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AMER. ZOOL.,19:195-209 (1979).
Significance of Skewness in Ectotherm Thermoregulation
CALVIN B. D E W I T T
AND
ROBERT M. FRIEDMAN
Institute for Environmental Studies, University of Wisconsin, Madison, Wisconsin 53706
SYNOPSIS. Body temperature distributions of ectotherms which are free to select any environmental temperature throughout a wide range usually are negatively skewed, that is, a
wider range of selected temperature falls below the median than above it. This negative
skewness can be understood as the consequence of the regulation of a temperaturedependent rate process whose rate, as an exponential function of temperature, is maintained within a normal curve of error. Analysis of frequency distributions of body temperature show that they can be characterized by the median and two additional parameters.
One of these can be used to approximate the Q10 of the hypothetical rate process. Mathematical and graphical techniques for the description of body temperature distributions are
developed, and are proposed as useful tools for comparative and experimental studies of
body temperature regulation.
INTRODUCTION
Body temperature distributions have
been published for a wide array of ectotherms studied using experimental
thermal gradients. Despite the lack of uniformity in procedures and techniques
many of these distributions display a common characteristic: they are negatively
skewed (Fig. 1). Negative skewness, characterized by a longer tail on the low temperature side of the distribution, is so
commonly observed as to warrant careful
attention and study. The purpose of this
paper is to describe and analyze this feature of negative skewness, to present and
to test an hypothesis on its physiological
basis, and to illustrate how a recognition of
the pervasiveness of negatively skewed
distributions can provide some insights
both for the description of the level and
precision of thermoregulation and for
comparative and experimental studies.
We acknowledge the assistance of Alan Goodwin in
obtaining data from the literature. Early development
of the work presented in this paper was supported by
N.S.F. Fellowships 20490, 21365, and 32231 to
C.B.D. and publication costs were provided from
N.S.F. Grant No. PCM78-05691 to W. W. Reynolds.
R. M. Friedman's current address is: Office of
Technology Assessment, U.S. Congress, Washington,
D.C. 20510
THE PHYSIOLOGICAL BASIS FOR
NEGATIVELY-SKEWED TEMPERATURE
DISTRIBUTIONS
An hypothesis
We hypothesize that the negatively
skewed temperature distributions have as
their basis the regulation, not of temperature per se, but rather of a physiological rate
process whose rate is an exponential function of temperature. If this temperature
dependent rate process is regulated such
that its frequency distribution is symmetrical, approximating a normal curve of error,
which we assume to be the case, then the
corresponding distribution of body temperatures would be negatively skewed (Fig.
2). Curve A (Fig. 2) shows the exponential
relationship of the regulated temperature
dependent rate process to temperature.
Curve B shows a normally distributed curve
for that rate process. And curve C shows
the corresponding negatively skewed frequency distribution of body temperature.
This hypothesis appears to be consistent
with what is known about the responses of
physiological rate processes to temperature, most of which are approximated by
exponential functions describable by exponential parameters such as the Q10 or Arrhenius ft. (See Precht et ai, 1973, and
195
196
C. B. D E W I T T AND R. M. FRIEDMAN
Adesmia
clathrata
A
I\
Rana cafes beiano
May
I
r
Zophosls punctata
Rana
pipiens,
April
Blatta oriantalis
o
z
Gerrhosaurus
A
Rana pipiens
June
/
r>
a
Ambystoma
'
Dry
flavigulpris
J\
Chameleo dilepis
/
Calathus fuscipes
•
J
/v
UJ
LU
Amphibolurus barbatus •
^
\
i
tigrinum
June
Cyprinus carpio
J
,
1
r
/
/ '
Calathus
/
, / '
\
fuscipes
Moist
Ambystoma
,
.
. y
Solmol gairdneri
tigrinum
July
,/
10
20
30
40
50
10
20
/'
30
TEMPERATURE
40 SO
10
" x
20
30
40 SO
(DEG. C)
FIG. 1. Temperature preference distributions for a
variety of ectotherms. Data from Gunn, 1934, for
Blatta orientalis; from Bodenheimer, 1931, for other
insects (remainder of first column); from Lucas and
Reynolds, 1967, for amphibians (second column);
from Lee and Badham, 1963, for Amphibolurm barbatus: from Stebbins, 1961, for Gerrhosaurus jlavigularis and Chameleo dilepis; fiom Pitt el al., 1956, for
Cyprinus carpio and from Javaid and Anderson, 1967,
Johnson et al., 1954, for discussion of
temperature dependent rate processes.)
