideration of two large ser-
ies of C.v. viridis. with respect to the head-length to body-
length relationship, we have shown that (l) the correlation
is linear; (2) the dispersion about the regression line is
substantially normal; (3) the dispersion is nearly uniform
on a percentage basis, increasing slightly amongst the adults,
as compared with the Juveniles. While we supply no analytic
proofs beyond these series, our graphic surveys of forty
other species, subspecies, and geographic groups indicate
the same conclusions to be valid. These is no indication of
subgeneric or group departures from linear regression, for
there is as frequent a tendency to bow slightly above a
straight line as to sag below it, and these departures seem
to be the result of random sampling fluctuations. Similarly
the proportionate type of dispersion seems to be indicated
in all the available series.
Head Length as a Diagnostic Character
Before proceeding to illustrations of the use of
head-length statistics in taxonomy, it is advisable to dis-
cuss certain phases of the method which has been developed.
Assume three subspecies to be investigated, and that the re-
gression lines of their head-body relationships have been
determined and are found to be the lines X, Y and Z as il-
lustrated in Fig. 10. Pictvire these as the back-bones or
ridges of dispersion surfaces. The solid parts of the lines
represent the range of length which each subspecies passes
through in actual life; the dashed continuations are magni-
tudes either smaller or larger than the snakes ever attain
in nature and are only included to illustrate points of
intersection.
In choosing the specimens of any two groups to be
20
FIGURE 10
EXAMPLE REGRESSION LINES
compared we find ourselves faced with conflicting alterna-
tives. The accuracy of the parameters of an unknown popu-
lation is improved by having large samples; therefore we
should use as many specimens as possible in our calculations.
On the other hand, we have already pointed out that a wide
range (in length) of specimens tends to accentuate errors
resulting from any or all of the following: (l) inaccurate
location of the regression line; (2) incorrect character of
the regression line, e.g., curved, rather than straight, as
assumed; or (3) inaccuracy of the theory that deviations
from normal remain constant as percentages during life. Fur-
thermore, assume that two species have regression lines
which intersect in the adolescent range, as X and Y inter-
sect at M in Fig. 10. It is clearly evident that if we
should happen to choose a standard length at M as a basis of
comparison, the means would be equal and we would inevitably
find the difference below the level of significance. On the
other hand, if the two lines intersect at or close to H - 0,
as do Y and Z at N, then we will not meet with this diffi-
culty, and the choice of the length-limits of the individu-
als to be compared will be less important.
Having all of these
usually be advisable to restri
the available adults within a
In cases where head length is
this is the range in which the
strated statistically, for we
weight to an adult difference
that tends to disappear in lat
considerations in
ct the specimens c
rather narrow leng
of real importance
difference should
should naturally g
than to a Juvenile
er life. The two
mind it will
ompared to
th range,
in taxonomy,
be demon-
ive more
difference
forms must
21
be snakes growing to about the same ultimate length; if one
form is knovm to be smaller, or stunted, we must use certain
special calculations to separate the mere effect of dwarfing
from the real specific difference, if there be one. (See
p. 34). I shall return again to the head characteristics of
stunted races, as compared to the forms from which they have
been derived, and will show that they do not normally follow
the same regression lines as their prototypes. To summarize:
by taking a narrow adult length range, the statistical accu-
racy which we sacrifice by limiting our investigation to
fewer specimens, is more than counterbalanced by the elimin-
ation of possible errors involved in translating specimens
over too great a distance along their regression lines.
Example Taxonomic Problems
We now proceed to work out some taxonomic problems
involving head length to demonstrate the application of the
method previously developed.
As we observe in captivity several cages of Cro-
talus mitchellii, some from the Cape Region of Lower Cali-
fornia, others from San Diego County and western Arizona,
it appears that the Mexican snakes have proportionately
smaller heads than those from California; whether there is
a difference between the San Diego County and Arizona mater-
ial is uncertain. It is desired to determine whether these
are real differences, or whether they are only imagined by
the observer. Such differences, if present, are often more
apparent in live than in preserved specimens.
We first survey the available material and find
that moderately plentiful series of adults from each area
are available. It appears that we can restrict the range
to be included in our comparison to snakes no less than 700
nor more than 950 mm. in body length and still have adequate
numbers. We decide on an approximate midpoint, or 350 mm.
as the standard length at which the geographical groups will
be compared. We now determine the regression equation for
each group.* With some experience this can usually be done
graphically with sufficient accuracy to satisfy the necessi-
ties of this problem. To illustrate the method of transpos-
ing a head length to a standard body length, let us take an
individual snake, a mitchellii from San Diego County with a
body length of 755 mm., and a measured head length of 35.2
mm. It is desired to find the hypothetical head length at
o\ir arbitrarily selected standard body length of 350 mm.
From a study of the available specimens from San Diego
County, we first determine the regression equation to be
H = 0.033L + 6.85, that is, a = 0.038 and b = 6.85. Then
b/a = 180. Lq is 350, and Lq + b/a is 1030. From the
formula Hq = H (Lq + b/a)/(L + b/a) we have the equation
Ho = 35.2 (l030)/(:755 + 180) = 38.8, which is the estimated
head length in millimeters that our snake would have at the
standard body length of 350 mm. Standardized head lengths,
thus computed for each available specimen, are tnen gathered
into an array (separate for each geographical group which
we are investigating), and their statistics are computed in
* In determining this equation, I prefer to use all avail-
able specimens of whatever age, rather tnan only tnose in
the restricted length-class being compared.
22
the usual way. In the example mitchellli problem, we have
the following :
Cape San
Lucas
Central
Arizona
San Diego
County
-45
18
29
732-939
700-9-43
730-931
850
850
850
0.220
0.339
0.93
1.82
2.4
4.7
Niimber of specimens
Body length range, mm.
Standard body length, mm.
Mean head length equated
to standard body length, mm. 32.04 39.33 39.24
Standard error of the mean, mm. 0.224
Standard deviation, mm. 1.50
Coefficient of variation, per cent 4.7
Using the ordinary formula to determine the stand-
ard error of the difference between the means of the parent
populations, we find, comparing the San Diego County with
the Cape San Lucas specimens, that the difference between
the means (39.24 - 32.04 = 7.20 mm.) is more than seventeen
times its standard error (O.412), which is, of course, highly
significant. On the other hand, comparing the Arizona with
the San Diego County specimens, we find a difference between
the means of 0.09 mm. t 0.4II. Here the difference is only
one-fourth of its standard error, and is, therefore, without
significance. We reach the conclusion that, in head length
proportionality, the Cape San Lucas specimens are signifi-
cantly different from those of the other two areas (which do
not differ from each other); and, especially if confirmed by
other characteristics, at least a subspecific segregation is
warranted. Thus, the revival of Cope's name pyrrhus for the
southern California and Arizona specimens is Justified.
