Robust sparse Bayesian regression for longitudinal gene-environment interactions

In longitudinal studies, repeated measure analysis of variance (ANOVA) is a classical analysis where selecting important main and interaction effects for accurate estimation and prediction is among one of its central goals. With high-dimensional genetic factors, ANOVA leads to a sparse longitudinal gene-environment ( $G\times E$ ) interaction problem that has not been thoroughly investigated so far, partially due to the challenges to incorporate robustness against skewed phenotypic measurements, intra-cluster correlations among longitudinal observations, and structured sparsity arising from the ANOVA design. We have developed a novel robust sparse Bayesian mixed model to tackle these challenges. Outliers and inter-relatedness among repeated measurements can be efficiently accommodated. Meanwhile, the proposed model conducts robust Bayesian variable selection accounting for main and interaction effects via structured spike-and-slab priors. We have developed Gibbs samplers and MCMC algorithms for fast computation and posterior inference. The advantage of the proposed method over benchmarks in variable selection and estimation has been established through extensive simulations. In the case study, we have analysed longitudinal lipidomics data with repeatedly measured body weight of CD-1 mice from a cancer prevention study. The proposed model has identified main and interactions with important implications and led to better prediction performance over alternative methods.