Principal interactions analysis for repeated measures data: application to gene-gene and gene-environment interactions.

Many existing cohorts with longitudinal data on environmental exposures, occupational history, lifestyle/ behavioral characteristics, and health outcomes have collected genetic data in recent years. In this paper, we consider the problem of modeling gene-gene and gene-environment interactions with repeated measures data on a quantitative trait. We review possibilities of using classical models proposed by Tukey (1949) and Mandel (1961) using the cell means of a two-way classification array for such data. Although these models are effective for detecting interactions in the presence of main effects, they fail miserably if the interaction structure is misspecified. We explore a more robust class of interaction models that are based on a singular value decomposition of the cell-means residual matrix after fitting the additive main effect terms. This class of additive main effects and multiplicative interaction models (Gollob, 1968) provide useful summaries for subject-specific and time-varying effects as represented in terms of their contribution to the leading eigenvalues of the interaction matrix. It also makes the interaction structure more amenable to geometric representation. We call this analysis 'principal interactions analysis'. While the paper primarily focuses on a cell-mean-based analysis of repeated measures outcome, we also introduce resampling-based methods that appropriately recognize the unbalanced and longitudinal nature of the data instead of reducing the response to cell means. We illustrate the proposed methods by using data from the Normative Aging Study, a longitudinal cohort study of Boston area veterans since 1963. We carry out simulation studies under an array of classical interaction models and common epistasis models to illustrate the properties of the principal interactions analysis procedure in comparison with the classical alternatives.