We now pursue a similar method in two other contro-
versial problems. Some herpetologists still fail to distin-
guish between Crotalus tigris of southern Arizona and Sonora,
and the form which I have called Crotalus mitchellii Steph-
ens i. an inhabitant of east-central California and southern
Nevada. These snakes differ in certain scale characters;
notably, the latter has sutured supraoculars, while those of
the former are entire. Ordinarily, no one would suggest that
a character as cumbersome as head size is seen to be, should
be employed in diagnosis, when other, more readily determined
and evaluated differences are apparent. But in the present
instance, where doubt has been indicated by the non-recogni-
tion of the new form in some check lists, let us see whether
the head size (which seems to me to be notably small in
tigris and normal in stephensi) will reinforce the other
differences. Using only adult specimens, we have the follow-
ing statistics:
Tigris Stephensi
Number of specimens
Body length range, mm.
Standard body length, mm.
Mean head length equated to
standard body length, mm.
Standard error of the mean, mm.
Standard deviation, mm.
Coefficient of variation, per cent
23
30
37
615-815
610-796
700
700
26.55
31.58
0.165
0.241
0.90
1.47
3.41
4.65
From the above statistics we find the difference
between the means to be 16.9 times its standard error. As
a difference of only twice its standard error indicates,
with a strong probability, that the difference is real, and
not the result of a chance divergence in the samples, there
can be no question as to the real and extensive difference
in head proportionality between tlgris and stephensl.
To cite one more example: The neglected species,
scutulatus. which was so long confused with cinereous, is
another case in point. These species differ in head scales,
hemipenes, pattern, color, and other characteristics; yet
even today scutulatus has not been universally recognized
as valid. The head-size comparison lends additional force
to the differentiation, for scutulatus has the smaller head.
It appears desirable to compare Arizona scutulatus with
Arizona cinereous, since the two forms occupy virtually co-
incident ranges in that state, and are not greatly dissim-
ilar in size, and thus any difference which may be found
cannot be interpreted as a mere territorial or racial diver-
gence; the two forms are either identical or separate spe-
cies. I have also Included, for comparison with the Arizona
cinereous, a Texas series of cinereous, to show that this
method may be as useful in disclosing identities as differ-
ences.
Cinereous Scutulatus
Texas Arizona Arizona
30
112
115
00-1100
700-1100
700-1100
900
900
900
40.75
40.17
36.74
0.328
0.137
0.142
1.79
1.46
1.52
4.4
3.6
4.1
Number of specimens
Body length range, mm.
Standard body length, mm.
Mean head length equated to
standard body length, mm.
Standard error of the mean, mm.
Standard deviation, mm.
Coefficient of variation, per cent 4.4
From the above statistics we find that the Arizona
scutulatus-cinereous means differ by 17.2 times the standard
error of the difference, which is significant beyond any rea-
sonable doubt; the Texas-Arizona cinereous ratio is only
1.61, and thus is of doubtfiil significance. The difference
in the latter case is probably due to the different ultimate
sizes to which this species grows in the two areas, as will
be explained later.
Validity of Head Length as a Diagnostic Character
All of these cases show a significance ratio above
4. In the present instance, while any difference is of in-
terest in showing relationship or subspecific trends, I would
not say, arbitrarily, that a difference between the means of
twice the standard error is significant, even though the
chances are only 4.6 in 100 that a difference of this extent
is the result of random sampling, rather than an actual dif-
ference of the parent populations. But any significance
ratio above 4 (indicating about 6 chances in 100,000) may
24
certainly be taken seriously, for this margin will more than
compensate for slight errors in the assiunptions or in the
graphic selection of the regression lines.
A study of the method will show that, when the
arbitrarily selected standard length is taken as the approx-
imate midpoint of a narrow length-zone, errors in the slope
or position of the regression line are relatively lonimport-
ant. This line influences only the translation of points to
a standard size; in such translations the increments or de-
crements of the actual measurements are relatively small,
and errors in the regression line result in minor errors in
increments rather than in the basic figure to which they are
added or from which they are subtracted. However, it is es-
sential that the two forms compared grow to approximately
the same ultimate size. If they do not, certain fiu-ther ad-
justments must be made, as pointed out in the discussion of
dwarf races (p. 34-).
One may make a fairly good guess from a mere in-
spection of the regression lines whether the difference is
likely to be significant, provided we know something of the
nature of the variation about the regression lines. In the
rattlesnakes the coefficient of variation of head length,
when equated to a standard body length, has been found to be
somewhat below 5 per cent. If about 25 specimens of each
form are available, a difference of 2-1/2 per cent in the
means of the head lengths is likely to approach significance.
Five per cent, or 2 mm. in head lengths of about 4-0 mm., is
almost certainly significant.
In considering the method thus far demonstrated,
I fully realize its relative impracticability, since many
specimens are required and the computations are laborious.
But the fault primarily lies in the natvire of the head-length
proportionality, not in the method. It is unfortunate, for
simplicity of calculation, that L/H is not constant through-
out life; the fact remains that it is not, at least amongst
the rattlesnakes, and we must be guided accordingly.
These complications in dealing statistically with
head length, or head proportionality, as a taxonomic criter-
ion may be taken as illustrative of the great advantage
which herpetologists and ichthyologists have in their work
of classification, as compared, for example, with mammalo-
gists and ornithologists. For, in the finer classifications
(i.e., species and subspecies), the latter are dependent
largely on items of pattern, color, and the proportions of
body parts; and these last are beset with the practical dif-
ficulties so evident in this head-length study, since these
proportions usually do not remain constant during the life
of a single individual. But the student of reptiles and
fisn has available a considerable variety of scale quanti-
ties and arrangements which can be counted; these are ex-
pressible as numbers and remain invariant during ontogeny.
Thus, quantities are readily available in a form most useful
for statistical study, particularly with respect to the sig-
nificance of differences.
Short Cut Methods
We have reason to believe that head length, in-
volving skeletal proportions as it does, is a relatively
25
stable character. The narrow dispersion within a species or
subspecies, bears this out. We should, therefore, put it to
use, where other less certain differences are to be validat-
ed, even though it is not easily manipulated.
It must be remembered that these statistical meth-
ods only determine the probability that two populations dif-
fer in a certain character. They do not prove that the dif-
ference is real, but they may show such a high probability
of reality that the taxonomist accepts it as proof, beyond
any reasonable doubt. But the final acceptance is a matter
of Judgment, not proof.
Refinement of method, then, leads only to the
strengthening of a probability. If we can devise means of
simplifying the method, without too strongly affecting the
resulting probability, we may still have a useful working
tool. Even the rather involved procedure hitherto used
would, I am sure, draw criticism from the professional sta-
tistician because of some of the assumptions and the method
of development. But it is an approach to accuracy and at
least it has been useful in demonstrating the nature of the
head to body relationship, and the inacciiracy of comparing
simple L/H ratios without restriction. After all, we should
not accept a difference of this kind as warranting a speci-
fic or subspecific distinction unless the evidence of a
single character is overwhelming, or unless there be the
cumulative weights of several independent characters all
pointing to the same conclusion. Thus, refinement of proba-
bility, in terms of the ratio of a difference to its stand-
ard error, to the second or third decimal place, is not the
goal of a calculation of this kind. But we would certainly
feel more justified in making the segregation if the ratio,
by what seem to be fairly accurate methods, is found to be
in excess of three than if it turns out to be less than one.
