Data Transformations and the Linear Model

Abstract

The familiar application of the normal linear model involves a response variable that is assumed normally distributed with constant variance and with location linear in a vector parameter. In other applications a response variable may occur in a form that suppresses an underlying normal linear structure. Sometimes in these applications the context may suggest a logarithmic or square root transformation which reveals the normal linear form. Box and Cox (1964) consider a parametric class of transformations on the response variable and derive a method for estimating the class parameter based on Bayesian and likelihood techniques. In this paper a more comprehensive statistical model is proposed; it is a revision of the structural model (Fraser 1966). It gives stronger inference statements in the context for the linear model. It can handle normal or nonnormal error. And in the context for the transformed linear model it gives directly a method for estimating the class parameter. This estimation avoids an approximation in the Bayesian prior and it includes additional sensitivity to the data.