The Box–Cox Transformation: Review and Extensions

Abstract

The Box-Cox power transformation family for non-negative responses in linear models has a long and interesting history in both statistical practice and theory, which we summarize. The relationship between generalized linear models and log transformed data is illustrated. Extensions investigated include the transform both sides model and the Yeo-Johnson transformation for observations that can be positive or negative. The paper also describes an extended Yeo-Johnson transformation that allows positive and negative responses to have different power transformations. Analyses of data show this to be necessary. Robustness enters in the fan plot for which the forward search provides an ordering of the data. Plausible transformations are checked with an extended fan plot. These procedures are used to compare parametric power transformations with nonparametric transformations produced by smoothing.