The multivariate regression model is a useful tool to explore complex associations between multiple response variables (e.g. gene expressions) and multiple predictors (e.g. SNPs). When the multiple responses are correlated, ignoring such dependency will impair statistical power in the data analysis. Motivated by an integrative genomic data, we propose a new methodology-sparse multivariate factor analysis regression model (smFARM), in which the covariance of the response variables is modeled by a factor analysis model with latent factors. This proposed method not only allows us to address the challenge that the number of genetic predictors is larger than the sample size, but also to adjust for unobserved genetic and/or non-genetic factors that potentially conceal the underlying real response-predictor associations. The proposed smFARM is implemented efficiently by utilizing the strength of the EM algorithm and the group-wise coordinate descend algorithm. In addition, the identified latent factors are explained by the means of gene enrichment analysis. The proposed methodology is evaluated and compared to the existing methods through extensive simulation studies. We apply smFARM in an integrative genomics analysis of a breast cancer dataset on the relationship between DNA copy numbers and gene expression arrays to derive genetic regulatory patterns relevant to breast cancer.