An explicit mean-covariance parameterization for multivariate response linear regression

TL;DR

This paper introduces a novel parametric link between regression coefficients and error covariance in multivariate linear regression, enabling improved estimation and analysis of complex biological data.

Contribution

It proposes a new explicit mean-covariance parameterization, a non-convex optimization criterion, and an efficient algorithm, with applications to microRNA and cancer drug activity data.

Findings

Effective modeling of error correlations based on coefficient angles

Application to microRNA and drug activity data

Open-source R package available for implementation

Abstract

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. We highlight two models under which this parameterization arises: the latent variable reduced-rank regression model and the errors-in-variables regression model. 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. Our method is used to study the association between microRNA expression and cancer drug activity measured on the NCI-60 cell lines. An R package implementing our method, MCMVR, is available at github.com/ajmolstad/MCMVR.