Gary Packard

Colorado State University, Biology, Emeritus

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Frequency distributions, dimorphisms, and allometric variation in size of the weapon on male harvestmen (Arthropoda, Arachnida, Opiliones

Evolutionary Ecology, 2023

Males of numerous species of harvestman have been reported to be dimorphic with respect to size o... more Males of numerous species of harvestman have been reported to be dimorphic with respect to size of the weapon that is used in aggressive encounters with conspecific males. The claim of dimorphism was based in every instance on finite mixture modeling of the frequency distribution for size of the weapon, with the detection of bimodality being equated with the detection of dimorphism. I used the Hartigan dip test and frequency histograms to reexamine the original data for size of the weapon in males of five species of harvestman. Distributions for two of the species are unimodal by both dip test and graphical display; distributions for two others have an appearance of bimodality in frequency diagrams but do not depart significantly from unimodality by the dip test; and the distribution for the remaining species is multimodal by the dip test but appears to be unimodal in graphical display. Patterns of bivariate (allometric) variation follow sigmoidal curves for all five species, with the steep part of the curve corresponding with the low point in the histogram for the two species that give an appearance of bimodality. Thus, bimodality-when it can be detected and confirmed-is most likely to be a consequence of an unexpected pattern of allometric variation and not dimorphism.

Data transformation and model selection in bivariate allometry

Biology Open, 2024

Students of biological allometry have used the logarithmic transformation for over a century to l... more Students of biological allometry have used the logarithmic transformation for over a century to linearize bivariate distributions that are curvilinear on the arithmetic scale. When the distribution is linear, the equation for a straight line fitted to the distribution can be back-transformed to form a two-parameter power function for describing the original observations. However, many of the data in contemporary studies of allometry fail to meet the requirement for loglinearity, thereby precluding the use of the aforementioned protocol. Even when data are linear in logarithmic form, the two-parameter power equation estimated by back-transformation may yield a misleading or erroneous perception of pattern in the original distribution. A better approach to bivariate allometry would be to forego transformation altogether and to fit multiple models to untransformed observations by nonlinear regression, thereby creating a pool of candidate models with different functional form and different assumptions regarding random error. The best model in the pool of candidate models could then be identified by a selection procedure based on maximum likelihood. Two examples are presented to illustrate the power and versatility of newer methods for studying allometric variation. It always is better to examine the original data when it is possible to do so.

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What is complex allometry

Biology Open, 2023

Complex allometry describes a smooth, curvilinear relationship between logarithmic transformation... more Complex allometry describes a smooth, curvilinear relationship between logarithmic transformations of a biological variable and a corresponding measure for body size when the observations are displayed on a bivariate graph with linear scaling. The curvature in such a display is commonly captured by fitting a quadratic equation to the distribution; and the quadratic term is typically interpreted, in turn, to mean that the mathematically equivalent equation for describing the arithmetic distribution is a two-parameter power equation with an exponent that changes with body size. A power equation with an exponent that is itself a function of body size is virtually uninterpretable, yet numerous attempts have been made in recent years to incorporate such an exponent into theoretical models for the evolution of form and function in both plants and animals. However, the curvature that is described by a quadratic equation fitted to logarithms usually means that an explicit, non-zero intercept is required in the power equation describing the untransformed distributionnot that the exponent in the power equation varies with body size. Misperceptions that commonly accompany reports of complex allometry can be avoided by using nonlinear regression to examine untransformed data.

Evolutionary allometry of horn length in the mammalian family Bovidae reconciled by non-linear regression

Biological Journal of the Linnean Society, 2018

I used the equivalent of ANCOVA for non-linear observations to reexamine data for allometric vari... more I used the equivalent of ANCOVA for non-linear observations to reexamine data for allometric variation in length of the horns in male and female bovid mammals. The data came from an earlier investigation that arrived at several important, but weakly supported, conclusions about the relationship between horn size and body size across species. The new analysis on untransformed observations revealed that patterns of variation in horn length are described better by three-parameter power equations than by two-parameter power functions that were the basis for the original study. The Y-intercepts and allometric exponents estimated in the new analysis are identical for both sexes, but the allometric coefficient is larger for males than for females, thereby accounting for the higher trajectory for the curve fitted to data for males. The fitted equations point to sexual dimorphism in length of the horns that increases in concert with increases in body size. However, the function describing horns in males may have been unduly influenced by statistical outliers, so the pattern of allometric variation in males is not settled. The analytical protocol described here is more versatile than conventional and phylogenetic ANCOVAs on log transformations.