The first modification toward workability, which
may be suggested in this method, is one which involves group-
ing and simpler calculations, rather than a change in theory.
Take the two species or subspecies to be compared and plot
the coordinates of each available specimen on quadrille-
ruled cross section paper. Draw the regression line of each
by eye. Divide the available specimens of each species into
groups by length; 10 mm. intervals will usually be found sat-
isfactory, as for instance, all the specimens from 840 to
849 mm., inclusive, fall into one group.* Pick off from the
regression line the normal head length at the mean body
length of each group; thus, in the above example, pick off
the head length corresponding to a body length of 844-5 mm.
Then, in each group, determine the ratio of the actual head
length of each specimen to this normal length. For example,
assume that the normal head length, at a body length of
844-5 mm. is found, from the regression line, to be 38.4 lam-,
* If a large number of specimens are available, say over
100, it will be simplest to group and average, in drawing
the regression line, instead of plotting individual speci-
mens. Where there are many specimens and this method is
usea, the results of selecting the regression line by eye
are quite accurate. Thus, in the Platteville series a gra-
phic determination produced H = 0.0355L + 7.0, wnile a cor-
relation table resulted in H = 0.0355L + 6.97. Tne corres-
ponding equations for the Pierre series were: Graphic
H = 0.0333L + 8.3; analytic H = 0.0335L + 8.36.
26
and that we have three specimens (in the 84.O-84.9 mm. body-
range) with head lengths measuring 36.6, 38.0, and 4.0. 6 mm.;
then the ratios to normal of these three specimens are re-
spectively, .953, .990, and 1.057. Gather these ratios for
all available specimens into one statistical array and com-
pute the standard deviation and coefficient of variation.
The mean will be close to 1.0 if the regression line has
been properly located. From the criteria discussed on p. 20,
determine the standard length at which the two species under
consideration are to be compared. From each regression line
pick off the normal head length corresponding to the standard
body length. From the known coefficient qf variation for
each species, as above determined, compute the standard de-
viation from the equation 0"= VM/100. For example, if we
decide to compare our two forms at a standard body length of
900 mm., and we find, from the regression line of one of our
species, that the normal head length for a snake of that
size is 4.0.4 mm., and the coefficient of variation is 3.2
per cent, then the standard deviation is 1.22 mm. Thus, we
have (for one of the two species) both of the statistics ne-
cessary to evaluate the difference between the two species.
Proceed similarly with the other species and then determine
the ratio of the difference between the means to its stand-
ard error as the test of significance.
It is necessary to re-emphasize the desirability
of limiting the individuals to be studied to a rather short
adult length-range for reasons previously mentioned, although
this limit need not be applied in determining the regression
lines. The determination of what range to use is best found
from a study of the regression lines and the availability of
specimens, as shown by the points on which the curves are
based. Usually the zone selected should be wholly within the
adult range, since a difference in head proportionality in
the adult stage (regardless of the juvenile situation) will
be more impressive than a difference between Juveniles which
fails to be manifest later in life.
Another approximate method is to draw up a stand-
ard linear correlation table* for each of the species, using
a restricted length-range. From the resulting statistics
calculate the two equations in the form H = aL + b and de-
termine the difference in the normal values of H at a stand-
ard head length near the center of range. Use the two stand-
ard errors of estimate as standard deviations and thus deter-
mine the significance of the head length differences. The
assumption thus introduced, that deviations do not vary with
length, is not important if the length range is narrow. How-
ever, this method is equally laborious, and is not particu-
larly to be recommended.
Use of L/H Ratio
What may be said with respect to the use of the
far simpler L/BP ratio as a basis of comparison? The answer
* For example, see Mills, 1924, p.378; Pearl, 1930, p. 378.
^ This is a preferable form to the reciprocal H/L since it
is easier to visualize "the number of times the head is con-
tained in the body length," than the "ratio of head to body."
But either may be used without in any way changing the dis-
cussion.
27
of coiirse depends on the nature of the regression ciirve.
Assuming a straight line relationship, in any genera of
snakes in which b in the equation H = aL + b is relatively-
small, we can use the L/H ratio directly in oiar comparisons
and all necessity for the complicated procedure of equating
to a standard body length is obviated. This possibility
should be investigated by plotting sample regression lines.
But if b is found important (as is the case with the rat-
tlers), then the use of the L/H ratio is only an approxima-
tion, for we would have variations of L/H even if the linear
correlation between L and H were perfect within each species.
(See Fig. 8). However, if we restrict our specimens to a
rather narrow body-length range, and if they are evenly dis-
tributed throughout' that range, then moderately reliable re-
sults can be secured, especially if the true significance is
well above the usually recognized border line of signifi-
cance. But it must be remembered that it is not sufficient
merely to determine the differences between the two averages
of L/H; the significance of this difference must be deter-
mined also, by comparing the difference with its standard or
probable error.
For an example we may take the 30 specimens of C.
tigris and the 37 specimens of C.m. stephensl previously
treated by the more elaborate method on p. 23, and investi-
gate the significance of the difference in their L/H ratios.
We set these ratios, separately calculated for each speci-
men, in a pair of the usual statistical arrays and from them
we obtain the following statistics:
Number of specimens
Body length range, mm.
L/H mean
L/H range
L/H standard error of the mean
L/H standard deviation
Tigris
30
Stephensi
37
615-815 610-796
26.38 22.40
23.9-29.1 20.0-24.2
0.423 0.173
2.32 1.05
L/H coefficient of variation, per cent 8.79
4.70
From these statistics we find the difference of
the means (26.30 - 22.40 = 3.93) to be 8.6 times the stand-
ard error of the difference (O.465). This difference is
therefore significant. The ratio thus determined (8.6) may
be compared with 16.9 as previously found by the more elab-
orate method. It is seen that while significance is indi-
cated in both cases, it is less marked when the approximate
method is used. It will be noted that the coefficients of
variation are increased, as compared to the more acciorate
method; this result is to be expected since a constant L/H
ratio assumes a regression line with a different and less
accurate slope than the straight line of best fit, and the
points do not cluster as closely around the former line as
the latter. V/ith the precautions outlined, the simple L/H
ratio seems to be entirely sufficient to prove the difference
between these two species. But it must be remembered that
in this instance the difference is so marked as to be almost
self-evident; the simplified method would not be so reliable
28
in a more doubtful case. It is essential in using this meth-
od that snakes of the same ages be compared, as may be pre-
sumed if it be know-n that they grow to about the same adult
length; otherwise a difference in L/H may merely prove that
snakes of different ages have been compared, rather than the
existence of a real difference.