An alternative hypothesis, implicit in
methodology which assumes normal distributions of body temperatures, is that
temperature per se, or some linear function
thereof, is regulated according to a normal
curve of error. Such a linear model would
assume that organisms respond to temperature in a way similar to that of the volumes of liquids, solids, and gases. We believe, however, that there is no biological
evidence of physiological processes so responding to temperature, and thus view
this alternative hypothesis as less tenable.
In the next section, and in a later section
we put our hypothesis to three initial tests.
The first of these is based upon the expectation that, if the hypothesis is at all valid,
the distribution of an observable and
quantified temperature dependent rate
process should be normal when computed
for a negatively skewed distribution of
body temperature. The second is based
upon the expectation that body temperature frequency distribution data should
reasonably fit whatever theoretical model
is used to quantitatively describe the
hypothesis. And the third test is based
upon the expectation that the QM, of the
hypothetical rate process, if it can be determined by analysis of a negatively
for Salmo gairdneri.
SKEWNESS IN ECTOTHERM THERMOREGULATION
Frequency
/
/
i
A
UJ h
<
\
S\
BODY
i
i
TEMPERATURE
FIG. 2. Graphical description of hypothesis, showing a rate process as an exponential function of temperature (Curve A), the regulation of this rate process
at a rate M with a normal curve of error (Curve B),
and the corresponding negatively skewed body temperature distribution (Curve C).
skewed temperature distribution should be
within reasonable range of the commonly
observed values at or near 2.5.
197
contrast to the body temperature distribution which is negatively skewed as shown
by the concave curve. This is in contrast to
the body temperature distribution which,
as shown by the concave curve, is negatively skewed.
A similar linear plot of cumulative oxygen consumption is obtained when the
same analysis is applied to the pooled body
temperature distributions for 33 days of
records for 11 desert iguanas (temperature
distributions from DeWitt, 1967), although
with different slope. Thus for individuals
as well as for groups of desert iguanas, the
computed frequency distributions of oxygen consumption are normally distributed.
We see in these results nothing to cause
us to reject the hypothesis, although its
validity is by no means made unquestionable by this analysis. The second and third
tests of the hypothesis must await development of the necessary analytical
methods. We thus proceed next to further
develop a theory of negative skewness,
from which methods can be derived.
OXYGEN CONSUMPTION (ml/g hr)
.20
.24
.28
.32
.36
Distributions of oxygen consumption and body
temperatures in the desert iguana: A first initial
test of the hypothesis
If the hypothesis on negative skewness
were to be tested, we ideally would identify
the assumed physiological rate process
being regulated and determine whether or
not it is normally distributed. Lacking
knowledge on what this process might be,
we have taken the exponential relationship
between minimal oxygen consumption
34
36
38
rates and temperature described for the
TEMPERATURE
(DEG.
C)
desert iguana, Dipsosaunis dorsahs (Dawson
and Bartholomew, 1958), together with FIG. 3. Probability plot for body temperature and
the body temperature frequency distribu- oxygen consumption of desert iguanas showing contion of minimal rates of oxygen consump- cavity (negative skewness) for body temperature and
tion. A cumulative probability plot (Fig. 3) linearity (normal distribution) for oxygen consumpData on temperature from curve A of Figure 6
shows this corresponding oxygen con- tion.
from DeWitt (1967) and data on minimal oxygen consumption distribution to be normally dis- sumption computed from Dawson and Bartholomew
tributed as indicated by its linearity, in (1958).
198
C. B. D E W I T T AND R. M. FRIEDMAN
THEORETICAL DEVELOPMENT
formed to —5° and +5°, respectively. The
reason for doing this is to simplify the calGeneral theory
culations. Returning to equation (1), at the
If the negative skewness of body tem- median temperature, the expression reperature distributions is in fact a conse- duces to:
quence of the regulation of a temperature
R — xo e — xo
(2)
dependent rate process, it should be possible to approximate the characteristics of The reference rate xo is defined as the rate
the rate-temperature curve (curve A in at the preferred temperature, or alternaFig. 2). This curve may be described using tively, the reference rate is the preferred
two parameters: xo, the value of the rate value of the rate process. We shall quantify
process at some reference temperature; the value of the rate in a later step.
and k, a parameter which defines how
2. Determine the value of k, the
rapidly the rate process increases with in- parameter which defines how rapidly the
creasing temperature. The rate-tempera- rate process increases with increasing
ture curve may be expressed mathemat- temperature.
ically as:
We have assumed that the rate process is
symmetrically distributed around a preferred value. This preferred value would
R = xo ek
(1) be the mean of the observations of the rate
process if we could measure the process.
where R is the rate process, dependent We may substitute equation (1) into the
upon T, the temperature, x0, the value of quation calculating the mean of the rate
the rate process at a reference tempera- irocess to obtain the following expression:
ture, and k, the change in the value of the
i
n
1
rate process with a change in temperature.
