Gary Packard

• Doctor of Philosophy
• Professor Emeritus at Colorado State University

A recent article in this journal presented a new analysis of ontogenetic variation in metabolic rate vs. body mass in tobacco hornworms (Manduca sexta) as part of an overall defense of the concept of polyphasic, loglinear allometry. The analysis applied a method that is widely used in contemporary studies of allometric variation: transform the orig...

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...

Several investigations in recent years have reported patterns of discontinuous, biphasic, loglinear variation in the metabolic allometry of aquatic animals. These putative shifts in pattern of allometry have been attributed to changes in the primary site for gas exchange from cutaneous to branchial as animals undergo ontogenetic changes in size, sh...

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;...

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 equate...

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 neede...

Authors of a recent report concluded that different patterns of metabolic allometry characterize juvenile and subadult stages in the life cycle of American eels (Anguilla rostrata). This conclusion was based on a comparison of straight lines fitted to logarithmic transformations of the original observations for metabolic rate and body mass, with th...

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...

I used linear and nonlinear regression to re-examine 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...

I re‐examined 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 each species give the appearance of two...

I used the equivalent of nonlinear analysis of covariance (ANCOVA) to re-examine relative growth by the horns on males and females of alpine ibex (Capra ibex) and mouflon sheep (Ovis gmelini). A prior study of allometric growth by the horns on these animals described a pattern of biphasic allometry for both sexes, with two different mathematical eq...

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 logarithmi...

The conventional allometric method entails fitting a straight line to logarithmic transformations of the original bivariate data and then back‐transforming the resulting equation (at least implicitly) to form a two‐parameter power function, Y = a × Xb, on the arithmetic scale. Although the protocol is widely used in contemporary research, it common...

In 1932, Julian Huxley introduced biologists around the world to a simple method for fitting two-parameter power equations, \( Y\, = \,b \times \, X^{k} \), 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 l...

I re-examined the scaling of metabolic rate versus body mass in eight species of ant (static allometry) to illustrate one of the ways by which contemporary concepts of metabolic allometry have been negatively impacted by the widespread use of a standardized procedure for analyzing bivariate data. The procedure in question is the one promoted by Jul...

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 foll...

I used linear and nonlinear regression to re-examine published data on the scaling of metabolic rate vs. body mass in an ontogenetic series of black carp (Mylopharyngodon piceus (Richardson, 1846)). My objective was to expose shortcomings of the conventional procedure for fitting statistical models to bivariate observations (i.e., the procedure tha...

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 ne...

A novel statistical routine is presented here for exploring and comparing patterns of allometric variation in two or more groups of subjects. The routine combines elements of the analysis of variance (ANOVA) with non-linear regression to achieve the equivalent of an analysis of covariance (ANCOVA) on curvilinear data. The starting point is a three-...

Studies performed over the last 20 years have repeatedly documented a slight convex curvature (relative to the x-axis) in double-logarithmic plots of basal metabolic rate (BMR) versus body mass in mammals. This curvilinear pattern has usually been interpreted in the context of a simple, two-parameter power function on the arithmetic scale, y = a ×...

The allometric method, which is widely (but somewhat inaccurately) attributed to Julian Huxley, is at the heart of some of the most important theoretical models in the field of evolutionary biology (e.g. the Metabolic Theory of Ecology). The procedure entails fitting a straight line to logarithmic transformations of the original bivariate data and...

Nonlinear regression was used to fit power functions with different forms for random error to data for length of the upper jaw versus length of the rest of body in two species of gars. Growth by the jaws of these species was reported earlier to conform to a pattern of polyphasic loglinear allometry indicative of relatively rapid growth by the jaw i...

White and Seymour examined the scaling of central arterial blood pressure against body mass in mammals ranging in size from a 30 g mouse to a 4080 kg elephant. Exponents in power functions fitted to each of three datasets (systolic, diastolic, and mean arterial pressure) were reported to be significantly greater than zero and indistinguishable from...

A perplexing problem confronting students of metabolic allometry concerns the convex curvature that seemingly occurs in log-log plots of basal metabolic rate (BMR) vs. body mass in mammals. This putative curvilinearity has typically been interpreted in the context of a simple power function, Y=a*Xb, on the arithmetic scale, with the allometric expo...