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Julian Huxley and the quantification of relative growth

Zoomorphology, 2020

In 1932, Julian Huxley introduced biologists around the world to a simple method for fitting two-... more In 1932, Julian Huxley introduced biologists around the world to a simple method for fitting two-parameter power equations, Y = b × X k , to bivariate observations that follow a curvilinear path on the arithmetic (= linear) scale. The method entails fitting a straight line to logarithmic transformations of the data and then (at least implicitly) back-transforming the resulting equation to the arithmetic domain. This general approach continues to be in wide use by students of biological allometry despite its several limitations and shortcomings. For example, Huxley's allometric method requires that the bivariate data of interest follow the path of a straight line in log domain (i.e., that the data be loglinear). However, many datasets that are otherwise suitable for allometric analysis do not meet the requirement for loglinearity and consequently are beyond the scope of Huxley's method. Such data can usually be examined in the untransformed state by nonlinear regression, and the regression approach enables investigators to fit models with different functional form and random error. Moreover, by combining nonlinear regression with a categorical variable, investigators can compare sets of observations that follow curvilinear paths on the arithmetic scale. The regression protocol is illustrated by reexamining data for relative growth by the internal hinge ligament in a bivalve mollusk and by the elongated snout in garfish (Actinopterygii). The methodology promoted by Huxley has played a major role in development of the field of biological allometry, but the procedure has been superseded by nonlinear regression.

The logarithmic transformation in bivariate allometry

Biological Journal of the Linnean Society, 2023

The field of biological allometry has been dominated since early in the last century by the logar... more The field of biological allometry has been dominated since early in the last century by the logarithmic transformation, which is widely perceived to be necessary for the proper analysis of bivariate data relating the size of a structure or the intensity of a process to some measure of body size. Some investigators argue that transformation is needed to align the analysis with underlying theory; others assert that transformation is required to describe multiplicative growth in living substance; and still other workers believe that transformation is necessary to accommodate multiplicative variation in the response variable (heteroscedasticity) and/or a lognormal distribution for residuals from the fitted equation. None of these beliefs is true. Moreover, constraints imposed by logarithmic transformation typically result in data being 'fitted' to a predetermined statistical model instead of a model being fitted to the data, thereby leading in many instances to erroneous perceptions of pattern in the data, misinterpretation of the findings and misdirection for future research. Robust statistical models with different functional form and different assumptions about random error can be fitted directly to the original data by non-linear regression, thereby obviating transformation altogether. The utility of the regression protocol is illustrated in a re-analysis of published data.

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Bimodal frequency distribution as a criterion for dimorphism in male rhinoceros beetles (Coleoptera: Scarabaeidae: Dynastinae

Biological Journal of the Linnean Society, 2022

Bateson & Brindley (1892) reported that males of the rhinoceros beetle Xylotrupes gideon are dimo... more Bateson & Brindley (1892) reported that males of the rhinoceros beetle Xylotrupes gideon are dimorphic (i.e. occurring in two distinct forms) with respect to size of the prominent cephalic horn. This claim was based on a comparison of frequency distributions for size of the horn (bimodal) and size of the body (unimodal). Bateson and Brindley did not realize, however, that a bivariate plot of horn length against body size follows a steep, S-shaped trajectory, or that steepness of such a curve-and not dimorphism-underlies the bimodal frequency distribution for size of the horn. The bivariate distribution for horn size vs. body size is continuous and without breaks or interruptions, so the beetles merely exhibit a pattern of allometric variation that went unrecognized at the time. Other species of dynastine beetle also display patterns of sigmoid allometry, but only those with the steepest slopes for the allometric curve also have bimodal frequency distributions for size of the horn. The concept of male dimorphism in X. gideon and other dynastine beetles needs to be reconsidered in the light of findings reported here and elsewhere.

When perception isn't reality: allometric variation in the exaggerated mandibles of male stag beetles (Coleoptera: Lucanidae

Biological Journal of the Linnean Society, 2021

A variety of protocols have been used to study allometric variation in size of the exaggerated ma... more A variety of protocols have been used to study allometric variation in size of the exaggerated mandibles on male stag beetles. Many of these protocols entail logarithmic transformation of the original measurements followed by numerical analysis of the transformations by linear regression or some conceptual extension thereof. I reanalysed data from four such studies to show how these protocols can lead investigators to conclusions that are not well supported by the original observations. One of the data sets was originally reported to conform to simple loglinear allometry, with untransformed observations that presumably follow the path of a two-parameter power function; one was said to represent biphasic, loglinear allometry, with two distinctive morphs having different scaling relationships on the arithmetic scale; and two were originally described as cases of discontinuous, loglinear allometry caused by dimorphisms. My analyses, which were based on graphical analysis and nonlinear regression of untransformed observations, revealed that all the data sets form S-shaped distributions and that each of the distributions is well described by a four-parameter sigmoid function. None of the bivariate distributions reveals a discontinuity or dimorphism. Thus, the original authors unknowingly offered descriptions and interpretations for patterns of variation that do not exist in their data.

