Some aspects of response variable selection and estimation in multivariate linear regression

Multivariate linear regression analysis is an important technique for modeling the predictive relationships of multiple related response variables on a set of common predictor variables. Numerous studies have been conducted on situations where response variables are given and only predictor variables are subject to variable selection. In practice, however, some response variables do not depend on any of the predictor variables and have very small regression coefficients, implying that response variables need to be selected. Several methods have been proposed for response variable selection in multivariate linear regression. Examples include Bonferroni selection, linear step-up selection, adaptive linear step-up selection, multiple-stage linear step-up selection, response best-subset selection and sparse envelope selection. In this article, we address some aspects of response variable selection focusing on the above-mentioned examples concerning methodological developments, theoretical properties and computational algorithms. We address their performances under the recall rate or true positive rate, true negative rate, precision rate, F-measure, model size and their standard deviations via simulation studies. We also highlight two issues that require further study.