Lemaitre and colleagues used conventional allometric methods to examine the relationship between length of antlers and mass of the body in males of 31 species in the mammalian family Cervidae [[1][1]]. Logarithmic transformation failed to linearize the bivariate distribution, so the authors fitted

Xiao and colleagues re-examined 471 datasets from the literature in a major study comparing two common procedures for fitting the allometric equation y = axb to bivariate data (Xiao et al., 2011). One of the procedures was the traditional allometric method, whereby the model for a straight line fitted to logarithmic transformations of the original...

Premise of research. The hypothesis of diminishing returns posits that growth by a leaf creates new surface for capturing light but that size of the leaf is ultimately limited by disproportionately large increases in mass of the petiole, the midrib, and/or supporting elements in the lamina itself. The concept is based on numerous investigations of...

The traditional allometric method, which is at the heart of research paradigms used by comparative biologists around the world, entails fitting a straight line to logarithmic transformations of the original bivariate data and then back-transforming the resulting equation to form a two-parameter power function in the arithmetic scale. The method has...

The Metabolic Theory of Ecology (MTE) transformed the field of biological allometry from a discipline that is focused on description to a discipline that is focused more on formulating and testing theory. However, much of the empirical research providing essential background for the MTE – as well as research to test predictions of the theory – is b...

The ongoing debate about methods for fitting the two-parameter allometric equation y = ax(b) to bivariate data seemed to be resolved recently when three groups of investigators independently reported that statistical models fitted by the traditional allometric method (i.e., by back-transforming a linear model fitted to log-log transformations) typi...

Professor Geraert and I agree on the importance of examining arithmetic data instead of logarithmic transformations but we disagree on other issues pertaining to the fitting of statistical models in bivariate allometry. The most important of our differences concerns the utility of quadratic

The allometric equation, y = ax b, is commonly fitted to data indirectly by transforming predictor (x) and response (y) variables to logarithms, fitting a straight line to the transformations, and then back‐transforming (exponentiating) the resulting equation to the original arithmetic scale. Sometimes, however, transformation fails to linearize th...

The allometric method, which often is attributed to Julian Huxley, entails fitting a straight line to logarithmic transformations of the original bivariate data and then back-transforming the resulting equation to form a power function in the arithmetic scale. Development of the technique was strongly influenced by Huxley's own research on growth b...

Three data sets from the recent literature were submitted to new analyses to illustrate the rotational distortion that commonly accompanies traditional allometric analyses and that often causes allometric equations to be inaccurate and misleading. The first investigation focused on the scaling of evaporative water loss to body mass in passerine bir...

Parameters in the two-parameter allometric equation are commonly estimated by fitting a straight line to logarithmic transformations of the original data and by back-transforming the resulting model to the arithmetic scale. However, log transformation distorts the relationship between the predictor and response variables, and this distortion may be...

Eggshells of the Australian bearded dragon consist of a layer of calcite overlying a multilayered, fibrous shell membrane. Some layers of the shell membrane are comprised of tightly woven feltworks of fibers, whereas other layers consist of more open mats of fibers with numerous spaces among the fibers. The shell membrane is organized into a series...

We monitored temperatures during the winter of 1995–1996 inside 18 nests containing hatchling painted turtles (Chrysemys picta). The study was performed at the Valentine National Wildlife Refuge in north-central Nebraska to assess survival of neonatal turtles in relation to the thermal environment inside their hibernacula. Minimum temperatures in t...

We performed an experiment at a field site in north-central Nebraska, U.S.A., to assess the importance of the nest environment as a determinant of body size and condition in hatchling snapping turtles (Chelydra serpentina). The contents of newly constructed nests were manipulated by reciprocal transplant so that each of several nests received a com...

We assessed the ability of hatchling painted turtles (Chrysemys picta) from northern Minnesota to resist freezing when exposed to conditions like those occurring in natural hibernacula (natal nests). We placed animals individually into artificial hibernacula constructed in jars of damp, loamy sand and then lowered the temperature to approximately -...

