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On the use of log‐transformation versus nonlinear regression for analyzing biological power laws

Packard, Gary C.
Biological journal of the Linnean Society 2014 v.113 no.4 pp. 1167-1178
allometry, data collection, equations, heteroskedasticity, least squares, statistical models
Xiao and colleagues re‐examined 471 datasets from the literature in a major study comparing two common procedures for fitting the allometric equation y = axᵇto bivariate data (Xiao et al., 2011). One of the procedures was the traditional allometric method, whereby the model for a straight line fitted to logarithmic transformations of the original data is back‐transformed to form a two‐parameter power function with multiplicative, lognormal, heteroscedastic error on the arithmetic scale. The other procedure was standard nonlinear regression, whereby a two‐parameter power function with additive, normal, homoscedastic error is fitted directly to untransformed data by nonlinear least squares. Xiao and colleagues articulated a simple (but explicit) protocol for fitting and comparing the alternative models, and then used the protocol to examine each of the datasets in their compilation. The traditional method was said to provide a better fit in 69% of the cases and an equivalent fit in another 15%, so the investigation appeared to validate findings from a large majority of prior studies on allometric variation. However, focus for the investigation by Xiao and colleagues was overly narrow, and statistical models apparently were not validated graphically in the scale of measurement. The present study re‐examined a subset of the cases using a larger pool of candidate models and graphical validation, and discovered complexities that were overlooked in their investigation. Some datasets that appeared to be described better by the traditional method actually were unsuited for use in an allometric analysis, whereas other datasets were not described adequately by a two‐parameter power function, regardless of how the model was fitted. Thus, conclusions reached by Xiao and colleagues are not well supported and their paradigm for fitting allometric equations is unreliable. Future investigations of allometric variation should adopt a more holistic approach and incorporate graphical validation on the original arithmetic scale.