To set at rest any doubt as to the importance of
the variation of the L/H ratio within a single species, I
present herewith the statistics of 200 snakes selected at
random from the Platteville series of C.v. viridis, 100 be-
ing Juveniles from 270 to 300 mm. long, the other 100 adults
from 720 to 760 mm.
Juveniles
Number of specimens 100
Body length range, mm. 270-300
L/H mean 16.85
L/H range U. 9-18. -4
L/H standard error of the mean O.O664.
L/H standard deviation O.664
L/H coefficient of variation, per cent 3-97
Ad lilts
100
720-760
22.4.8
20.6-24.2
0.0674
0.674
3.00
The mean difference (5.63) is found to be about
sixty times its standard error (0.0946). Thus, there can be
no question as to the real difference between head-size pro-
portionality in Juveniles and adults, the significance of
which is evident even without the formality of calculation.
The lack of overlap is noteworthy; not a single Juvenile has
as large an L/H ratio as the adult having the smallest ratio.
Of course this is only a roundabout proof of the importance
of b. With this extensive difference within a single species
at different ages, it is shown how useless it is to use so
variable a ratio as L/H to demonstrate Interspecies differ-
ences, unless the size range of the specimens considered be
severely restricted. But this size restriction imposes the
necessity of many specimens, otherwise the samples will be
too small for reliability.
Dwarfed Forms
What are the relationships between stunted races
and their full sized relatives in their H/L correlations?
Do the two tend to follow the same regression lines, or is
there some other consistent relationship between their re-
spective equations?
We have 5 pairs or couplets of these forms. Their
taxonomic relationships are generally obvious from similari-
ties of scutellation and pattern, and from territorial con-
siderations. In each case the stunted form seems to be an
offshoot of the larger. These pairs are as follows:
Crotalus cinereous and C. tortu^ensls. The latter
is a stunted species found only on Tortuga Island, Gulf of
California. It is obviously derived from C. cinereous of
29
the mainland. (See Bull.Zool.Soc.S.D. ,No.6,1930) .
Crotalus ruber and C. exsul. The latter is a
stunted form found only on Cedros Island. Ruber is a main-
land species of the Californias. (Op.cit.)
Crotalus viridis vlridis and C.v. nuntius. Inter-
gradation between these two forms is undoubted. The first
is widely distributed in the western Missouri-Mississippi
basin; the second is restricted to northeastern Arizona and,
in its most extreme form, is found only in the basin of the
Little Colorado River near Winslow. Specimens from this
vicinity only were used in working out the correlation be-
tween these two forms. (For fiorther data on the relation-
ship between these subspecies see Trans. S.D.Soc. Nat. Hist . ,
Vol. 8, No. 13, pp. 75-90, 1935) .
Crotalus viridis lutosus and C.v. concolor. The
latter form is intermediate in several characters between
C.v. lutosus and C.v. viridis. and is territorially related
to both. For the purposes of the present discussion I have
assumed it to be an offshoot of lutosus.
Crotalus v. oreganus (San Diego County) and Cro-
talus V. or eg anus (Coronados Islands). The stunted island
form is clearly a derivative of the larger snake on the ad-
jacent mainland.
From our studies of series of snakes involved in
these several couplets we find the coefficients in the equa-
tion H = aL + b to be as follows:
Couplet
Cinereous-tortugensis
Ruber-exsul
Viridis*-nuntius
Lutosus -concolor
Oreganus-oreganus
(mainland-island)
We note here a definite directional relationship,
for in all but one case the derivative form iias an equation
lower in both constants a and b than the prototype from
which it has presumably evolved.
An off-hand presiomption might have led us to ex-
pect the two forms to follow the same H/L regression line.
A brief consideration, however, will show that if two races
are similar in body proportions at the same periods of life
they must have different equations. For, assume two such
races, one reaching approximately twice the average adult
length as the other. Obviously, if the two follow a single
regression line, an adult of the stunted form will have the
head-to-body proportions of an adolescent of the larger sub-
species. But we know from the previous discussion that
young specimens have proportionately larger heads than
adults; therefore, the two forms must follow different re-
gression lines, unless dwarfed races have proportionately
larger heads than their prototypes at corresponding ages.
* Average of Platteville and Pierre series,
30
Values
of a
Values
of b
Parent
Dwarf
Parent
Dwarf
0.0351
0.0315
8.7
8.1
0.0374
0.0354
8.4
8.2
0.0345
0.0324
7.7
7.3
0.0337
0.0321
7.5
6.7
0.0361
0.0367
3.2
7.4
Fig. 11 shows as an example the relationship of the regres-
tion lines of one of these pairs of species.
10
FIGURE II
EXAMPLE H-L RELATIONSHIP OF A
DWARF SPECIES WITH ITS PROTOTYPE
200
400
600
800 1000 1200
BODY
LEirslGXH IN MM
1400 1600 1800
V/e next endeavor to find whether there is a fixed
relationship between the equations of the couplets --whether
there is any method whereby the equation of a stunted form
may be determined from that of the prototype. We first test
the head-length ratios at certain ages to see whether the
dwarfs have the same ratios as their prototypes at the same
period of life. If they do, we will have established a
scheme of derivation. To illustrate the method of calcula-
tion, we take cinereous and tortugensis as examples. We
start with tne equation of the first as previously found.
H = 0.0351L + 8.7. From Table 11 (see Occ.Pap.No.3,p.28)
we ascertain that the average size of this form at birth is
310 mm., and the largest males are about 1700 mm. long. At
these body lengths we find from the equation that the head
lengths are 19.6 mm. and 59.7 mm. respectively. This gives
L/H values of 15.8 at birth and 24.. 9 at full maturity.
We find from the same table that tortu>;ensis is
about 250 mm. long at birth and 1000 mm. at full male growth.
Since this species is to have the same bodily proportions as
cinereous, we apply the cinereous head ratios at these ages
and find that tortugensis will have a head 15.8 mm. long at
birth and 4O.2 mm. at full maturity. From these two fixed
points we determine the theoretical tortugensis equation to
be H = 0.0325L + 7.7.
31
Similarly, the other couplets were tested and we
find the following comparative figiires for the constants in
the equation H = aL + b*:
Dwarfed Race
Values of a
Values of b
Actual
Computed
Actual
Computed
Tortugens
is
0.0315
0.0325
8.1
7
7
Exsul
0.0354
0.0358
8.2
7
1
Nuntius
0.0324
0.0332
7.3
5
.8
Concolor
0.0321
0.0319
6.7
6
.2
Oreganus
(i£
sland)
0.0367
0.0336
7.4
5
.8
We see that the results for constant a are moder-
ately consistent with the facts but the computed values of b
are imiformly low. Young specimens affect b more than a;
from this it appears that a system of morphological similar-
ity is not so closely followed in youth as in age. Of course
the deviations may be the result of inaccurate assumptions
with respect to average size at birth and at full maturity,
for these affect the calculations. In some instances our
supply of young specimens is quite inadequate to determine
with accuracy either relative sizes or regression trends.