Eggs of a desert lizard (Callisaurus draconoides) were incubated under hydric conditions simulating those of natural nests. Eggs increased in mass during the first half of incubation owing to a net uptake of water, and declined in mass during the second half of incubation owing to a net loss of water. Patterns of net water exchange were related to...

Sex ratios of hatchling painted turtles (Chrysemys picta) are influenced by the hydric environment when eggs are exposed to fluctuating temperatures similar to those encountered in natural nests. When temperature varies between 18 and 30 °C over the course of a single day, nearly equal numbers of males and females hatch from eggs held on wet substr...

Most hypotheses addressing the significance of intraspecific variation in size of reptilian eggs have focused on potential benefits accruing to the large hatchlings that usually emerge from large eggs. We find, however, that large eggs of painted turtles (Chrysemys picta) have a significantly higher probability of hatching than do small eggs incuba...

Water potential and temperature are interrelated variables that must be studied simultaneously to gain insight concerning the water relations of reptilian eggs incubating in subterranean nests. We measured these variables inside nests of common snapping turtles (Chelydra serpentina) using thermocouple psychrometry. Water potentials in nests were hi...

Small amounts of ammonia and urea were present in freshly laid eggs of Japanese quail (Coturnix coturnix), but neither substance increased appreciably in quantity during the 17 days between oviposition and pipping. In contrast, urate was not detected in eggs before day 10 of incubation, but accumulated rapidly thereafter. Over 90% of the waste nitr...

We measured temperatures inside seven natural nests of painted turtles (Chrysemys picta) at our study site in north central Nebraska during the winter of 1987–1988. Although the lowest temperature recorded in some nests was only slightly below 0 °C, the minimum temperature measured in other nests was as low as −6.2 °C. Viable hatchlings were presen...

The effect of temperature on blood pH in embryonic snapping turtles (Chelydra serpentina) was examined to determine whether the blood pH changes in the same manner as the neutral pH of pure water. Eggs were incubated on moistened vermiculite (water potential of −150 or −950 kPa) at 26 or 27 °C. On day 59 of incubation, eggs were placed in individua...

We injected eggs of softshell turtles (Trionyx spiniferus) with solutions of urea at the midpoint of incubation to induce different levels of uremia in developing embryos. The experiment was undertaken as a test of the hypothesis that urea inhibits intermediary metabolism of embryos and thereby causes a reduction in their rates of growth. The injec...

Several attempts have been made in recent years to formulate a general explanation for what appear to be recurring patterns of allometric variation in morphology, physiology, and ecology of both plants and animals (e.g. the Metabolic Theory of Ecology, the Allometric Cascade, the Metabolic-Level Boundaries hypothesis). However, published estimates...

Rigid-shelled eggs of softshell turtles (Trionyx spiniferus) incubated at 29d̀C under hydric conditions simulating those at the interior of natural nests (i.e. eggs not contacting the substrate) declined in mass during incubation owing to the continuous escape of water vapour to air trapped inside the chambers, with eggs incubated in relatively dry...

We use data from the literature to compare two statistical procedures for estimating mass (or size) of quadrupedal dinosaurs and other extraordinarily large animals in extinct lineages. Both methods entail extrapolation from allometric equations fitted to data for a reference group of contemporary animals having a body form similar to that of the d...

The traditional approach to allometric analysis entails the fitting of a straight line to logarithmic transformations of the data, after which parameters in a two-parameter allometric equation are estimated by back-transformation to the original scale. We re-examined published data for dimensions of the limbs in 22 species of varanid lizards to ill...

The field of biological allometry was energized by the publication in 1997 of a theoretical model purporting to explain 3/4-power scaling of metabolic rate with body mass in mammals. This 3/4-power scaling exponent, which was first reported by Max Kleiber in 1932, has been derived repeatedly in empirical research by independent investigators and ha...

We re-examined data for field metabolic rates of varanid lizards and marsupial mammals to illustrate how different procedures for fitting the allometric equation can lead to very different estimates for the allometric coefficient and exponent. A two-parameter power function was obtained in each case by the traditional method of back-transformation...

The standard approach to most allometric research is to gather data on a biological function and a measure of body size, convert the data to logarithms, display the new values in a bivariate plot, and then fit a straight line to the transformations by the method of least squares. The slope of the fitted line provides an estimate for the allometric...

