HAYES, JP; SHONKWILER, JS; University of Nevada, Reno; University of Nevada, Reno: Allometry, antilog transformations, and the perils of prediction on the original scale

Allometric equations in the form of power functions, Y = aM ^b, have been used in thousands of papers. Important uses of allometric equations include predicting Y for species or individuals which have not yet been measured and determining whether a measured Y is higher or lower than might be predicted for a species or individual of that mass. Allometric equations are fitted by detransforming the regression on log transformed data back to the original scale. It is well established that this detransformation leads to bias, such that predictions from allometric equations typically underestimate the true value of Y. What has not been well established is whether the bias is large enough to be of concern to evolutionary physiologists. We analyzed 20 interspecific and 10 intraspecific data sets, examined how biased the allometric predictions were, and compared the performance of four other estimators: a consistent estimator, a minimum variance unbiased estimator (MVUE), and an approximate MVUE that all depend on the raw scale data being lognormally distributed, and a non-parametric estimator. Detransformation bias was minor for the intraspecific data, but substantial for many of the interspecific data sets. For half the interspecific data sets, the mean predicted from the allometric equation differed from the actual mean by 10% or more. For mammals, the predicted mean BMR was 32% lower and the predicted mean FMR 38% lower than the actual means of the data, so uncorrected predictions from interspecific allometric equations may be seriously in error. Comparison of the bias of the alternative estimators suggests that differences among them typically were small.