Is complex allometry in field metabolic rates of mammals a statistical artifact?

Data transformation and model selection in bivariate allometry

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 log-linearity, 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.

Evaluating the muscle attachment hypothesis for sagittal cresting in Gorilla and Pongo

Primate mandibular morphology is often associated with jaw functionality of the masticatory complex in the context of variation in diets. Recent research into the disparities between the diet and jaw functionality in male and female hominoids is inconclusive and suggests that sexual dimorphism in the mandible may be influenced by external factors such as temporalis and masseter muscle morphology, which in turn may be influenced by sexual selection. As the muscles associated with mastication (i.e., the type of chewing exhibited by primates and other mammals) encompass the mandible as well as the neurocranium, including the sagittal crest among some individuals, this study investigates sex-specific associations between regions of the mandibular ramus and neurocranium associated with mastication in a dentally mature sample of Gorilla and Pongo. A total of four cranial and mandibular variables were measured in two Gorilla taxa (Gorilla gorilla gorilla and Gorilla beringei graueri) and one Pongo taxon (Pongo pygmaeus pygmaeus) (n = 220). For all three taxa, we investigate (a) whether the degree of sexual dimorphism in cranial regions associated with sagittal cresting (sagittal crest size (SCS) and temporalis muscle attachment area (TMAA)) is proportional to the degree of mandibular ramus area (MRA) and coronoid process height (CPH) sexual dimorphism, (b) whether there are sex differences in scaling relationships between TMAA and MRA, and (c) whether there are sex differences in the strength of association between TMAA and CPH. We show that for G. g. gorilla, variables associated with sagittal cresting show higher sexual dimorphism values than our two mandibular ramus variables, which is not the case for G. b. graueri or for P. p. pygmaeus. All three taxa show similar sex-specific scaling relationships between TMAA and MRA, where for males this relationship does not diverge from isometry, and for females there is a negative allometric relationship. Our findings also show intraspecific sex differences in allometric slopes between MRA and TMAA for all three taxa. Only G. g. gorilla shows a significant association between TMAA and CPH, which is observed in both sexes. Although there are some statistical associations between the cranial and mandibular regions associated with mastication, our results show that among male gorillas and orangutans, patterns of variation in the sagittal crest, TMAA, mandibular ramus and the coronoid process cannot be explained by the muscle attachment hypothesis alone. These findings have implications surrounding the associations between social behaviour and the morphology of the craniofacial complex.

What is complex allometry?

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 distribution - not 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.

Ventilation Frequency Reveals the Roles of Exchange Surface Areas in Metabolic Scaling

The surface area (SA) theory proposes that resting metabolic rate (RMR) scales with body mass, which parallels the exchange SA of organisms, and that a species with a larger scaling exponent of exchange SA has a larger scaling exponent of RMR. However, the effects of exchange SA on metabolic scaling may be eclipsed because oxygen transfer across the respiratory surface is determined not only by the exchange SA but also by ventilation. We hypothesize that the scaling of both gill surface area (GSA) and ventilation frequency (VF) positively affects the scaling of metabolic rate. In six closely related species of carp maintained under the same experimental conditions, the scaling exponents of RMR and GSA were analyzed. In the goldfish, RMR scaled with body mass by an exponent significantly lower than that of GSA but not different from the exponents of GSA in the remaining five species. The scaling exponent of RMR was positively related to those of both GSA and VF among the species. In addition, the VF-corrected metabolic scaling exponent was positively related to the scaling exponent of GSA among the species. These results suggest that variations in GSA scaling and in VF scaling among species mutually affect metabolic scaling.

Modeling allometric variation: lessons from the metabolic allometry of black carp (Mylopharyngodon piceus)

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 that is widely attributed to J.S. Huxley) and simultaneously to outline a more general and utilitarian protocol for analyzing bivariate data in studies of allometry. Authors of the original study on carp reported exponents of 0.83 and 0.78 for two-parameter power functions fitted to observations for resting metabolism and maximum metabolism, respectively. However, metabolic scaling in these fishes actually is described best by straight lines having positive intercepts with the Y axis. The allometric exponent is 1 for a straight line, so interpretations from the current analyses differ substantially from those reached in the original investigation. Contemporary theories for the evolution of optimal body size (e.g., the Metabolic Theory of Ecology) are based on patterns of metabolic allometry that have been estimated by the conventional analytical method. Thus, the current investigation raises questions about generally accepted patterns of metabolic allometry and theoretical models based upon them.

Examination of the Effects of Curvature in Geometrical Space on Accuracy of Scaling Derived Projections of Plant Biomass Units: Applications to the Assessment of Average Leaf Biomass in Eelgrass Shoots

Conservation of eelgrass relies on transplants and evaluation of success depends on nondestructive measurements of average leaf biomass in shoots among other variables. Allometric proxies offer a convenient way to assessments. Identifying surrogates via log transformation and linear regression can set biased results. Views conceive this approach to be meaningful, asserting that curvature in geometrical space explains bias. Inappropriateness of correction factor of retransformation bias could also explain inconsistencies. Accounting for nonlinearity of the log transformed response relied on a generalized allometric model. Scaling parameters depend continuously on the descriptor. Joining correction factor is conceived as the partial sum of series expansion of mean retransformed residuals leading to highest reproducibility strength. Fits of particular characterizations of the generalized curvature model conveyed outstanding reproducibility of average eelgrass leaf biomass in shoots. Although nonlinear heteroscedastic regression resulted also to be suitable, only log transformation approaches can unmask a size related differentiation in growth form of the leaf. Generally, whenever structure of regression error is undetermined, choosing a suitable form of retransformation correction factor becomes elusive. Compared to customary nonparametric characterizations of this correction factor, present form proved more efficient. We expect that offered generalized allometric model along with proposed correction factor form provides a suitable analytical arrangement for the general settings of allometric examination.

