Relative Growth by the Elongated Jaws of Gars: A Perspective on Polyphasic Loglinear Allometry

An extended multiplicative error model of allometry: Incorporating systematic components, non-normal distributions, and piecewise heteroscedasticity

Allometry refers to the relationship between the size of a trait and that of the whole body of an organism. Pioneering observations by Otto Snell and further elucidation by D'Arcy Thompson set the stage for its integration into Huxley's explanation of constant relative growth that epitomizes through the formula of simple allometry. The traditional method to identify such a model conforms to a regression protocol fitted in the direct scales of data. It involves Huxley's formula-systematic part and a lognormally distributed multiplicative error term. In many instances of allometric examination, the predictive strength of this paradigm is unsuitable. Established approaches to improve fit enhance the complexity of the systematic relationship while keeping the go-along normality-borne error. These extensions followed Huxley's idea that considering a biphasic allometric pattern could be necessary. However, for present data composing 10 410 pairs of measurements of individual eelgrass leaf dry weight and area, a fit relying on a biphasic systematic term and multiplicative lognormal errors barely improved correspondence measure values while maintaining a heavy tails problem. Moreover, the biphasic form and multiplicative-lognormal-mixture errors did not provide complete fit dependability either. However, updating the outline of such an error term to allow heteroscedasticity to occur in a piecewise-like mode finally produced overall fit consistency. Our results demonstrate that when attempting to achieve fit quality improvement in a Huxley's model-based multiplicative error scheme, allowing for a complex allometry form for the systematic part, a non-normal distribution-driven error term and a composite of uneven patterns to describe the heteroscedastic outline could be essential.

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.

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

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.

Back to the basics: allometric growth by the horns of bovid mammals

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 equations being required to capture the pattern of variation over the full range in body size. However, the investigation in question used conventional analytical methods based on logarithmic transformations, which alter bivariate distributions and commonly introduce problems with analysis and interpretation. My new analyses of data for both species revealed that untransformed observations for both males and females are monophasic and that they are described quite well by three-parameter power equations with negative intercepts. Equations for males follow a steep upward trajectory whereas those for females follow much shallower paths. The negative intercepts indicate that males and females of both species must attain a minimum body size before horns begin to develop. Conclusions from the earlier investigation were based on inaccurate perceptions of pattern in the data. Future studies should be based on graphical and analytical analysis of observations expressed on the original arithmetic scale.

Allometric growth in mass by the brain of mammals

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 discrete mathematical distributions, but untransformed observations nonetheless conform to a single distribution that is well described by a single, nonlinear equation. The concept of biphasic, allometric growth by the brain consequently is an artifact of transformation. The notion of Rapid and Slow phases in relative growth by the brain also is an artifact, because the notion is based explicitly on the concept of biphasic growth allometry. Relative growth by the brain in sheep, pigs, cows, and horses follows the path of a power curve with an exponent less than 1, so relative growth declines progressively as animals grow to their maximum size, at which point growth effectively ends for both brain and body. Relative growth by the brain in rats and cats follows the path of an exponential curve and consequently is more like relative growth by the brain of odontocoete cetaceans and primates, with the brain growing rapidly relative to the body early in ontogeny and attaining maximum (cats) or near-maximum (rats) mass well before the body reaches its maximum. An exponential pattern of relative growth by the brain appears to have evolved independently in rodents, carnivores, odontocoetes, and primates.

Assessment of a Takagi-Sugeno-Kang fuzzy model assembly for examination of polyphasic loglinear allometry

Background

The traditional allometric analysis relies on log- transformation to contemplate linear regression in geometrical space then retransforming to get Huxley’s model of simple allometry. Views assert this induces bias endorsing multi-parameter complex allometry forms and nonlinear regression in arithmetical scales. Defenders of traditional approach deem it necessary since generally organismal growth is essentially multiplicative. Then keeping allometry as originally envisioned by Huxley requires a paradigm of polyphasic loglinear allometry. A Takagi-Sugeno-Kang fuzzy model assembles a mixture of weighted sub models. This allows direct identification of break points for transition between phases. Then, this paradigm is seamlessly appropriate for efficient allometric examination of polyphasic loglinear allometry patterns. Here, we explore its suitability.

Methods

Present fuzzy model embraces firing strength weights from Gaussian membership functions and linear consequents. Weights are identified by subtractive clustering and consequents through recursive least squares or maximum likelihood. Intersection of firing strength factors set criterion to estimate breakpoints. A multi-parameter complex allometry model follows by adapting firing strengths by composite membership functions and linear consequents in arithmetical space.

Results

Takagi-Sugeno-Kang surrogates adapted complexity depending on analyzed data set. Retransformation results conveyed reproducibility strength of similar proxies identified in arithmetical space. Breakpoints were straightforwardly identified. Retransformed form implies complex allometry as a generalization of Huxley’s power model involving covariate depending parameters. Huxley reported a breakpoint in the log–log plot of chela mass vs. body mass of fiddler crabs (Uca pugnax), attributed to a sudden change in relative growth of the chela approximately when crabs reach sexual maturity. G.C. Packard implied this breakpoint as putative. However, according to present fuzzy methods existence of a break point in Huxley’s data could be validated.

Conclusions

Offered scheme bears reliable analysis of zero intercept allometries based on geometrical space protocols. Endorsed affine structure accommodates either polyphasic or simple allometry if whatever turns required. Interpretation of break points characterizing heterogeneity is intuitive. Analysis can be achieved in an interactive way. This could not have been obtained by relying on customary approaches. Besides, identification of break points in arithmetical scale is straightforward. Present Takagi-Sugeno-Kang arrangement offers a way to overcome the controversy between a school considering a log-transformation necessary and their critics claiming that consistent results can be only obtained through complex allometry models fitted by direct nonlinear regression in the original scales.

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.

The fallacy of biphasic growth allometry for the vertebrate brain

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.

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.