(3)
R = - 2 R, = i
ek
n
n i=1
The parameters xo and k provide one
way to compare thermoregulation between
organisms. To determine the values of where R is the mean of the rate process,
these parameters, we need the following n is the number of observations, and Rj
information: 1) The observed temperature and T,' are corresponding values of the
preference distribution curve (curve C in rate process and temperature, with temFig. 2) and 2) A rate process preference perature expressed as deviation from the
distribution curve (curve B in Fig. 2), median.
We can now make use of the transforwhich we assume looks like the common
symmetrical distribution around some pre- mation explained in step 1. Using the new
ferred rate (at this point unknown). For temperature scale, x0 is equal to the value
mathematical convenience, we will further of the rate process at the median temperassume that the rate process preference ature. This is also the median rate, which
distribution is normal, a reasonable as- for a symmetrical distribution, is equal to
sumption given the number and variety of the mean. Therefore R is equal to xo and
processes which follow this distribution. expression (3) reduces to:
To fit the two parameters we must go
n = 2 ek
through a sequence of mathematical oper(4)
i=t
ations, described below:
1. Transform the temperature scale We can solve this expression for k either
from degrees Celsius to deviation from the numerically or graphically. (Note, there
median of the observed temperature dis- are two solutions for k: k = 0, the trivial
tribution curve in Celsius degrees.
solution, and k = some positive number,
For example, if the median (the temper- the solution of interest).
ature at which 50% of the observations
3. Determine the value of xo, the preoccur at a lower temperature) is 25°C, ferred value of the rate process. Because
observations at 20°C and 30°C are trans- we do not know the scale or units of the
SKEWNESS IN ECTOTHERM THERMOREGULATION
199
rate process, we shall adopt an arbitrary Effects oftheQi0 and k on the characteristics of
scale on the basis of the spread of the rate body temperature distributions
process distribution curve. We shall for
Using the general theory just presented
convenience define s, the standard deviation of the rate process distribution curve, we find that the effect of k, and thus also of
equal to 1. Thus, if we calculate that xo the Q I0 is an effect upon the spread of the
equals 4, this means that the preferred rate distribution and not an effect upon its
is 4 standard deviation units from zero symmetry. This effect is illustrated in Figrate, and that 95% of the observations (±2 ure 4 for Q10 values ranging from 1.5 to
3.5.
SD units) should fall between 2 and 6.
The mathematical relationship upon
We can again substitute equation 1 into a
description of the rate process distribution, which these curves are based is derivable
using in this step the expression for the from the general theory and takes the following form:
variance of the process:
-p+> - T ( - » _ 1 n (x» + s) ~ 1 n (xo ~ s)
1
2
2 (R, - R)
n-1
2(x o e
kT
(9)
'--x o )
2
(5)
where s is the standard deviation of the
rate process and all other variables are the
same as before.
Rearranging this equation, we obtain an
expression which allows us to solve for x()
using the temperature preference data
and the value of k derived previously:
which shows a measure of the spread of
the distribution, T (+) — T ( ~\ to be a linear
function of 1/k, a function plotted in Figure 5.
In order to deal more specifically with
the measure of spread, T<+) - T(~', a further explanation of the terms of this equation and some definitions are necessary.
20
1
Xn =
2
(6)
2kT
e
'i - n
4. The last point in this section is the
relationship between k and Q lo of the rate
process. The Qi0 is the ratio of the rates
measured at a 10C° interval. Again using
equation (1) we obtain
12
3?
8
o
UJ
o
Ul
cr
v
p
e
10
l0k
(7)
~ R,
xTi
Alternatively this may be expressed:
. _ ln(Q, 0 )
10
(8)
Thus, the value of the Q 10 of the
hypothetical regulated rate process is in
theory obtainable from an analysis of a
negatively skewed body temperature distribution.
20
30
40
TEMPERATURE (DEG. C)
50
FIG. 4. Effect of Q10 on body temperature preference distributions. Based on calculations from general theory, with xo set to 4-.0 for all curves and with k
values selected corresponding to Qi0 values of 1.5 to
3.5.
200
C. B. D E W I T T AND R. M. FRIEDMAN
viation units on either side of the mean for
the corresponding normally distributed
rate process.
Figure 5 shows the relationship of these
two ranges to the Q 10 and k of the
hypothesized normally distributed rate
process. When the parameter x0 is held
constant, the 68% and 95% ranges are seen
to decrease with increasing Q10. Stated differently, the spread of body temperature
distributions is greater for lower Q10 values, if x0 is held constant.
