Testing departure from additivity in Tukey's model using shrinkage: application to a longitudinal setting.

While there has been extensive research developing gene-environment interaction (GEI) methods in case-control studies, little attention has been given to sparse and efficient modeling of GEI in longitudinal studies. In a two-way table for GEI with rows and columns as categorical variables, a conventional saturated interaction model involves estimation of a specific parameter for each cell, with constraints ensuring identifiability. The estimates are unbiased but are potentially inefficient because the number of parameters to be estimated can grow quickly with increasing categories of row/column factors. On the other hand, Tukey's one-degree-of-freedom model for non-additivity treats the interaction term as a scaled product of row and column main effects. Because of the parsimonious form of interaction, the interaction estimate leads to enhanced efficiency, and the corresponding test could lead to increased power. Unfortunately, Tukey's model gives biased estimates and low power if the model is misspecified. When screening multiple GEIs where each genetic and environmental marker may exhibit a distinct interaction pattern, a robust estimator for interaction is important for GEI detection. We propose a shrinkage estimator for interaction effects that combines estimates from both Tukey's and saturated interaction models and use the corresponding Wald test for testing interaction in a longitudinal setting. The proposed estimator is robust to misspecification of interaction structure. We illustrate the proposed methods using two longitudinal studies-the Normative Aging Study and the Multi-ethnic Study of Atherosclerosis.