A Likelihood Based Framework for Data Integration with Application to eQTL Mapping

We develop a new way of thinking about and integrating gene expression data (continuous) and genomic information data (binary) by jointly compressing the two data sets and embedding their signals in low dimensional feature spaces with an information sharing mechanism, which connects the continuous data to the binary data, under the penalized log-likelihood framework. In particular, the continuous data are modeled by a Gaussian likelihood and the binary data are modeled by a Bernoulli likelihood which is formed by transforming the feature space of the genomic information with a logit link. The smoothly clipped absolute deviation (SCAD) penalty, is added on the basis vectors of the low dimensional feature spaces for both data sets, which is based on the assumption that only a small set of genetic variants are associated with a small fraction of gene expression and the fact that those basis vectors can be interpreted as weights assigned on the genetic variants and gene expression similar to the way the loading vectors of principal component analysis (PCA) or canonical correlation analysis (CCA) are interpreted. Algorithmically, a Majorization-Minimization (MM) algorithm with local linear approximation (LLA) to SCAD penalty is developed to effectively and efficiently solve the optimization problem involved, which produces closed-form updating rules. The effectiveness of our method is demonstrated by simulations in various setups with comparisons to some popular competing methods and an application to eQTL mapping with real data.