We may neglect the size at birth in our calcula-
tions by assuming another type of growth trend. We have
seen that, in any single homogeneous group, H varies with
L + b/a, rather than with L alone. Let us consider L + b/a
as a sort of basic length of a snake, as opposed to its true
length L. Assume further that dwarfed races have the same
value of b/a as their parental analogues and that when the
two forms are fully grown they are morphologically similar.
What then would be the regression lines of the stunted races
and how would these lines compare with those actually found
from the specimens available? We have the following results:
Dwarfed Race
Tortugensis
Exsul
Nuntius
Concolor
Oreganus (island)
Values of a
Actual Computed
Values of b
Actual Computed
0.0315
0.0354
0.0324
0.0321
0.0367
0.0323
0.0350
0.0314
0.0309
0.0318
8.1
8.2
7.3
6.7
7.4
8.0
7.9
7.0
6.9
7.2
It will be observed that these calculated results
agree more nearly with the facts than the assumption of mor-
phological similarity throughout life, particularly with
respect to the constant b. Considering the uncertainties
of some of the basic data, the closeness of the actual and
theoretical regression lines is rather striking. (See Fig.
11) . I do not say that a correlation system has been proven,
since the cases are too few, and the data too meagre to war-
rant such an assumption. It may be of interest, however, to
consider what would be the relationship of regression lines
and life stages in a group of related forms involving this
* Two sets of lengths are not obtainable from Table 11.
These are the lengths of the Pierre and Platteville series
of viridis , combined, which are assumed to be 250 mm. at
birth and 1000 mm. for a large male; also the corresponding
lengths of Coronados Islands oreganus are taken as 180 and
650 mm.
32
type of interrelation. Such a system is illustrated in
Fig. 12. If two forms of considerably different sizes are to
be compared for taxonomic purposes, the amount of the differ-
ence that may be expected from size alone may be ascertained
with a fair presumption of accuracy, by this method; and the
remainder may then be assayed to determine the real species
difference.
J^-
40^
.^^
/
/// CORRELATIONS
FIGURE 12
IN RELATED FORMS
^— Va L
Possibly it may be deemed fruitless to attempt to
find a formula wherewith to derive the regression curve of
one of two related forms from the other.* Disregarding this
phase of the study, we have at least shown, by taking the ac-
tual curves representing the stunted races and comparing
them with their prototypes, that the young of the dwarfed
races have proportionately larger heads than their analogues
when young, but that when the adult stage is reached morpho-
logical similarity is more nearly approached, although the
dwarf head is still slightly larger in proportion to body
* It may be thought that I have taken too much for granted
in assuming that the larger form comprises the prototype
from which the other sprang. As a matter of fact, the same
geometrical relationship holds in either direction. We may,
if we wish, assume the small forms to be ancestral, and cal-
culate the regression curves of the large therefrom. How-
ever, this may be said concerning the rattler couplets: in
every instance the smaller form is restricted in area (in
three cases on small islands) and therefore comprises an in-
significant population compared to the larger. More speci-
mens of the large forms are at hand; their regression curves
are therefore more accurately known and comprise a better
basis of calculation.
33
length. This is shown explicitly by the following schedule:
Values of L/H
At
Birth
Large
Male
Parent
Dwarf
Parent
Dwarf
15.8
15.4
24.9
24.6
15.3
U.3
23.1
22.5
15.3
13.4
23.7
22.2
16.1
15.5
24.9
24.5
15.2
13.3
23.3
22.3
Couplet
Cinereous-tortugensls
Ruber-exsul
Viridis-nuntius
Lutosus-concolor
Or eg anus - or eg anus
(mainland-island)
Here the dwarf juvenile ratios average 8.3 per cent lower
than the parents; but the adult ratios differ by only 3.1
per cent. This corroborates the finding under growth,* that
stunting is not a uniform reduction from germ cell to adult,
but is a force partly effective in the post-natal stage.
Effect of Ultimate Length on Species - Difference Calculations
I have previously stated (p. 25) that, in making tax-
onomic comparisons, based on head length, it is necessary that
the two forms have approximately the same total lengths over-
all. If they are considerably different in size, we may be
proving, in an exceedingly cumbersome manner, that one is a
stunted race. For the proof of taxonomic difference, based
on head length, is essentially the investigation of the sep-
aration of the two regression lines at some particular time
of life — usually the adult stage. Having shown that stunted
races have different regression lines from their prototypes,
it follows that the same race in two areas, where different
ultimate lengths are reached, will have different lines. In
making comparisons for taxonomic purposes, we must be pre-
pared to separate this factor from the total difference, to
determine the amount attributable to a real morphological
divergence. This presupposes some knowledge of the ultimate
lengths to which the forms grow — a necessity which is obvious,
for otherwise we cannot be sure that we are not comparing the
adolescents of one form with the adults of the other. Such a
comparison would lack significance.
We have seen that the relationship indicated in
Fig. 12 is closely followed by the stunted races which we have
tested; it is also a rather logical assumption aj represent-
ing the growth pattern of a single subspecies growing to dif-
ferent lengths in separate areas. While proof of this exact
relationship is lacking, we believe it an approach to accu-
racy to compute from it the correction (attributable solely
to length over-all) which should be applied to a difference
before testing for significance. Calling the correction D,
we have the following value as determined from the geometri-
cal relationship of the variables:
D = Hi(Lo + b/a)/(Li + b/a) - iEjL2/l^i) (l^o + b/a)/(L2 + b/a)
L-, and L? are the corresponding lengths of the larger and
smaller forms, at some uniform-age basis of comparison, say
a large male of each form as shown in Table 11. Ht is the
head size of the large form at length L-|_ determined from the
* Occ. Papers S.D.S*oc.Nat .Hist . ,No.3,p.31,1937.
34
known regression equation Hn = aL^ + b; b/a is the ratio of
the constants in the same equation, and L^ is the standard
length at which the comparison is made (700 mm., for example,
in the problem on p. 23, second table).
We now return to the three taxonomic problems pre-
viously discussed (p. 22), and calculate D by means of the
above equation, to see what effect this correction for ulti-
mate lengths would have on the previous conclusions:
Pyrrhus- Stephensi- Cinereous-
Mltchellii Tigris Scutulatus
Total difference in head length
at standard body length 7.20 5-03 3.-43
Attributable to difference in
length (value of D)
Net real difference
1.00
.80
2.70
6.20
4.23
.73
0.412
0.297
0.199
15.0
U.2
3.7
Standard error of difference
(previously determined)
Significance
We see that the differences with respect to the
first two pairs are still highly significant, the reductions
being slight. On the other hand, in the c inereous-scutulatus
couplet, the reduction resulting from adult length differ-
ences is more important, and the net separation of head
lengths, although still above the border line of significance,
has been decidedly reduced by the correction. Thus, this ex-
ample indicates the importance of making an adjustment of this
type.
