Grid-based Gaussian process models for longitudinal genetic data

Although various statistical methods have been developed to map time-dependent genetic factors, most identified genetic variants can explain only a small portion of the estimated genetic variation in longitudinal traits. Gene-gene and gene-time / environment interactions are known to be important putative sources of the missing heritability. However, mapping epistatic gene-gene interactions is extremely di ffi cult due to the very large parameter spaces for models containing such interactions. In this paper, we develop a Gaussian process (GP) based nonparametric Bayesian variable selection method for longitudinal data. It maps multiple genetic markers without restricting to pairwise interactions. Rather than modeling each main and interaction term explicitly, the GP model measures the importance of each marker, regardless of whether it is mostly due to a main e ff ect or some interaction e ff ect(s), via an unspecified function. To improve the flexibility of the GP model, we propose a novel grid-based method for the within-subject dependence structure. The proposed method can accurately approximate complex covariance structures. The dimension of the covariance matrix depends only on the number of fixed grid points although each subject may have di ff erent numbers of measurements at di ff erent time points. The deviance information criterion (DIC) and the Bayesian predictive information criterion (BPIC) are proposed for selecting an optimal number of grid points. To e ffi ciently draw posterior samples, we combine a hybrid Monte Carlo method with a partially collapsed Gibbs (PCG) sampler. We apply the proposed GP model to a mouse dataset on age-related body weight.