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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. 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