IO
8
32
4 3 2.4
1.7
2
1.5
28
24
o
d 20
LJ
O
~
16
UJ
(S
1 12
Effects ofxo on the characteristics of body
temperature distributions
0
4
8
12
16
20
24
28
l/k
FIG. 5. Relation of the spread of body temperature
distributions to the Q,o of the regulated rate process.
Spread is given as the 68% and 95% range of the body
temperature distribution and is plotted against the
reciprocal of k for various selected values of xo.
The term, s, in this equation is the selected
number of standard deviation units from
the median of the normally distributed rate
process, for which the corresponding temperature, T(*', above the median, and the
corresponding temperature, T ( ~\ below
the median are to be determined. The difference between these temperatures, T( + )
— T'"', depends upon the number of standard deviation units selected. Typically
selected standard deviations for the rate
process are plus and minus 1.0 SD, corresponding to the central 68% of the rate
process frequency distribution, and plus
and minus 2.0 SD corresponding to the
central 95% of this distribution.
For the first of these two typical selections the temperature difference T (+l —
T ( ~\ is defined as the 68% temperature
range with the boundary temperatures of
this range being designated as TBH(+) and
TfiH1"' respectively. If used together, but
without subscript designation, T'*1 and
T( ' refer to temperatures which correspond to an equal number of standard de-
Using the general theory further, we
find that the effect of x0 is an effect both
upon the spread and the skewness of the
temperature distribution (Fig. 6). The effect on spread is described by equation (9)
(Fig. 7A). In this plot the Qi0 is held at 2.5,
and thus k is held constant. The spread of
the temperature distribution is seen to increase with decreasing values of xo with
30
TEMPERATURE
(DEG. C)
FIG. 6. Effect of x,, on body temperature preference
distributions. Based on calculations from general
theory, with Qm set to 2.5 for all curves and with \ ,
values selected from 2.5 to 7.0.
201
SKEWNESS IN ECTOTHERM THERMOREGULATION
1.0
.9
8
12
16
20
8
12
16
20
*0
FIG. 7. Relation of the spread (7A) and asymmetry
(7B) to xo. Spread is given as the 68% and 95% range
of the body temperature distribution and asymmetry
is given as the 68% and 95% temperature ratios.
greater sensitivity to change of x(, at lower
values of x,,.
The effect of xo on skewness is expressed by the following equation, derivable
from the general theory:
The determinants and physiological significance
ofx,,
Two parameters define the shape of
body temperature distributions according
to the general theory which has been pres)-lnx(,
(10) sented. Of these, k and its physiological
M l n x , , - ln(x,,-s)
significance is well discussed in the literawhere the terms are defined the same as ture, particularly in the form of the Qi»,
before, where M is the median tempera- derivable from k using equation (7). The
ture, and where the ratio (T( + ) -M)/(M - other parameter, x(>, and particularly its
T'~') is used as an expression of the sym- physiological significance, needs further
metry of the temperature distribution.
discussion. Referring once again to Figure
Based on the earlier discussion and 2, in the context of the general theory redefinitions we define the ratio (T(1K' + > - lating to xo presented earlier, we should
M)/(M - TBK'""1) as the 68% temperature ratio note that equal distances between the horiand (T9r>( + I - M)/(M - T,,-,'-') as the 957c zontal lines in this figure represent stantemperature ratio. The relationship between dard deviation units of 1.0. The line desigthese ratios to xo is presented in Figure 7B, nated M passes through the median (and
showing increasing ratios, and thus in- mean) of the normally distributed rate
creasing symmetry, with increasing x(1.
process at a specific level above the rate of
202
C. B. DEWITT AND R. M. FRIEDMAN
zero, and its height above a rate of zero can
be given in standard deviation units. It is
this height, so expressed, that is designated
xo in the general theory.
We have seen that the temperature distribution, as illustrated by curve C of Figure 2 is affected by xo in both its spread
and its skewness. And, as a consequence of
this importance of x0, it is worthwhile to
consider what determines its numerical
value. There are two such determinants:
the breadth (or standard deviation) of the
rate frequency distribution (curve B) and
the distance that the median of this distribution is from the zero rate level. Thus a
reduction in x0 and thus also of the spread
and skewness of the body temperature
distribution, can be achieved by a narrowing of the rate frequency distribution
(curve B) or by increasing the level at
which the rate is regulated (moving curve
B upward on the vertical axis). Body temperature distribution data as they currently
exist are insufficient to resolve these two
determinants of x0.
