Combining the Box-Cox power and generalised log transformations to accommodate nonpositive responses in linear and mixed-effects linear models

Transformation of a response variable can greatly expand the class of problems for which the linear regression model or linear mixed-model is appropriate. Beginning with the fundamental work of Box and Cox, maximum-likelihood-like estimation has been applied to select a transformation from among a family of transformations, with the possible goals of achieving approximate normality, removing nonlinearity in a mean function, or stabilizing variance. The Box-Cox power family (BC) of transformations is by far the most common with the Box-Cox methodology, and it requires a strictly positive response. In this article we introduce a new family of transformations that we call the Box-Cox power with nonpositives (BCN) family that allows inclusion of a few nonpositive values. The BCN family is a modification of the basic power family that is inspired by the generalised log, or glog transformation, proposed for use with the more limited goals of stabilizing variance or achieving approximate normality. The glog transformation is itself a special case of the Johnson SU transformation, and we show that the BCN family derived from it is in turn a simple modification of the BC family. Computer code for implementing this family is included in the car package in R (Fox and Weisberg, 2011). The methodology is illustrated using a problem in clinical chemistry