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Is Allometric Variation in the Cephalic Horn on Male Rhinoceros Beetles Discontinuously Dimorphic

Evolutionary Biology, 2021

I used linear and nonlinear regression to reexamine the allometric relationship between length of... more I used linear and nonlinear regression to reexamine the allometric relationship between length of the cephalic horn and width of the pronotum (a measure of body size) for males of 28 species of rhinoceros beetle (Coleoptera, Scarabaeidae, Dynastinae). An earlier investigation reported that variation in relative size of the horn conforms in most of the 28 species to a pattern of discontinuous, loglinear allometry, which implies, in turn, that two different power equations are needed to describe the full range in untransformed data. My analyses indicate, however, that allometric variation is continuous in all 28 species and that pattern in each of the original bivariate distributions is well described by a single mathematical function. Allometric variation on the original, arithmetic scale is linear for some species; either linear or slightly sigmoidal for other species; and decidedly sigmoidal in still other species. So-called major and minor morphs merely represent the extremes in absolute and relative size for the horn on animals drawn from the same, continuous distribution. The different findings reported in the earlier investigation resulted from insidious problems introduced by logarithmic transformation of the original bivariate data coupled with an overly complex statistical protocol for classifying individual males as either "major" or "minor." Attempts at classification lead inevitably to oversimplification and misrepresentation of the pattern of allometric variation. Focus needs to return to describing and interpreting pattern in untransformed observations because the relationship between the original variables is the relationship of interest.

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Allometric growth in mass by the brain of mammals

Anatomical Record, 2021

I reexamined published data for ontogenetic change in relative mass of the brain in six species o... more I reexamined published data for ontogenetic change in relative mass of the brain in six species of mammal (i.e., sheep, pig, cow, horse, rat, cat) to illustrate an insidious problem with conventional analyses of brain-body allometry. Graphical displays of logarithmic transformations of the original data for

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The fallacy of biphasic growth allometry for the vertebrate brain

Biological Journal of the Linnean Society, 2019

The concept of biphasic, loglinear growth of the vertebrate brain is based on graphical displays ... more The concept of biphasic, loglinear growth of the vertebrate brain is based on graphical displays of logarithmic transformations of the original measurements. Such displays commonly give the appearance of two distinct mathematical distributions-one set of observations following a steep trajectory at the low end of the size range and another set following a shallow trajectory at the high end. However, the appearance of two distributions is an artefact resulting from the logarithmic transformations. Observations of brain mass vs. body mass in each of the eight vertebrate species examined in the current investigation conform to a single mathematical distribution that is well described by a single equation fitted to the original, untransformed data by non-linear regression. Data for carp, chickens, kangaroos and rabbits are described by three-parameter power equations whereas those for dolphins and primates are described by exponential functions that rise rapidly to a maximum. The brain continues to grow throughout life in carp, chickens, kangaroos and rabbits but not in dolphins and primates. Future investigations of relative growth of the brain should be based on graphical and analytical study of observations expressed on the native mathematical scale.

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The misuse of ratios to scale physiological data that vary allometrically with body size

New Directions in Ecological Physiology (eds. M. E. Feder et al.), 1987

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The use of percentages and size-specific indices to normalize physiological data for variation in body size: wasted time, wasted effort

Researchers commonly compute percentages or size-specific indices in an attempt to remove effects... more Researchers commonly compute percentages or size-specific indices in an attempt to remove effects of body size from physiological data. Unfortunately, such ratios seldom eliminate the influence of body size on a physiological response and the ratios introduce major (but often unrecognized) problems with respect to statistical analysis and interpretation of the data. Indeed, these shortcomings of ratios frequently lead investigators to arrive at incorrect conclusions in otherwise flawless experiments. A superior alternative to using ratios combines graphical analysis and the analysis of covariance, which is a widely available statistical routine that uses least-squares regression to remove effects of body size from physiological data. Accordingly, we counsel researchers to discontinue forming ratios in an attempt to normalize physiological data for variation in body size and to adopt a reliable alternative. We also advise readers of scientific research not to place great confidence in results of studies that use ratios for scaling.

