An explicit mean-covariance parameterization for multivariate response linear regression
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We develop a new method to fit the multivariate response linear regression model that exploits a parametric link between the regression coefficient matrix and the error covariance matrix. Specifically, we assume that the correlations between entries in the multivariate error random vector are proportional to the cosines of the angles between their corresponding regression coefficient matrix columns, so as the angle between two regression coefficient matrix columns decreases, the correlation between the corresponding errors increases. This assumption can be motivated through an error-in-variables formulation. We propose a novel non-convex weighted residual sum of squares criterion which exploits this parameterization and admits a new class of penalized estimators. The optimization is solved with an accelerated proximal gradient descent algorithm. Extensions to scenarios where responses are missing or some covariates are measured without error are also proposed. We use our method to study the association between gene expression and copy-number variations measured on patients with glioblastoma multiforme. An R package implementing our method, MCMVR, is available online.
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