Joint Mean and Correlation Regression Modelling for Multivariate Data

Abstract

In the analysis of multivariate or multi-response data, researchers are often not only interested in studying how the mean (say) of each response evolves as a function of covariates, but also and simultaneously how the correlations between responses are related to one or more similarity/distance measures. To address such research questions, we propose a novel joint mean and correlation regression model that simultaneously regresses the mean of each response against a set of covariates and the correlations between responses against a set of similarity measures, which can be applied to a wide variety of correlated discrete and (semi-)continuous responses. Under a general setting where the number of responses can tend to infinity with the number of clusters, we demonstrate that our proposed joint estimators of the regression coefficients and correlation parameters are consistent and asymptotically normally distributed with differing rates of convergence. We apply the proposed model to a dataset of overdispersed counts of 38 Carabidae ground beetle species sampled throughout Scotland, with results showing in particular that beetle total length and breeding season have statistically important effects in driving the correlations between beetle species.