Eggs of Blanding's turtles (Emydoidea blandingii) were incubated in the laboratory under hydric conditions eliciting different patterns of net water exchange between eggs and surrounding air and substrate. Eggs incubated on wet and intermediate substrates increased in mass during the first half of incubation and decreased in mass during the second...

Female zebra-tailed lizards (Iguanidae: Callisaurus draconoides) lay roughly ovoid eggs with thin, highly extensible shells. The outer surface of the eggshell is a thin, calcareous crust of calcium carbonate in the calcite morph. Immediately beneath the crystalline matrix is a shell membrane composed of multiple layers of fibres organized into an u...

We performed an experiment at a field site in north-central Nebraska to assess the role of the nest environment in inducing variation in bone mineral content in hatchling painted turtles Chrysemys picta (Schneider 1783). The contents of several newly constructed nests were manipulated by reciprocal transplant, after which the eggs were allowed to i...

Painted turtles (Chrysemys picta) typically spend their first winter of life in a shallow, subterranean hibernaculum (the natal nest) where they seemingly withstand exposure to ice and cold by resisting freezing and becoming supercooled. However, turtles ingest soil and fragments of eggshell as they are hatching from their eggs, and the ingestate u...

Hatchling painted turtles (Chrysemys picta) typically spend their first winter of life in a shallow, subterranean hibernaculum (the natal nest), where they may be exposed for extended periods to ice and cold. The key to their survival seems to be to avoid freezing and to sustain a state of supercooling. As temperature declines below 0 degrees C, ho...

Hatchlings of the North American Painted Turtle, Chrysemys picta (Schneider 1783) typically spend their first winter of life inside a shallow, terrestrial hibernaculum (the natal nest) where they commonly are exposed for extended periods to ice and cold. Current evidence indicates that turtles withstand such exposure by resisting freezing and becom...

Many physiologists believe that hatchling painted turtles (Chrysemys picta) provide a remarkable, and possibly unique, example of 'natural freeze-tolerance' in an amniotic vertebrate. However, the concept of natural freeze-tolerance in neonatal painted turtles is based on results from laboratory studies that were not placed in an appropriate ecolog...

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 conti...

Baby Painted Turtles, Chrysemys picta (Schneider 1783), ingest soil and fragments of eggshell as they are hatching from their eggs. The ingested soil typically contains bacterial nucleating agents that can cause water in the gut to freeze at high subzero temperatures. If ice forms in the gut, however, the crystals penetrate the wall of that organ a...

Hatchlings of the North American painted turtle (Family Emydidae: Chrysemys picta) typically spend their first winter of life inside a shallow, subterranean hibernaculum (the natal nest) where life-threatening conditions of ice and cold commonly occur. Although a popular opinion holds that neonates exploit a tolerance for freezing to survive the ri...

Neonatal Painted Turtles, Chrysemys picta (Schneider 1783), typically spend the first winter of their life inside the shallow, subterranean nest where they completed incubation the preceding summer. This behaviour commonly causes hatchlings in northerly populations to be exposed in mid‐winter to life‐threatening conditions of ice and cold. Neonates...

We dissected hearts from near-term embryos and hatchlings of common snapping turtles (Chelydridae: Chelydra serpentina) whose eggs had incubated on wet or dry substrates, and then dried and individually weighed the heart and yolk-free carcass from each animal. Hearts and carcasses of prenatal and neonatal animals grew at different rates, and the pa...

The contents of newly constructed nests of Painted Turtles, Chrysemys picta (Schneider 1783), were manipulated by reciprocal transplant so that each of several nests received a complement of eggs from each of several females. The eggs were recovered from nests after 8 weeks and allowed to complete their incubation under standard conditions in the l...

Hatchlings of the North American painted turtle (Chrysemys picta) typically spend their first winter of life inside the shallow, subterranean nest where they completed incubation the preceding summer. This facet of their natural history commonly causes neonates in northerly populations to be exposed in mid-winter to ice and cold, which many animals...