However, it should be pointed out that the extent
of the reduction in the scutulatus-clnereous comparison is
partly the result of the large size (1700 mm.) assumed for
cinereous, as compared with 1100 mm. for scutulatus. This is
not really a fair basis of comparison. Over much of the cen-
tral and northwestern sections of Arizona, cinereous is not
as large a snake as in Texas, and the figure of 1700 mm. for
a large male in Table 11 was based in part on Texas specimens.
If we premise our conclusions exclusively on Arizona material
we may use a length of 1350 mm. as being more fairly compar-
able with 1100 mm. for scutulatus. Also, we determine a new
regression equation based only on Arizona cinereous, of
H = 0.0350L + 8.3. Using these new data, we find the differ-
ence attributable to different ultimate sizes to be reduced
to 1.30 mm. The net species difference becomes 2.13 and the
significance 10.7, which thus validates the difference beyond
any reasonable doubt. As I have pointed out before, this
head length difference between scutulatus and cinereous is
presented merely as an example. There are other conclusive
differences, particularly of lepidosis, which can be demon-
strated much more readily.
It might be thought that I have assumed too much in
apparently taking for granted a racial or subspecific rela-
tionship between the forms being compared. As a matter of
fact this has not been done. I have merely determined the
difference (as an effect of relative size) which would na-
turally exist if one were a stunted form of the other, to see
35
if the remainder, which represents a true morphological dif-
ference,still retains significance.
Effect of Ultimate Length on Dispersion
One more point remains to be mentioned. It is ob-
vious from the fact that two races, one stunted (or gigantic)
and one normal, do not follow the same head-length equations,
that if we study a widespread population of a species, or
subspecies, involving sub-populations reaching different ul-
timate lengths, the dispersion of the individuals about the
line of regression will be greater than in a homogeneous ser-
ies, because the several group-elements of the population
really follow slightly different regression lines. Thus,
even if the correlation in a local homogeneous population
were perfect, a widespread population (assuming ultimate size
differences) would show dispersion.
Carrying this further, it is evident that in a lo-
cal population, having the usual variations in ultimate length
common to all organisms, if L/H is to be a constant at any
fixed age (full adult size, for example) then each individual
must follow, during growth, a different regression line from
his fellows. Thus, we have a tendency of uniformity in L/H
.to produce nonuniformity in H = aL + b. Furthermore, this
would require the several individuals to approach the type of
relationship indicated in Fig. 12, so that we have some theo-
retical reason to expect head length deviations from the rat-
tlesnake mean regression line to vary with L + b/a, as was
initially shown to be closer to fact than any assumption of a
constant difference in H in terms of millimeters, or other
absolute measurement.
We could test this theory without difficulty if we
had a group of rattlers of assured age equality. As we have
seen that stunted races do not closely follow their proto-
types at birth in L/H, this test should be made upon adults.
If we found in such a limited group that in their particular
regression line the constant b tended to disappear, the
theory would be proven. The sexes would have to be tested
separately. With the material available, I have been unable
to devise an accurate check of this relationship.
Sexual Dimorphism in L/H
The theory of racial consistency in L/H, which has
been indicated by the dwarfed species, would naturally lead
us to expect the sexes to have different regression lines,
since they reach different ultimate lengths, the males grow-
ing to a larger size than the females, except in cerastes.
Our class comparisons (p. 3) failed to discover such differ-
ences. But if the sexes do have approximately the same re-
gression lines, it follows that females have proportionately
larger heads than males of similar age.
In order to check which of these relationships is
more nearly adhered to (i.e., uniformity of regression, or
morphological uniformity), we employ the Platteville series
as an example. We assume that the largest males and females
have reached the same average age. We take as representative
of these groups, the 26 largest males (out of a total of 314
adults and adolescents) and the 24 largest females (out of
36
287), and find that we have the following data:
Males Females
Length range, mm. 860-9-^9 791-857
Mean body length, mm. 899 814
Mean head length, mm. 38.3 35.9
Mean value of L/H 23.47 22.73
Now, if the females tend to have the same L/H ratio
as the males, the average head size of the females shoixld be
814/23. 47, or 34.7 mm. If, on the other hand they tend to
follow the male regression line (which has been separately
determined from the males alone) the head size would be 35.7
mm. Actually, the mean head size is found to be 35.9 mm.,
as shown in the original data. Hence it is clear that of
the alternatives, the trend is toward the two sexes following
the same regression line rather than morphological similarity
at corresponding ages. Plotting the line which the females
would have to follow, to be like the males in L/H at all ages,
places the female line below actuality in every instance
tested.
Another investigation of possible sexual dimorphism
in virldis was made by the preparation of correlation tables
of the Plattevllle and Pierre series, separating the sexes
and using adults only. The regression equations were found
to be as follows:
Plattevllle Pierre
Males H = 0.03326L + 8.25 H = 0.03229L + 9.25
Females H = 0.03351L + 8.12 H = 0.03258L + 9.23
It will be noted that in each instance a is
slightly higher in the female equation than in the male; the
contrary is true of the constant b. Taking snakes 900 mm.
long and finding the head lengths from these equations, we
find the sexual differences to be Just below the level of
significance. But, as the two series give the same result,
I am inclined to believe that the difference is real, al-
though of such slight magnitude as to be evident only in
very large series. Ovir ignoring of sexual dimorphism in our
calculations is still Justified. These calciilations are
based on a coefficient of variation of 3.6 per cent, which
figure has previously been found to represent the dispersion
of the adults of these series.
From these two investigations we reach the conclu-
sion that female rattlers of this species do not have the
same L/H values as the males at corresponding ages. On the
contrary, the sexes follow the same regression lines (or the
female line may be very slightly above the male) which means
that the females have larger heads at comparable ages. Less
complete investigations of other races indicate the same con-
clusions, as did the original comparisons of average lengths
by zones (p. 3).
Crotalus cerastes seems to be the only species de-
viating from this normal rattler relationship, for it is the
only species in which the males and females seem to follow
different regression lines, the females having the larger
heads at the same body lengths as the males. The difference
is so considerable that it can hardly result from inadequate
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data. Nor is an explanation wanting. It will be remembered
that this is the only species in which the females are con-
sistently larger than the males (Occ.Pap.No.3,p.27) . These
two deviations from the rattlesnake normal appear interrelat-
ed. For it is evident that if males and females of normal
species tend to follow the same regression lines, thus making
the female heads proportionately larger at corresponding ages,
we would expect that if a species developed larger females,
and this head proportionality were retained, different regres-
sion lines would result, with a decided female superiority.
This seems actually to have come about in cerastes. To show
why this follows, let us take again as an example of the nor-
mal rattlesnake relationship the Platteville series of viridis.