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Mobilization of calcium, phosphorus, and magnesium by embryonic alligators (Alligator mississippiensis)

American Journal of Physiology-Regulatory, Integrative and Comparative Physiology, 1989

The yolk of an alligator (Alligator mississippiensis) egg contains approximately 115 mg of Ca, 18... more The yolk of an alligator (Alligator mississippiensis) egg contains approximately 115 mg of Ca, 180 mg of P, and 17 mg of Mg at oviposition. This compartment is the primary (or sole) source of P and Mg for the forming embryo. Both of these elements are depleted rapidly from the yolk once the embryo enters the growth phase of development. The yolk also is an important source of Ca for the embryo, particularly during the first two-thirds of incubation. During the last trimester of development, however, the embryo supplements Ca from the yolk with Ca mobilized from the eggshell. Indeed, more Ca is withdrawn from the eggshell during the last 2 wk of incubation than can be used by the embryo in skeletogenesis. The excess Ca is stored in the yolk, thereby causing net transfer of Ca to shift from withdrawal from the yolk to deposition in the yolk. Consequently, the residual yolk in the hatchling alligator contains a substantial reserve of Ca to support skeletal growth during the neonatal pe...

Effect of Water Balance on Growth and Calcium Mobilization of Embryonic Painted Turtles (Chrysemys picta)

Physiological Zoology, 1986

Flexible-shelled eggs of painted turtles (Chrysemys picta) were incubated in wet (-150 kPa water ... more Flexible-shelled eggs of painted turtles (Chrysemys picta) were incubated in wet (-150 kPa water potential) and dry (-750 kPa water potential) substrates to elicit different patterns of net water exchange between eggs and their environment. Embryos exposed to wet substrates consumed yolk more rapidly, grew faster, and incubated longer than did embryos exposed to dry substrates. As a result, hatchlings emerging in wet environments were larger and contained less residual yolk than hatchlings emerging in dry environments. Embryos exposed to wet and dry conditions relied on both yolk and eggshell to supply the calcium required for development. However, embryos exposed to wet substrates used the eggshell for a larger proportion of their calcium (56%) than did embryos exposed to dry substrates (40%). Residual yolk of hatchlings in both experimental groups contained less than 1 mg of calcium. Consequently, residual yolk cannot support growth of neonates at or near levels characteristic of late-term embryos. Thus, the larger energy reserve available to small hatchlings in the form of a large residual yolk probably can be used only to support maintenance metabolism, and young presumably must begin to forage soon after emerging from the nest if growth of both soft and hard tissues is to continue.

Comparative aspects of calcium metabolism in embryonic reptiles and birds

Respiration and metabolism of embryonic vertebrates, 1984

Eggs of oviparous, amniotic vertebrates must be endowed at oviposition with all of the organic an... more Eggs of oviparous, amniotic vertebrates must be endowed at oviposition with all of the organic and most of the inorganic components required for embryonic growth. A major inorganic constituent of these eggs is calcium used for ossification of the skeleton. The two main sources of this element are the yolk and the eggshell, but the proportion of calcium supplied by these compartments varies among species. Some of the calcium absorbed from the eggshell by embryos of domestic fowl is stored in the yolk. causing the calcium content of this compartment to increase appreciably during incubation. In contrast, the calcium content of yolk declines throughout incubation in eggs of reptiles, and the yolk that is withdrawn into the abdominal cavity just prior to hatching contains only small quantities of this element. Thus, major differences in the pattern of calcium metabolism characterize avian and reptilian embryos, and studies of embryos of domestic fowl may not provide a broadly-based model with which to characterize calcium metabolism in embryos of other species. Control of calcium transport across the cellular epithelia (the yolk sac and chorioallantois) that separate embryos from their sources of calcium (yolk and eggshell) represents one aspect of control of calcium metabolism during embryogenesis, but this process has been examined only in eggs of domestic fowl and only in the chorioallantois. Calcium transport across the chorioallantois of embryonic chicks is influenced by a vitamin K-dependent calcium-binding protein, carbonic anhydrase, the level of calcium to which the chorioallantois is exposed, and vitamin D. However, a complete story concerning control of calcium transport across the chorioallantois and its relationship to calcium regulation in embryos of domestic fowl is not yet possible.

