Why you cannot transform your way out of trouble for small counts.

Abstract

While data transformation is a common strategy to satisfy linear modeling assumptions, a theoretical result is used to show that transformation cannot reasonably be expected to stabilize variances for small counts. Under broad assumptions, as counts get smaller, it is shown that the variance becomes proportional to the mean under monotonic transformations g(·) that satisfy g(0)=0, excepting a few pathological cases. A suggested rule-of-thumb is that if many predicted counts are less than one then data transformation cannot reasonably be expected to stabilize variances, even for a well-chosen transformation. This result has clear implications for the analysis of counts as often implemented in the applied sciences, but particularly for multivariate analysis in ecology. Multivariate discrete data are often collected in ecology, typically with a large proportion of zeros, and it is currently widespread to use methods of analysis that do not account for differences in variance across observations nor across responses. Simulations demonstrate that failure to account for the mean-variance relationship can have particularly severe consequences in this context, and also in the univariate context if the sampling design is unbalanced.