Optimal power transformations for analysis of sperm concentration and other semen variables

The nongaussian (or nonnormal) distribution of sperm concentration, and variables deriving from it, is a common practical problem in the statistical evaluation of semen data. Yet it has been little studied, and its importance to data analysis, as well as to practical remedies, is not widely appreciated. Inappropriate use of the raw scale of measurement produces inflated estimates of mean and variance, leading to false-negative (underpowered) statistical comparisons and excessive sample size estimates. This study employs the Box-Cox family of power transforms to illustrate by a simple graphical method how to identify optimal power transforms for semen data variables. Using robust statistical methods, it is shown that the nongaussian distribution is due to right skewing rather than multimodality or influential outliers. The optimal power transform, typically in the region of 0.15 to 0.35 (most easily implemented as a cube-root transformation), usually performs better than the logarithmic transformation in normalizing the data. In addition, the power transformation has an important practical advantage over the logarithmic transformation in the appropriate handling of zeros (azoospermia), a regular and important features of such data sets in practice.