Are the surface areas of the gills and body involved with changing metabolic scaling with temperature?

The metabolic-level boundaries (MLB) hypothesis proposes that metabolic level mediates the relative influence of surface area (SA)- versus volume-related metabolic processes on the body-mass scaling of metabolic rate in organisms. The variation in the scaling of SA may affect how metabolic level affects the metabolic scaling exponent. This study aimed to determine the influence of increasing metabolic level at a higher temperature on the metabolic scaling exponent of the goldfish and determine the link between metabolic scaling exponents and SA parameters of both gills and body. The SA of gills and body and the resting metabolic rate (RMR) of the goldfish were assessed at 15°C and 25°C, and their mass scaling exponents were analyzed. The results showed a significantly higher RMR, with a lower scaling exponent, in the goldfish at a higher temperature. The SA of the gills and the total SA of the fish (TSA) were reduced with the increasing temperature. The scaling exponent of RMR (b_RMR) tended to be close to that of the TSA at a higher temperature. This suggests that temperature positively affects metabolic level but negatively affects b_RMR The findings support the MLB hypothesis. The lower scaling exponent at a higher temperature can be alternatively explained as follows: the higher viscosity of cold water impedes respiratory ventilation and oxygen uptake and reduces metabolic rate more in smaller individuals than in larger individuals at lower temperature, thus resulting in a negative association between temperature and b_RMR.

A new research paradigm for bivariate allometry: combining ANOVA and non-linear regression

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-parameter power equation to which a categorical variable has been added to identify membership by each subject in a specific group or treatment. The protocol differs from earlier ones in that different assumptions can be made about the form for random error in the full statistical model (i.e. normal and homoscedastic, normal and heteroscedastic, lognormal and heteroscedastic). The general equation and several modifications thereof were used to study allometric variation in field metabolic rates of marsupial and placental mammals. The allometric equations for both marsupials and placentals have an explicit, non-zero intercept, but the allometric exponent is higher in the equation for placentals than in that for marsupials. The approach followed here is extraordinarily versatile, and it has wider application in allometry than standard ANCOVA performed on logarithmic transformations.

Ejection fraction as a statistical index of left ventricular systolic function: the first full allometric scrutiny of its appropriateness and accuracy

Left ventricular ejection fraction (EF) is a ratio that is deemed to accurately normalize stroke volume (SV) to end-diastolic volume (EDV). Ratios are now well-recognized for not normalizing the numerator, in this case SV, consistently for the denominator, EDV. We aimed to provide the first allometric-based scrutiny of the conventional assumptions that underpin the EF ratio. We allometrically modelled untransformed SV and EDV measurements from 112 preclinical heart failure patients in the Multi-Ethnic Study of Atherosclerosis (MESA), and 864 chronic heart failure patients in the Treatment of Preserved Cardiac Function Heart Failure with an Aldosterone Antagonist (TOPCAT) study. An information-theoretic approach was adopted to assess the relative quality of twelve candidate models for normalizing SV to EDV. None of the conventional underlying assumptions for accurate ratio normalization, for example an allometric exponent ≈1, were upheld for EF. A two-parameter power function with normal, heteroscedastic error was the best model for scaling SV to EDV in both samples. The allometric exponent (95% confidence interval) was 0·776 (0·682 to 0·869) in MESA, and 0·860 (0·857 to 0·864) in TOPCAT. EF was inversely correlated with EDV in MESA (r = -0·67, 95% CI: -0·76 to -0·55) and TOPCAT (r = -0·41, 95% CI: -0·46 to -0·35). Consequently, for fundamental statistical reasons, EF was biased low for people with generally larger EDVs, and vice versa. For the first time, we have demonstrated that EF is an inaccurate statistic for scaling SV to EDV, leading to potential biased inferences for research and individual patients.

A comprehensive allometric analysis of 2nd digit length to 4th digit length in humans

It has been widely reported that men have a lower ratio of the 2nd and 4th human finger lengths (2D : 4D). Size-scaling ratios, however, have the seldom-appreciated potential for providing biased estimates. Using an information-theoretic approach, we compared 12 candidate models, with different assumptions and error structures, for scaling untransformed 2D to 4D lengths from 154 men and 262 women. In each hand, the two-parameter power function and the straight line with intercept models, both with normal, homoscedastic error, were superior to the other models and essentially equivalent to each other for normalizing 2D to 4D lengths. The conventional 2D : 4D ratio biased relative 2D length low for the generally bigger hands of men, and vice versa for women, thereby leading to an artefactual indication that mean relative 2D length is lower in men than women. Conversely, use of the more appropriate allometric or linear regression models revealed that mean relative 2D length was, in fact, greater in men than women. We conclude that 2D does not vary in direct proportion to 4D for both men and women, rendering the use of the simple 2D : 4D ratio inappropriate for size-scaling purposes and intergroup comparisons.