One possible experimental approach to
this resolution is the study of situations in
which the level of the regulated rate process is moved upward or downward while
the spread of the rate distribution (curve
B) and the relation of the rate to temperature (curve A) remains the same. Studies
that would be similar to those of Kluger
(1979) and Reynolds et al. (1978) on fever,
if extended to include analysis of the shape
of body temperature distribution curves
hold promise in this regard. And, if the
theory is applicable to body temperature
distributions of endotherms, studies of the
shape of body temperature distribution
curves for hibernators both during and
following hibernation may be helpful.
METHODS FOR DETERMINATION OF
DISTRIBUTION PARAMETERS
Mathematical method
There are two methods which can conveniently be used to determine the distribution parameters xo and k for negatively skewed distributions. The first and
basic method is mathematical and re-
quires: (1) solving for k in equation (4) by
an iterative method such as the NewtonRaphson iterative solution (see Adby and
Dempster, 1974), and (2) solving for x0
using equation (6). These calculations are
easily performed on a digital computer,
which when accompanied by appropriate
graphics routines, can be used to plot the
original body temperature frequency distributions as well as analytical results. In
the results section which follows this has
been the method employed for analysis of
15 different temperature distributions
taken from the literature. Criteria for
selection of these distributions are discussed in that section.
Graphical method
The second method, which is based
upon the mathematical method, employs
graphical techniques. This method is more
generally useful when but a single body
temperature distribution is analyzed. This
method, in abbreviated form, consists of
the following steps: (1) converting frequency data into a cumulative frequency
distribution; (2) arbitrarily selecting x0 and
the corresponding specially prepared probability graph paper; (3) plotting the cumulative frequency data; (4) selecting a
larger x0 if the resulting curve is convex or
selecting a smaller x0 if concave; (5) repeating steps (2) through (4) until the plot
is linear, noting the final value; and (6)
computing the value of k.
Preparation of graph paper required in
step (2) includes: (1) placing a vertical line
through the selected value of xo on Figure
8; (2) at points where this vertical line intersects the frequency lines plotted in Figure 8, drawing onto an adjacent sheet of
paper, horizontal lines labeled with the indicated cumulative frequencies; and (3),
adding an appropriate linear scale of temperature along the abscissa of this adjacent
sheet.
Computation of k from the final plot can
be done by: (1) extending the fitted
straight line to intersect the 97.73% and
2.27% lines of the prepared graph paper,
(2) recording the temperatures corresponding to these intersections, and (3)
SKEWNESS IN ECTOTHERM THERMOREGULATION
203
bles available for this purpose (see Weast,
1978) or using an approximation formula
(see Abramowitz and Stegun, 1968, or
Hewlett-Packard, 1974); (3) adding an arbitrary value for x0 to these standard deviation values; (4) plotting these sums on
the log scale of semi-log paper; (5) selecting a larger xo if the resulting curve is convex (such as the lower curve of Fig. 9) or a
smaller value of x0 if concave (such as the
upper curve of Fig. 9); (6) repeating steps
(3) through (5) until the plot is linear, noting the final value of xo; (7) finding the
temperature, T95(+), corresponding to 2 SD
above the median and the temperature,
TJS'" 1 , corresponding to 2 SD below the
median and using these values in equation
(11) to compute k.
RESULTS
2.27
12
II
FIG. 8. Probability scale generator, showing the
vertical displacement from the baseline of various
probabilities as a function of x0. A vertical line placed
through any x0 on this plot establishes a probability
scale, with the indicated frequency values being assigned to points of intersection.
A purpose of this paper is to test the
stated hypothesis that the frequently observed negatively skewed body temperature distributions have as their basis the
8.0
7.0
6.0
5.0
calculating k from a rearrangement of
equation (9) as follows:
ln(x,, + 2.0) - l n ( x o - 2 . 0 )
(11)
k =
T ( + l — T (1
!I5
l
4.0
IK
T9Sl + )
where
is the higher of the two temperatures just selected and T ^ " 1 is the
lower.
Q- 2
V)
Alternative graphical method
O
A modification of the graphical method
(Fig. 9) allows for plotting several frequency distribution curves on the same
graph, but has the disadvantage that
cumulative frequency distribution data
must be converted to corresponding standard deviation units. The procedure for
this alternative method consists of: (1) converting frequency data into cumulative
frequencies; (2) converting the cumulative
frequencies into standard deviation units
using any of several standard statistical ta-
t I
z
|
.8
.7
.6
.5
34
36
38
TEMPERATURE
40
42
(DEG.C)
FIG. 9. Semi-log plot of a cumulative body temperature distribution for desert iguanas for various values of xo, showing concave plots at values of xo above
4 and convex plots at values of xo below 4. Data are
the same as used in Figure 3.
204
C. B. DEWITT AND R. M. FRIEDMAN
regulation of a rate process whose rate is
an exponential function of temperature.
