Misconceptions about logarithmic transformation and the traditional allometric method

Highlights

• Logarithmic transformation is widely believed to be necessary in allometry.

• Logarithmic transformation has limited application and is not required.

• Nonlinear regression is superior to the traditional allometric method.

Abstract

Logarithmic transformation is often assumed to be necessary in allometry to accommodate the kind of variation that accompanies multiplicative growth by plants and animals; and the traditional approach to allometric analysis is commonly believed to have important application even when the bivariate distribution of interest is curvilinear on the logarithmic scale. Here I examine four arguments that have been tendered in support of these perceptions. All the arguments are based on misunderstandings about the traditional method for allometric analysis and/or on a lack of familiarity with newer methods of nonlinear regression. Traditional allometry actually has limited utility because it can be used only to fit a two-parameter power equation that assumes lognormal, heteroscedastic error on the original scale. In contrast, nonlinear regression can fit two- and three-parameter power equations with differing assumptions about structure for error directly to untransformed data. Nonlinear regression should be preferred to the traditional method in future allometric analyses.

Introduction

The early part of the 20th century was marked by widespread interest among biologists in the use of simple power functions of the form

y = a _* x^b

to describe pattern in bivariate observations that follow a curvilinear path on the arithmetic (=linear) scale. The predictor variable (x) in the two-parameter equation typically was a measure of body size (e.g., body length or body mass), and the response variable (y) was some measurement taken on the structure, organ, or process of special concern. Some investigators at the time apparently fitted the equation directly to scatterplots of untransformed data by a series of trial-and-error approximations (e.g., Nomura, 1926, Kleiber, 1932), or by fitting a curve by eye and then reading from the graph (e.g., Hecht, 1913, Hecht, 1916, Crozier and Hecht, 1914, Kleiber, 1932). Other workers, however, estimated the slope and intercept of a straight line drawn by hand on a graph displaying logarithmic transformations (or on a graph with logarithmic coordinates) and then took antilogs for the coefficients to obtain parameters in the power equation (e.g., Pearsall, 1927, Huxley, 1927a, Huxley, 1927b, Huxley, 1932, Kunkel and Robertson, 1928). And yet a fourth group of investigators fitted straight lines to logarithmic transformations by ordinary least squares regression and then back-transformed the resulting equation to the arithmetic scale (e.g., Clark, 1928, Galtsoff, 1931, Brody and Proctor, 1932, Green and Green, 1932). This last approach to fitting the power function continues, for all intents and purposes, to be in general use today (Warton et al., 2006, White et al., 2012, White and Kearney, 2014) and has come to be known as the traditional allometric method (e.g., Packard, 2014).

The traditional allometric method has had its critics over the years (e.g., Thompson, 1943, Smith, 1980, Smith, 1984, Lovett and Felder, 1989, Bales, 1996), and it recently has come under renewed scrutiny (e.g., Lagergren et al., 2007, Sartori and Ball, 2009, Packard, 2014, Packard, 2015, Packard, 2016, Packard, 2017a, Packard, 2017b). Supporters of the protocol are understandably concerned that a large body of published research might be undermined if criticisms of the method were taken seriously, and they consequently have mounted a spirited defense of their research paradigm (e.g., Klingenberg, 1998, Nevill et al., 2005, Kerkhoff and Enquist, 2009; White et al., 2012; Ballantyne, 2013; Glazier, 2013; Lai et al., 2013; Mascaro et al., 2014; Niklas and Hammond, 2014; Lema&tre et al., 2015). However, the defense is based in many instances on ill-defined arguments and/or misunderstanding of various statistical methods. Here I examine four of the most common misconceptions.