We have seen that at one age-equality (ultimate average growth)
the L/H ratios are: males 23.-47, and females 22.73; hence on
this basis of age equality the female head will exceed the
male in the ratio of 23.47/22.73, or about 3 per cent. The
same proportionate condition occurs in other normal species
of rattlers. Now, assume this relation to be maintained in a
species in which the females grow to a length exceeding the
males by 4. per cent, as is the case in cerastes. Then at
equal lengths the two heads will differ by about 7 per cent
and separate regression lines will result. This is approxi-
mately the difference in size between the heads of cerastes
at an adult length of 650 mm. This relationship is subject
to further verification as more adult specimens become avail-
able; the present investigation was based on 84. adult females
and 135 males, but more very large specimens are necessary to
smooth the curves. The approximate equations for cerastes
with the sexes separated are: Males, H = 0.0391L + A. 9; fe-
males, H = 0.0409L +5.5. It will be noted that the differ-
ences in the constants are much greater than were found in
viridis.
Species Differences - General Discussion
I have placed the general discussion of species
differences after that covering the position of dwarf forms,
since the latter, if not first explained, would continually
arise in any general comparison. I have endeavored to clari-
fy the relationship which seems to exist, as a result of size
differences, between the regression lines of related forms.
Having disposed of this we may now proceed to a survey of the
regression lines of H on L based on data available in all
species and subspecies. These were set up on uniform coordi-
nates and the values of the constants determined graphically.
The derived statistics are set forth in Table 19.
The number of specimens measiored, as shown in the first col-
umn, may be taken as some indication of the reliability of
the regression constants.
A discussion of species differences and relation-
ships, as indicated by these data, is beset with difficulties.
The regression lines, although essential in any taxonomic de-
terminations, do not in themselves, or in terms of their con-
stants, permit species comparisons, or indicate subgeneric
trends. And it is difficult to select a single quantity, or
relationship, which may be readily used as a lucid basis of
comparison. The employment of these regression lines is so
intimately connected with considerations of agis and size
that their use is dependent on the availability of data with
respect to the latter.
39
HEAD LENGTH IN MM.
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the head width and length co'drdinates of our specimens, we
find once more a straight line relationship, W = a'H - b',
but in this case b' assumes a negative value. (The accents
are used to avoid confusion with the letters employed in the
previous equations of the relationship, H = aL + b).
Before proceeding with the detailed analysis of
the Platteville data, it may be stated that this type of
equation of 1ft and H seems to be quite characteristic of the
rattlers. Giving due regard to the scatter resulting from
the difficulties of measurement previously discussed, in
every case where sufficient material is available, a straight
line of this type is clearly indicated.
Since the equation W = a'H - b' may be converted
into W/H = a' - b»/H, and since b»/H is negative and de-
creases as H increases, it follows at once that W/H increases
as rattlers grow. In other words, rattlers have proportion-
ately wider heads as they mature; young rattlers have slimmer
heads than adults.
Returning to the Platteville series of 233 speci-
mens, a correlation analysis results in the following statis-
tics. The straight line of best fit by the method of least-
squares is found to be W = 0.808H - 2.55, both W and H being
expressed in millimeters. The original data are presented
in Table 20 as a visual exposition. The scatter is by no
means excessive. The coefficient of correlation (r) is high,
being 0.982. This, however, is not of great importance,
since it is self-evident that the larger snakes will have
larger heads in both dimensions. The Harris correlation cri-
terion,* r^g is 0.4.99. The standard error of estimate is
1.06 mm.
Since W = 0.808H - 2.55 and from our former inves-
tigations, H = 0.0355L + 6.97, we can derive the equation of
W and L; this is found to be W = 0.0287L + 3.O8.P Hence, as
in the case of head length, young snakes have larger heads
(using the width dimension as a criterion) than adults. From
these two equations we find the following proportionality in
C.v. viridis between head length and width, at birth, and for
a large adult male:
L H W Ratio W/H
At birth 250 15.9 10.3 0.65
Large male 1000 A2.5 31.9 0.75
♦ Treloar, op.cit., p. 69.
^ It is obvious that these three equations are interrelat-
ed, for if H = aL + b and W = a'H + b« (in this particular
case b' is negative), then the relation of W and L must be
of the form W = aWL + b". For if we replace H in the second
equation with the value of H in the first, we have
W = a' (aL + b) + b» = a'aL + a'b + b'. Thus a" = a'a and
b" = a'b + b», and any one of the three equations is obtain-
able from the other two. It is equally evident that we
could have obtained the values of the constants a" and b"
in W = a"L + b" from the original data at hand, leaving H
entirely out of consideration. But being primarily inter-
ested in the head shape, that is, the ratio of width to
length, rather than the ratio of head width to body length,
I deemed it advisable to work directly with the relation-
ship of W and H.
47
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Thus, we see in figures that, with this type of
relationship, rattlesnake heads become proportionately wider
as the snakes age; the young have slimmer heads than the
adults. This same conclusion can be derived from a compari-
son of b/a with b"/a". The latter figure is smaller, indi-
cating that W is more nearly proportional to L than is H.
In order to prove that this trend is statistically
significant, we compare groups of Juveniles and young adults
within narrow body-length ranges. A group of 4,0 Juveniles
varying in head length from 16^ to 18t mm., were found to
have an average W/H ratio of 0.665*. Similarly, 26 adults
having heads from 35| to 38^ mm. long, average 0.74.0. Analy-
sis shows the difference to be 6.7 times its standard error;
therefore, there can be no question of its reality. The
average body lengths of the snakes compared were: Juveniles,
295 mm.; adults, 850 mm.
The coefficients of variation of the ratios were
4.. 7 per cent in the case of the Juveniles, while the adult
figure was 6.8 per cent. These quantities give an idea of
the scatter about the W-H line.
We previously found (Table 18) that, in groups with
narrow body-length restrictions, head lengths had a coeffi-
cient of variation of about 2* to 3i per cent in Juveniles,
and about 3 to 4 per cent in adults, this giving a good pic-
ture of the character of the scatter about the regression
line H on L. A similar check on oui" Platteville series shows
the Juvenile dispersion of the head width about the W/L re-
gression line to be about 5.1 per cent, and the adult 5.2 per
cent. This wider scatter is probably due in part to a really
greater variation in head width than head length, and in part
to the difficulties of accurate measurement previously men-
tioned. It will be noted that in this measurement the dis-
persion is practically constant from youth to age on a per-
centage, not absolute measurement, basis.
So much for the analysis of a single species. This
gives a sufficient picture of the extent of the variation of
head width. We now tiirn to other forms to see whether there
are important species differences.
In a number of species and subspecies, wherein good
series of measurements were available, a graphical study was
made to determine, separately for each form, the constants in
the equation W = a'H - b' and other statistics of interest.
The results are set forth in Table 21. Here are shown the
values of a', b», and the width-ratio W/H at birth and at
ultimate male size. The value of W at H = 30 mm. is also
given, so that the direction of each regression line may be
visualized. A few of these lines are shown in Fig. 16, which
includes several species which adhere closely to the rattle-
snake mode, as well as those which deviate most conspicuously
therefrom. Most of those not shown would fall between the
stephensi and adamanteus lines.