We have proposed three initial tests, one of
which we already have presented. This
first tests shows that the distribution of
minimal oxygen consumption rates of desert iguanas when calculated for the corresponding negatively skewed body temperature distribution is a normal distribution.
This test provided justification for development of the general theory, which we
now can subject to the two remaining tests
described earlier.
A second initial test of the hypothesis: The Jit of
body temperature frequency distribution data to
the theoretical model
We have used data published earlier
(DeWitt, 1967) for desert iguanas to conduct this test. The data used are those of
11 animals during occupancy of a concentric thermal gradient on three consecutive
days. The records used exclude data recorded during the initial period of warming at the beginning of the day as well as
the data recorded following the time at
which individuals retreated to shelter the
last time on a particular day. Despite exclusion of these data for initial warming and
final cooling, the frequency distribution is
significantly (P<.Q\) negatively skewed,
with the measure of skewness, g, equal to
-0.904 (DeWitt, 1967).
When these data are subjected to the
graphical analytical methods described
previously, the result is the fit shown in the
middle plot of Figure 9. Our interpretation is that (1) the fit is sufficiently good
that the hypothesis on the basis of the fit
should not be rejected. The resulting plot
has an xo of 4.0, a k of O.I 1 and a Q lo of
3.1, determined by the graphical method.
Following this test on a single species,
the test was next applied to temperature
frequency distributions gathered from the
published literature for 35 species of ectotherms. Because the analysis is basically
one requiring the careful description of
the shape of the distributions, a minimal
total number of points clearly is necessary
for the description. Furthermore, for very
broad distributions, a large number of
points in the far tails of the distribution are
not nearly as helpful in describing the
shape as are points in the region of the
peak of the distribution. Thus a minimal
number of points is required here also. On
the basis of these considerations, totals of
10 and 5 points for the central 95% and
55%, respectively, were set as minimum
criteria necessary to be fulfilled if the shape
of a distribution is to be adequately described. Following application of these criteria distributions for only 12 of the initial
35 species remained. This relatively small
proportion of the total is not surprising
since the purpose of most published work
was to determine the mean of the temperature distribution rather than to define its
shape.
In the analysis (Fig. 10) three of the 12
species are described for two different
conditions, two for different seasons (Rana
pipiens larvae for April and June; Amlrystorna tignnum larvae for June and July) and
one for two different conditions (Calathus
fuscipes in dry and moist atmospheres). Of
the total of 15 plots presented, 12 are
shown to be linear and thus what appear to
be good fits to the model.
Inspection of these plots is useful in a
number of ways. The middle sections of
the plots for April Rana pipiens and for
Salrno gairdnen show a steeper slope than
the two ends. Do these curves tend to support rejection of the hypothesis? We think
they do not. Rather, they point to experimental procedures which do not necessarily provide data useful for analysis of the
shape of body temperature distributions.
The procedure used in both of these was to
measure water temperature rather than
body temperature. It thus allows inclusion
of temperatures in the distribution which
represent excursions of short duration into
water at high temperature or low temperature without the body temperature ever
reaching these extremes. This is reflected
also in the corresponding frequency distributions for these two cases in Figure 1,
both of which appear particularly peaked
(leptokurtotic). Actual body temperatures
were in fact recorded only for the reptilian
species of Figures I and 10. All othei ice-
205
SKEWNESS IN ECTOTHERM THERMOREGULATION
Adesmlo ckjthrata
/•
Rana catesbeiana /'
May
Zophosls punctata
Amphlbolurus bar bat us
/
Rana pi plans
/
April
Gerrhosaurus fktvfgularis J
/
s'
Rana pipitns
Blatia oriental/* I
y
June
jr
Chameleo dilapis
/
y
o
en
Dry
CD
<
CD
O
June
/
•/
/
/
IT
o_
Cyprlnus carpio
Ambystoma tigrinum/
Calathus fusclpes /
Moist
0
10
July
/
20
30
So/mo galrdneri^
Ambystoma tigrinum/
Calathus fusclpes Y
40
50
10
/
20
30
40
50
10
20
30
40
50
TEMPERATURE (DEG. C)
FIG. 10. Cumulative body temperature distributions arbitrarily assigned an xo of 4.0. Horizontal temperfor a variety of ectotherms. Plots are similar to Figure ature scales are uniform for all plots. Vertical scales
vary for each plot, but the upper and lower limit of
9, but use the final computed value for xo. Exceptions, for which a single x0 could not be calculated are each are the same, at plus and minus 47.5 percent of
distributions for Rana pipiens — AprW, Ambystoma the distribution from the median. Arrangement and
tigrinum—June, and Salmo gairdneri, all of which were data sources are the same as for Figure 1.
ords are environmental temperature records. Environmental temperatures may
be adequate for determining mean body
temperatures (Reynolds et «/., 1976) but
they may not be adequate for describing
the shape of body temperature distributions. It would seem that they would be
adequate for this purpose only when they
at all times closely approximate body temperatures, such as might be the situation
for animals of small size that have minimal
thermal lag as well as for animals that are
relatively inactive. Again, we find that pro-
cedures used to determine mean body
temperatures are not necessarily appropriate for analysis of the shape of body
temperature frequency distributions.
