Fitting statistical models in bivariate allometry

Abstract

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 for parameters in allometric equations often are inaccurate, owing to undetected bias introduced by the traditional method for fitting lines to empirical data. The traditional method entails fitting a straight line to logarithmic transformations of the original data and then back-transforming the resulting equation to the arithmetic scale. Because of fundamental changes in distributions attending transformation of predictor and response variables, the traditional practice may cause influential outliers to go undetected, and it may result in an underparameterized model being fitted to the data. Also, substantial bias may be introduced by the insidious rotational distortion that accompanies regression analyses performed on logarithms. Consequently, the aforementioned patterns of allometric variation may be illusions, and the theoretical explanations may be wide of the mark. Problems attending the traditional procedure can be largely avoided in future research simply by performing preliminary analyses on arithmetic values and by validating fitted equations in the arithmetic domain. The goal of most allometric research is to characterize relationships between biological variables and body size, and this is done most effectively with data expressed in the units of measurement. Back-transforming from a straight line fitted to logarithms is not a generally reliable way to estimate an allometric equation in the original scale.

Random sampling of skewed distributions implies Taylor’s power law of fluctuation scaling

“…2 had statistically significantly nonzero linear coefficient b (with P < α; here α = 0.05), and if a least-squares quadratic regression between the independent variable log(mean) and dependent variable log(variance) did not yield a statistically significant quadratic coefficient c (P > α). The use of the doubly logarithmic scale in the testing of TL and other bivariate allometric relationships (e.g., scaling of metabolic rate with body mass) has been questioned (39,(42)(43)(44) and defended (45,46).…”

Female reproductive cycle and batch fecundity in the central-southern Adriatic population ofEngraulis encrasicolus(Osteichthyes: Engraulidae)

“…The lack of homoscedasticity of the data in the linear model represents an important drawback, as pointed out also in the original discussion of the DEPM (Daily Egg Production Method) method Lasker 1985). In addition, Packard et al (2011), who discuss the abuse of the allometric model, support the power curve model as an appropriate means to achieve data homoscedasticity. In the present case, the selection of the power curve model was furthermore feasible because of the virtual sameness of the MSE values derived from both the linear and the power curve models.…”

Allometry for Biomass Estimation in Jatropha Trees Planted as Boundary Hedge in Farmers’ Fields

“…The imprecision of the model observed in this study might have been introduced by the rotational distortion that accompanies regression analyses performed on logarithms. Because of the nonlinear relationship between values expressed in arithmetic and logarithmic scales, small arithmetic values for predictor and response have a much greater influence than large values on parameters (scaling coefficient, β 0 , and exponent, β 1 ) of the linear equations fitted on logarithmic transformations [36].…”

“…The most recent allometric research in trees and forests tends to focus on explaining the scaling of structural and functional properties of trees with measures of their body sizes (e.g., diameters, heights) [36]. West et al [29] suggested that the scaling exponent (β 1 ) should scale against tree diameter with a universal exponent β 1 ≈ 2.67 on trees whose height increments are at their maximum.…”

“…Converting arithmetic data to logarithms, however, is a nonlinear transformation that fundamentally alters the relationship between predictor and response variables [35]. The change in the relationship is such that influential outliers in the arithmetic domain may be drawn toward the centre of the distribution, and can thereby go undetected, potentially introducing bias into the analysis [36]. In order to avoid this in our study, the detection of influential outliers was first done on data expressed in arithmetic scale as recommended by Packard et al [36].…”