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.

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.

A Revision of the Traditional Analysis Method of Allometry to Allow Extension of the Normality-Borne Complexity of Error Structure: Examining the Adequacy of a Normal-Mixture Distribution-Driven Error Term

Huxley’s model of simple allometry provides a parsimonious scheme for examining scaling relationships in scientific research, resource management, and species conservation endeavors. Factors including biological error, analysis method, sample size, and overall data quality can undermine the reliability of a fit of Huxley’s model. Customary amendments enhance the complexity of the power function-conveyed systematic term while keeping the usual normality-borne error structure. The resulting protocols bear multiple-parameter complex allometry forms that could pose interpretative shortcomings and parameter estimation difficulties, and even being empirically pertinent, they could potentially bear overfitting. A subsequent heavy-tailed Q-Q normal spread often remains undetected since the adequacy of a normally distributed error term remains unexplored. Previously, we promoted the advantages of keeping Huxley’s model-driven systematic part while switching to a logistically distributed error term to improve fit quality. Here, we analyzed eelgrass leaf biomass and area data exhibiting a marked size-related heterogeneity, perhaps explaining a lack of systematization at data gathering. Overdispersion precluded adequacy of the logistically adapted protocol, thereby suggesting processing data through a median absolute deviation scheme aimed to remove unduly replicates. Nevertheless, achieving regularity to Huxley’s power function-like trend required the removal of many replicates, thereby questioning the integrity of a data cleaning approach. But, we managed to adapt the complexity of the error term to reliably identify Huxley’s model-like systematic part masked by variability in data. Achieving this relied on an error term conforming to a normal mixture distribution which successfully managed overdispersion in data. Compared to normal-complex allometry and data cleaning composites present arrangement delivered a coherent Q-Q normal mixture spread and a remarkable reproducibility strength of derived proxies. By keeping the analysis within Huxley’s original theory, the present approach enables substantiating nondestructive allometric proxies aimed at eelgrass conservation. The viewpoint endorsed here could also make data cleaning unnecessary.

Is relative growth by the mammalian heart biphasic or monophasic?

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.

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.

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.

Mass, phylogeny, and temperature are sufficient to explain differences in metabolic scaling across mammalian orders?

Whether basal metabolic rate-body mass scaling relationships have a single exponent is highly discussed, and also the correct statistical model to establish relationships. Here, we aimed (1) to identify statistically best scaling models for 17 mammalian orders, Marsupialia, Eutheria and all mammals, and (2) thereby to prove whether correcting for differences in species' body temperature and their shared evolutionary history improves models and their biological interpretability. We used the large dataset from Sieg et al. (The American Naturalist174, 2009, 720) providing species' body mass (BM), basal metabolic rate (BMR) and body temperature (T). We applied different statistical approaches to identify the best scaling model for each taxon: ordinary least squares regression analysis (OLS) and phylogenetically informed analysis (PGLS), both without and with controlling for T. Under each approach, we tested linear equations (log-log-transformed data) estimating scaling exponents and normalization constants, and such with a variable normalization constant and a fixed exponent of either ⅔ or ¾, and also a curvature. Only under temperature correction, an additional variable coefficient modeled the influence of T on BMR. Except for Pholidata and Carnivora, in all taxa studied linear models were clearly supported over a curvature by AICc. They indicated no single exponent at the level of orders or at higher taxonomic levels. The majority of all best models corrected for phylogeny, whereas only half of them included T. When correcting for T, the mathematically expected correlation between the exponent (b) and the normalization constant (a) in the standard scaling model y = a x^b was removed, but the normalization constant and temperature coefficient still correlated strongly. In six taxa, T and BM correlated positively or negatively. All this hampers a disentangling of the effect of BM, T and other factors on BMR, and an interpretation of linear BMR-BM scaling relationships in the mammalian taxa studied.