Logistic Box-Cox Regression to Assess the Shape and Median Effect under Uncertainty about Model Specification

The shape of the relationship between a continuous exposure variable and a binary disease variable is often central to epidemiologic investigations. This paper investigates a number of issues surrounding inference and the shape of the relationship. Presuming that the relationship can be expressed in terms of regression coefficients and a shape parameter, we investigate how well the shape can be inferred in settings which might typify epidemiologic investigations and risk assessment. We also consider a suitable definition of the median effect of exposure, and investigate how precisely this can be inferred. This is done both in the case of using a model acknowledging uncertainty about the shape parameter and in the case of ignoring this uncertainty and using a two-step method, where in step one we transform the predictor and in step two we fit a simple linear model with transformed predictor. All these investigations require a family of exposure-disease relationships indexed by a shape parameter. For this purpose, we employ a family based on the Box-Cox transformation.