There is a long-standing interest in the evaluation of gene-gene and gene-environment interactions in cancer epidemiology and genetics studies. Statisticians generally define interaction as a departure from additivity in a linear model on a certain scale in which the outcome is measured. Certain types of interactions may be eliminated via an invertible transformation of the outcome so that the model is additive on the transformed scale. When the outcome is binary, a transformation corresponds to a suitable link function.
This seminar will examine the properties of three families of link functions for fitting additive models in the presence of a removable interaction. It is shown that: (1) when an interaction is removable, the logistic link function can be written as a systematic departure from the underlying additive model; and (2) in this form, the required transformation is intimately related to a specific family of link functions among those considered. These results and their practical implications are illustrated using case-control data from multiple cancer epidemiology studies.