On Use of Logarithms to Accommodate Scale

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Abstract
The logarithmic transformation can be utilized to equilibrate variances of traits of different size when these variances scale proportionally to the square of the trait means. Otherwise variances will not be equilibrated by log transformation. A simple model of ontogenetic growth is utilized to show that trait variances increase with the square of the mean during ontogeny when individual growth increments are perfectly correlated. Alternatively, if these individual growth increments are uncorrelated, trait variance accumulates only in direct proportion to the mean itself. For most actual ontogenies, the incremental growths would not be perfectly correlated, so log-transformed variances would be expected to decrease during ontogeny. The model was extended to address the comparison of variances between two traits differing in size. When two traits are highly correlated, the ratio of variances of the traits will be proportional to the square of the mean ratio. When two traits are uncorrelated, the ratio of variances scales directly to the ratio of means. Biological traits are usually characterized by varying degrees of intercorrelation (i.e., they exhibit multivariate structure). Since the appropriate transformation to accommodate scale depends upon the intercorrelation among a set of traits, it is unlikely that a single transformation would equilibrate variances (and covariances) among all traits. A similar caution applies to genetic variances and covariances in quantitative genetics. However, narrow-sense heritabilities and additive genetic correlations are both approximately invariant under a change of scale and can be compared across traits and/or populations with less concern about scale effect.