As the field of human genetics moves beyond simple GWAS, we are observing two clear trends in the types of datasets being collected, and the types of analysis used, to help elucidate disease etiology. Firstly, cohorts are collecting multiple disease and molecular phenotypes on the same samples. Secondly, heritability analysis in related samples is becoming more popular. In such datasets, it maybe of interest to infer the underlying network of dependence between phenotypes that has a genetic basis. Such networks summarize graphically the joint heritability of the measured phenotypes. Inference of such a network is complicated, as can we expect significant correlations between phenotypes, due to both the underlying network and observational noise. We model observations as a sum of two matrix normal variates, such that the joint covariance function is a sum of Kronecker products. This model, which generalizes the Graphical Lasso, assumes observations are correlated due to known genetic relationships and corrupted with non-independent noise. We have developed a computationally efficient EM algorithm to fit this model. On simulated datasets we illustrate substantially improved performance in network reconstruction by allowing for a general noise distribution.

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