## Introduction

Defined as the influence of body size on form and function (LaBarbera 1989), allometry has a rich and venerable history and continues to draw much attention as a consequence of its widespread use in behaviour, physiology, ecology and evolution (e.g. Peters 1983; Calder 1984; Schmidt-Nielsen 1984; Brown *et al.* 2004). The main reason for adopting an allometric approach is that log transformation linearizes a wide array of nonlinear biological relationships, thereby enabling simpler linear statistical analyses to be used. This, in turn, enables the calculation of confidence limits and, for example, statistical testing for homogeneity of slopes across groups. Allometric relationships are used not only to defend or refute particular theories by comparison of slope estimates (Farrell-Gray & Gotelli 2005; Glazier 2005; Reich *et al.* 2006; Chown *et al.* 2007; Duncan, Forsyth, & Hone 2007; White, Cassey, & Blackburn 2007), but also to predict, with a degree of accuracy far greater than random, a variable of interest based solely on body size (Wainwright & Richard 1995; Lindstedt & Schaeffer 2002). The predictive capacity is also one of the greatest strengths of allometry and has been used, for instance, in the pursuit of unifying theories of animal locomotion (Bejan & Marden 2006) and metabolic ecology (Allen, Brown, & Gillooly 2002; Brown *et al.* 2004; Gillooly *et al.* 2005; Brown & Sibly 2006).

A widely accepted problem when working with allometric equations is one associated with Jensen’s inequality, i.e. the discrepancy between the arithmetic and the geometric mean (e.g. Sprugel 1983; Blackburn & Gaston 1998). Allometric equations derived from linear regression using log-transformed data estimate the geometric and not the arithmetic mean, and predictions using the geometric mean on the original antilog scale can have a substantial effect on the outcomes of biological studies (Smith 1993; Hayes & Shonkwiler 2006). As a result, a series of correction estimators has been designed to mitigate this problem (Finney 1941; Heien 1968; Bradu & Mundlak 1970; Zellner 1971; Beauchamp & Olson 1973; Duan 1983). By intentionally transforming the *Y*-prediction from the geometric to the arithmetic mean, these estimators can thereby largely reduce prediction bias (i.e. a systematic difference in the estimator’s expectation and the true value of the parameter being tested), measured by the mean squared residual (MSR).

Although the relative performance of the correction estimators has been tested (e.g. Smith 1993; Hayes & Shonkwiler 2006), their performance relative to different line-fitting methods (LFMs), as well as the potential effects of the error distribution, sample size and variation in the data on their performance has not been explicitly assessed. Using systematic simulations, we demonstrate the effect of residual shape, sample size and the magnitude of residuals (or scatter, defined below) on the allometric regression. A general relationship among the LFMs and correction estimators, as well as a guideline for the selection of the LFM for allometric regression, are provided. Of particular importance, by using parameter landscapes we are able to demonstrate the relative performance of the LFM and correction estimators under different measures of bias [specifically, the MSR and the mean of the squared discrepancy of the frequency distribution of the *Y*-variable between predictions and observations, MSD]. Using this approach, we identify a serious bias when using correction estimators and the nonlinear LFM for prediction, and suggest a shift of focus in allometric regression from designing new LFMs to defining bias with appropriate measures.