ASSESSING SCALING RELATIONSHIPS: USES, ABUSES, AND ALTERNATIVES

Premise of research. Workers have relied on fitting a straight line to logarithmically transformed data to determine biological scaling relationships without testing the assumption that error is normal and additive on the logarithmic scale. Methodology. We review the history of this practice, the pros and cons of log transformation, and the use of model Type I and II regression protocols. Using standard statistical protocols and the Akaike Information Criterion, we then evaluate linear and nonlinear models applied to a large interspecific data set and a smaller intraspecific data set to reexamine the hypothesis called diminishing returns, which states that the surface areas of mature leaves may fail to increase one-to-one (isometrically) as lamina dry mass increases. Pivotal results. The error structures of both data sets were multiplicative and lognormal and thus complied with a linear model, which obtained log-log linear lines with slopes less than 1; i.e., the data were consistent with the hypothesis of diminishing returns. Conclusions. History shows that log transformation has always been a controversial practice. However, the extent to which linear or nonlinear models comply with a particular data set is generally transparent using standard statistical protocols (e. g., analysis of residuals). Previous scaling analyses using log-transformed data therefore are likely generally valid. Nevertheless, the error structure in every data set should be assessed to determine whether linear or nonlinear regression models are appropriate. Reliable algorithms are available for this purpose.