The remaining distribution which is not
linear on the plots of Figure 10 appears to
be a normal distribution. This plot, for
tiger salamander larvae (Ambystoma tigrinum) is based- upon water temperature
rather than body temperature data. Yet
the plot for the same species in July is
linear (Fig. 10). The reasons remain obscure.
206
C. B. D E W I T T AND R. M. FRIEDMAN
A third initial test of the hypothesis: Proximity of
the computed Qi0 to 2.5
spread (DeWitt, 1967; Reynolds and Casterlin, 1976). The arithmetic mean, for
example, a measure frequently used for
Table 1 presents the results of analysis central tendency, is quite sensitive to disfor the 15 temperature frequency dis- tribution asymmetry and is strongly aftributions. The basic parameters for de- fected by the presence of data in a tail of
scribing the distributions, the median the distribution (Sokal and Rohlf, 1969).
temperature, x0 and k are shown, followed Thus, relatively few data at the low end of
by the derived values for the Q,o. An ob- the distribution may cause an appreciable
jective of this analysis was to determine drop in the mean (see Table 1). Recogniwhether Q10 values derived from applica- tion of this problem has led some investion of the general theory would be in the tigators to discard the lower body temperproximity of 2.5. We conclude, given the atures in negatively skewed distributions,
variety of unspecified conditions under based on the assumption that all of these
which the original data for this analysis records are aberrant or have been rewere obtained, and given the use of en- corded in the "basking range" (see Cowles
vironmental rather than actual body tem- and Bogert, 1944; Soule, 1963). But since
peratures for most of these studies, that the negative skewness remains after elimithese values do not sufficiently deviate nation of records from the initial warming
from the expected 2.5 level to warrant re- or basking period of a given day (DeWitt,
jection of the hypothesis. We recognize 1967) an alternative must be found to
that this third initial test is far from conclu- solving this problem.
sive. First, one might reasonably argue that
We propose as a solution to this problem
QioS well in excess of normal values might
be involved in body temperature regula- the use of a measure of central tendency
tion. Thus the discovery of very high val- which is not sensitive to changes in the
ues for the Q l o might not be sufficient shape of frequency distribution: the megrounds for rejection. Second, we observe dian. In addition to its insensitivity to
that one of the distributions, that of June skewness and to single data points in the
Ambystoma tigrinum larvae is not negatively far tails, the median is identical to the
skewed and therefore does not even pro- mean for normal distributions. Thus for
vide the possibility of a Q,,) determination. normal distributions the median serves
But this case, as a basis for hypothesis re- equally as well as the mean for description
jection, is weakened by the observation of central tendency. But for asymmetric
that another data set for the same species is distributions its use is preferable because
negatively skewed with a computed Q1(, of of its better description of central tend1.35. The use of this case for rejection is ency. In the context of this study the mefurther questionable because of the data dian is additionally attractive as a measure
being in the form of water temperatures of the preferred body temperature since it
rather than body temperatures. As we corresponds with the mean and median of
have noted earlier, it seems that carefully the distribution of the hypothesized rate
designed studies specifically directed to the process.
shape of body temperature distributions
Negative skewness also poses problems
are necessary to more rigorously test the for the use of the standard deviation as a
hypothesis.
measure of the spread of a temperature
distribution. This measure defines the
temperatures which bound the central
68% of a normal distribution, temperaEXPRESSION OF BODY TEMPERATURE
tures which are at equal distances above
DISTRIBUTION CHARACTERISTICS
and below the mean. The use of the stanNegative skewness of body temperature dard deviation as a measure of spread for
frequency distribution creates some chal- a negatively skewed distribution imposes
lenges for an adequate description of both upon it characteristics of a normal distheir central tendency as well as for their tribution. We propose as a solution to this
TABLE 1. Descriptions of ectotherm temperature distributions and corresponding derivedQ10 values.