Accumulation of Lactate by Frozen Painted Turtles (Chrysemys picta) and Its Relationship to Freeze Tolerance

Physiological and Biochemical Zoology, 2004

Hatchling painted turtles (Chrysemys picta) survived freezing at -2 degrees C for 4 d, few recove... more Hatchling painted turtles (Chrysemys picta) survived freezing at -2 degrees C for 4 d, few recovered from freezing lasting 6 d, and none survived being frozen for 8 d. Whole-body glucose and lactate were low in animals that had not been subjected to cold and ice but increased precipitously in animals that were frozen for 2 d. Both metabolites continued to increase, but at a somewhat lower rate, in animals frozen for 4, 6, or 8 d. The increase in whole-body lactate reflects a reliance by frozen hatchlings on anaerobiosis, whereas the increase in glucose presumably results from mobilization of glycogen reserves to support anaerobic metabolism. Mortality of frozen hatchlings is correlated with the increase in whole-body lactate. Factors that may contribute to the observed correlation include a compromised capacity for individual organs to cope with the lactic acidosis that accompanies anaerobic metabolism and organ-specific depletion of energy reserves. Individual organs must rely on buffering and glucose reserves available in situ because blood of frozen hatchlings does not circulate. Thus, buffer from the shell cannot be transported to other organs, lactate cannot be sequestered in the shell, and glucose mobilized from liver glycogen is not available to supplement glucose reserves of other tissues. This integrated suite of physiological disruptions may limit tolerance of freezing to conditions with little or no ecological relevance.

Water Relations and Nitrogen Excretion in Embryos of the Oviparous Snake Coluber constrictor

Copeia, 1987

SAS Institute, Inc. Cary, North Carolina. TRACY, C. R., ANDJ. W. DOLE. 1969. Orientation of displ... more SAS Institute, Inc. Cary, North Carolina. TRACY, C. R., ANDJ. W. DOLE. 1969. Orientation of displaced toads, Bufo boreas, to their breeding sites. Copeia 1969:693-700. TWITTY, V. C. 1961. Experiments on homing behavior and speciation in Taricha, p. 415-459. In: Vertebrate speciation. W. F. Blair (ed.). University of Texas Press, Austin, Texas. . 1966. Of scientists and salamanders. W. H. Freeman & Co., San Francisco, California. WADE, M. 1982. Group selection: Migration and the olymorphism and systema ics in the genus Peroyscus I. Vari tion in the old-fie mouse (Peromyss polionotus). Univ. Texas Publ. 7103, Stud. Gedifferentiation of small populations. Evolution 36: 949-961.

Is relative growth by the mammalian heart biphasic or monophasic?

Journal of Anatomy, 2020

I re‐examined data for relative growth by the heart in four species of mammal to reconcile diverg... more I re‐examined data for relative growth by the heart in four species of mammal to reconcile divergent reports that appear in the literature. Raw data for heart and body mass for Horro sheep, humans, gray kangaroos, and tammar wallabies were studied by linear and nonlinear regression, thereby enabling me to avoid the confounding effects of logarithmic transformation and to evaluate multiple statistical models for describing pattern in each set of observations. My analyses indicate that relative growth by the heart is monophasic in all four species and either isometric or near isometric on the arithmetic scale. The heart in these mammals consequently grows in mass in approximate proportion to growth in mass by the body. The appearance of biphasic allometric growth in prior studies was an artifact resulting from logarithmic transformation. Although parturition in sheep and humans is accompanied by a change in the distribution of blood out of the heart and into pulmonary and systemic circuits, the challenge is met without marked increases in absolute or relative size of the heart.

Evolutionary allometry of horn length in the mammalian family Bovidae reconciled by non-linear regression

Biological Journal of the Linnean Society, 2018

I used the equivalent of ANCOVA for non-linear observations to re-examine data for allometric var... more I used the equivalent of ANCOVA for non-linear observations to re-examine data for allometric variation in length of the horns in male and female bovid mammals. The data came from an earlier investigation that arrived at several important, but weakly supported, conclusions about the relationship between horn size and body size across species. The new analysis on untransformed observations revealed that patterns of variation in horn length are described better by three-parameter power equations than by two-parameter power functions that were the basis for the original study. The Y-intercepts and allometric exponents estimated in the new analysis are identical for both sexes, but the allometric coefficient is larger for males than for females, thereby accounting for the higher trajectory for the curve fitted to data for males. The fitted equations point to sexual dimorphism in length of the horns that increases in concert with increases in body size. However, the function describing horns in males may have been unduly influenced by statistical outliers, so the pattern of allometric variation in males is not settled. The analytical protocol described here is more versatile than conventional and phylogenetic ANCOVAs on log transformations.

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