Hatchling painted turtles typically spend their first winter inside the shallow, subterranean nest where they completed incubation the preceding summer. This behavior often exposes animals in northerly populations in midwinter to life-threatening conditions of ice and cold. The hatchlings apparently withstand exposure to such conditions by avoiding...

Hatchlings of the North American painted turtle (Chrysemys picta) spend their first winter of life inside a shallow, subterranean hibernaculum (the natal nest) where they may be exposed for extended periods to ice and cold. Hatchlings seemingly survive exposure to such conditions by becoming supercooled (i.e., by remaining unfrozen at temperatures...

The equilibrium freezing point for body fluids of hatchling Blanding's turtles (Emydoidea blandingii) is near -0.7 C, but neonates that are held in a dry, ice-free environment can be (super)cooled to approximately -6 C before they begin to freeze spontaneously (i.e., by heterogeneous nucleation). Unfrozen turtles recover from exposure to -4 C, so t...

1. The contents of newly constructed nests of Painted Turtles, Chrysemys picta, were manipulated by reciprocal transplant, so that each of several nests received a complement of eggs from each of several females. The eggs were recovered from nests after 8 weeks and allowed to complete incubation under standard conditions in the laboratory. 2. Live...

Hatchlings of the North American painted turtle (Chrysemys picta) typically spend their first winter of life inside the shallow, subterranean nest where they completed embryogenesis the preceding summer. Neonates at northern localities consequently may be exposed during winter to subzero temperatures and frozen soil. Hatchlings apparently survive e...

We manipulated the amount of water that was available to prenatal and neonatal snapping turtles (Chelydra serpentina) in order to assess the impact of water on growth by different organs in these animals. Three treatments were used: (1) turtles that completed their incubation on a wet substrate, (2) turtles that completed their incubation on a dry...

Hatchling painted turtles (Chrysemys picta), Blanding's turtles (Emydoidea blandingi), slider turtles (Trachemys scripta), and snapping turtles (Chelydra serpentina) were inoculated with ice at temperatures near the equilibrium freezing point for their body fluids (ca. -0.7 C) and then frozen at -2.0 C. All animals survived freezing for up to 30 h,...

We assessed the ability of hatchling painted turtles (Chrysemys picta) from northern Minnesota to resist freezing when exposed to conditions like those occurring in natural hibernacula (natal nests). We placed animals individually into artificial hibernacula constructed in jars of damp, loamy sand and then lowered the temperature to approximately -...

The exchange of water between a chelonian egg and its subterranean environment is influenced by numerous factors, the most important of which are (1) structure of the calcareous layer of the eggshell, (2) water potential and temperature in the nest, and (3) fraction of the eggshell that actually contacts soil in the nest cavity. Eggs with relativel...

We performed an experiment at a field site in north-central Nebraska, U.S.A., to assess the importance of the nest environment as a determinant of body size and condition in hatchling snapping turtles (Chelydra serpentina). The contents of newly constructed nests were manipulated by reciprocal transplant so that each of several nests received a com...

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 interpret...

Hatchling painted turtles from a population in northcentral Nebraska (USA) reach their limit of supercooling at temperatures near - 17°C. However, animals usually die without freezing at temperatures between - 10°C and - 15°C, so turtles already are dead by the time they reach their limit of supercooling. Although hatchlings may exploit a capacity...

Neonatal painted turtles (Chrysemys picta) spend their first winter inside the shallow, subterranean nest cavity where they completed embryogenesis. Consequently, hatchlings at high latitudes may be exposed to ice and cold during the winter. This study was undertaken to determine how long hatchlings withstand freezing at temperatures slightly below...

The equilibrium freezing point for body fluids of hatchling slider turtles (Trachemys scripta) is approximately -0.6 C, but turtles in dry, ice-free environments supercool quite readily at least to -4 C. However, the integument of neonatal sliders offers little resistance to the penetration of ice into body compartments from the environment, and fr...

Miniature dataloggers were used during the winter of 1994-1995 to measure temperatures inside 14 nests containing hatchling painted turtles (Chrysemys picta). The study was conducted at the Valentine National Wildlife Refuge in northcentral Nebraska (1) to assess variation in temperature within and among nests, (2) to determine the duration of expo...