* The methods previously devised of equating to a standard
head length was used; that is, each specimen was deemed to
maintain a constant percentage deviation of W from the re-
gression line, in translation from its actual value H to a
standard H„.
49
WIDXH OF HEAD IN MM. (w)
We note here a certain uniformity in the equations;
there are no great variations in the constants a» and b « .
Particularly a', which is the more important of the two, is
rather stable.
The constants a" and b" in the equation W = a"L + b"
can be determined from the values of a and b given in Table
19 and the values of a» and b' given in Table 21, for we
have shown that a" = a'a and b" = a'b + b'. We will not
discuss the equations of W and L except to repeat, as was
stated in the viridis development, that, as b"/a" is always
less than b/a, L/W is more nearly a constant during ontogeny
than L/H; rattlesnake head widths are more nearly proportional
to body length than are head lengths.
Just as the discussion of interspecies comparisons
of head-length was difficult because of the changes during
growth, so we find the same trouble here. Taking the rattle-
snake mode to be represented by the important species viridis
and cinereous, it will be observed that the head width is
about 63 to 72 per cent of the head length at birth and in-
creases to 74. to 83 per cent when full growth is reached.
Of those which do not follow this normal trend,
polystictus is by far the most conspicuous, for it has a long
narrow head; W being only 51 per cent at birth, and 59 per
cent at maturity. Other species with narrow heads are enyo,
miliarius . willardl, klauberi, and horrldus, in the order
named, using the adult ratio as a basis of comparison. Nun-
tius tends to have the same W/H ratio as its prototype viri-
dis at full growth; therefore its values of a' and b' are
larger than those of viridis.
Mitchellil, pyrrhus , and cerastes are the only
species having heads distinctly wider than the rattlesnake
mode.
Head depth is even more difficult than width to
measure accurately. Only three species were tested, ciner-
eous, pyrrhus . and catenatus. Pyrrhus was thought to be
particularly flat-headed and is, on a basis of width, but
not length. The equations were found to be approximately
D = 0.4,2H for pyrrhus . D = O.44.H for cinereous, and O.46H
for catenatus. The scatter is considerable. At full growth
the head depth in pyrrhus (using the W/H ratio previously
found) is about 49 per cent of the width, while in cinereous
it is 56 per cent. This difference was not as great as had
been anticipated. Some of the smaller snakes have propor-
tionately deeper heads. Thus, catenatus depth averages 62
per cent of the width. The indications are that the ratio
D/H is more nearly constant amongst the rattlers than D/W.
In other words, narrow-headed rattlers approach squareness
when looked at head on, for although the head is narrowed
proportionate to its length, the depth is not corresponding-
ly reduced.
Summary and Conclusions
1. Of the head dimensions of rattlesnakes,
length is more accurately measurable than width
or depth. This paper is primarily a study of head
length (H) in relation to body length over-all (L) .
2. Head length amongst the rattlesnakes, as
51
a proportion of length of body, is virtually inde-
pendent of sex, except in C. cerastes. In this
species the females have larger heads than the males
of the same body length.
3. Among the rattlers the relationship of H
and L conforms closely to a linear equation of the
form H = aL + b. The constants a and b are in fair-
ly close agreement among the several species. As b
is always positive, Juvenile rattlesnakes have pro-
portionately larger heads than adults.
4. The straight line is the curve of best fit,
there being no subgeneric trends deviating from this
relationship. Variations in either direction away
from a straight line appear to be dictated by chance.
5. Tests on six colubrid species indicate that
two follow a straight line relationship, while four
adhere^ closely to exponential equations of the form
H = mn^. Neither amongst the rattlers, nor these
few colubrids, does any follow the simple ratio
H = pL. From this it follows that L/H is not con-
stant during life and interspecies comparisons can-
not be made by comparing L/H except under restric-
tions with respect to corresponding ages.
6. In the rattlers the correlation between H
and L is high, the correlation coefficient usually
being well above + 0.85. However, as the dispersion
is not constant in terms of absolute measurement,
statistics other than the standard error of estimate
must be devised in making comparisons.
7. The dispersion of H about the regression
line of H on L is found to be practically constant
on a percentage basis at all ages, althoiigh there
is a slight increase in variability in the final
adult stages. The coefficient of variation of H
about the H-L line is usually between 2| and 31
per cent.
8. Based on constant percentage deviations,
a method is devised for determining the equivalent
head length at any arbitrarily selected standard
body length. This permits the concentration of ma-
terial for dispersion study. The distribution about
the regression line is found to be substantially
normal.
9. Since H has a coefficient of variation of
about 3 per cent about the regression line, it con-
stitutes a rather consistent character and may be
useful in critical taxonomic problems, althoiAgh
such use is cumbersome compared with numerical char-
acters (scale counts) which do not change during on-
togeny. In using head length it is necessary to
study the regression lines of the species being com-
pared to determine at what length the difference
should be ascertained. Usually, narrow ranges in L
in the adult field should be adopted. The two forms
being compared must grow to approximately the same
ultimate length or a special correction is necessary,
the formula for which is given. Example taxonomic
problems are worked out.
52
10. Simple L/H ratios may be compared as a
vo\]gh Indication of differences. However, as the
L/H ratio changes within any species during ontogeny,
it is necessary to compare specimens at a rather re-
stricted age.
11. The regression equations of dwarfed forms
may be rather closely approximated by derivation
from their prototypes. This derivation is based on
a rmiform L/H at maturity, and a uniform value of
b/a, the ratio of the constants in the regression
equation of the parent form.
12. Individual length differences in them-
selves tend to produce variations in H = aL + b,
if the individuals are to have uniform L/H values
at maturity. Thus, the dispersion of H about the
H on L regression line may in part result from a
tendency toward uniformity in L/H.
13. Species differences in H are pointed out.
Tigris and m. mitchellii have conspicuously small
neads; molossus and adamanteus large. Small spe-
cies tend to have large heads; this is particularly
marked in the smallest forms.
14,. Studies of head width (v:) indicate a
straight line relationship witn H, W = a'H - b».
An additional straight line relationship,
W = a"L + b" follows from the two previously de-
rived. The constant b« is always negative, hence
adult rattlers have wider heads proportionate to
head length th^in juveniles. Head widths are more
nearly proportional to body length than are head
lengths. The coefficient of variation of W about
the W-L regression line is about 5 per cent.
15. Species comparisons with respect to W
are presented. Most forms are found to have a W/H
ratio of about 63 to 72 per cent at birth, and
about 74 to 33 per cent at full maturity. Poly-
stictus and enyo have conspicuously narrow heads;
mitchellii. pyrrhus , and cerastes are unusually
wide.
16. Head depth (D) is not measurable with
great accuracy. In general, it seems to follow
a simple ratio with H. D/H is more nearly constant
amongst the rattlers than D/W. Thus, narrow-headed
rattlers have deep heads compared to their widths,
and wide headed-rattlers, such as pyrrhus . have no-
tably shallow heads.
53
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