Evaluation of removable statistical interaction for binary traits

This paper is concerned with evaluating whether an interaction between two sets of risk factors for a binary trait is removable and, when it is removable, fitting a parsimonious additive model using a suitable link function to estimate the disease odds (on the natural logarithm scale). Statisticians define the term 'interaction' as a departure from additivity in a linear model on a specific scale on which the data are measured. Certain interactions may be eliminated via a transformation of the outcome such that the relationship between the risk factors and the outcome is additive on the transformed scale. Such interactions are known as removable interactions. We develop a novel test statistic for detecting the presence of a removable interaction in case-control studies. We consider the Guerrero and Johnson family of transformations and show that this family constitutes an appropriate link function for fitting an additive model when an interaction is removable. We use simulation studies to examine the type I error and power of the proposed test and to show that, when an interaction is removable, an additive model based on the Guerrero and Johnson link function leads to more precise estimates of the disease odds parameters and a better fit. We illustrate the proposed test and use of the transformation by using case-control data from three published studies. Finally, we indicate how one can check that, after transformation, no further interaction is significant.