Species
Description
by basic
parameters
AMPHIBIANS
Med.
x»
Rana catesbeiana
4.6
larvae—May
24.5
Rana pipiens
larvae—April
24.0 17.8
Rana pipiens
3.1
28.1
larvae—June
Ambystoma tignnum
26.3
larvae—June
Ambystoma tigrinum
7.4
25.4
larvae—July
REPTILES
2.5
35.2
Amphibolunis barbatus
2.9
Gerrhosaurus flavigularis 34.7
2.8
32.6
Chameleo dilepis
FISH
3.1
32.1
Cypnnus carpio
18.6
Salmo gairdneri
INSECTS
38.9 4.3
Adesmia clathrata
3.4
36.0
Zophosis punctata
5.4
25.8
Blatta orientalis
4.3
24.7
Calathus fuscipes — Dry
Calalhus fuscipes — Moist 24.2 10.0
Standard
description
(68%)
Q. 0
k
Alternative
standard
description
Med. (+34%, -34%)
Med. (+47.5%, -47.5%)
Temperature
ratio
68%
95%
Normal
description
x ±SD
.048
1.6
24.5 (+4.1, -5.1)
24.5 (+7.5, -11.8)
.80
.64
24.0 ± 4.7
.011
1.1
24.0 ( -
24.0 ( -
-
-
23.9 ± 5.0
.065
1.9
28.1 (+4.3, -5.9)
28.1 (+7.6, -15.7)
.73
.48
27.2 ± 5.3
-
-
26.3 (+3.5, -3.5)
26.3 (+6.9, -6.9)
1.00
1.00
26.3 ± 3.5
.030
1.4
25.4 (+4.3, -4.9)
25.4 (+8.1, -10.7)
.88
.76
25.0 ±4.7
.110
.112
.162
3.0
3.1
5.1
35.2 (+3.0, -4.6)
34.7 (+2.7, -3.8)
32.6 (+1.9, -2.7)
35.2 (+5.3, -14.3)
34.7 (+4.7, -10.6)
32.6 (+3.3, -7.7)
.65
.71
.70
.37
.44
.43
34.3 ±4.4
34.1 ±3.7
32.2 ±2.6
.116
3.2
.
32.1 ( + 2.4, -3.3)
18.6 ( - )
32.1 (+4.3, -8.9)
18.6 ( - )
.73
-
.48
-
31.6 ±3.0
18.8 ±4.6
2.1
2.1
2.3
1.9
1.3
38.9
36.0
25.8
24.7
24.2
.78
.74
.84
.79
.90
.62
.52
.68
.61
.82
38.5
35.3
25.5
24.2
24.0
-
.072
.076
.082
.063
.025
(+2.9,
(+3.4,
(+2.1,
(+3.3,
(+3.8,
- )
-3.7)
-4.6)
-2.5)
-4.2)
-4.2)
38.9
36.0
25.8
24.7
24.2
(+5.3,
(+6.1,
(+3.9,
(+6.1,
(+7.3,
- )
-8.6)
-11.7)
-5.7)
-10.0)
-8.9)
± 3.2
±4.4
±2.5
±3.9
±4.1
208
C. B. DEWITT AND R. M. FRIEDMAN
problem the use of the temperature interval between the median and plus and
minus 34% from the median, computed
from the general theory or determined by
using plots prepared in the graphical
analysis of distributions as previously described. Since, for normal distributions
these measures of spread are identical with
the standard deviation, this method presents no substantial departure from customary practice for distributions which in
fact are normal. But for asymmetric distributions it provides a measure of spread
that better fits the data. It also provides
data which when used with the general
theory, is sufficient for computing xo, k,
Q,o, the 68% range, and the 95% range of
the distribution.
Our proposal for description of central
tendency and dispersion of temperature
distributions has been followed under the
heading of "Standard Description" in
Table 1, in which the median is given followed by parentheses containing the temperature intervals between the median and
plus and minus 34% from the median. For
a normal distribution, such as that shown
for Ambystoma tigrinum—June, this description reduces to the one which is standard for a normal distribution. However,
for a negatively skewed distributiop, the
description provides sufficient information
to fully describe it, including computation
of Qio and 95% range. The alternative
standard description presented in Table 1
is one which is useful when the 95% range
of body temperature is of special interest.
This alternative description can, by application of the general theory, be translated
into the general description.
CONCLUSION
We draw two major conclusions from
this work: (1) that recognition of the
asymmetry of body temperature distributions and applications of the standard description to these distributions provides a
useful tool for comparative and experimental studies of body temperature regulation; and (2) that the general theory may
provide a useful tool for investigating Q)0,
k, and xo if accompanied by carefully de-
signed studies. Such studies should (1) include enough class intervals to define the
shape of the distribution; (2) be based
upon actual body temperature data or
equivalent; (3) include one or more replications; and (4) be conducted under well
understood and accurately specified thermal environments.
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