Sufficient dimension reduction for clustered data via finite mixture modelling

Abstract

Summary Sufficient dimension reduction (SDR) is an attractive approach to regression modelling. However, despite its rich literature and growing popularity in application, surprisingly little research has been done on how to perform SDR for clustered data, for example as is commonly arises in longitudinal studies. Indeed, current popular SDR methods have been mostly based on a marginal estimating equation approach. In this article, we propose a new approach to SDR for clustered data based on a combination of finite mixture modelling and mixed effects regression. Finite mixture models offer a flexible means of estimating the fixed effects central subspace, based on slicing the space up and probabilistically clustering observations to each slice (mixture component). Dimension reduction is achieved by having the mixing proportions vary only through the sufficient fixed effect predictors. We then incorporate random effects as a natural means of accounting for correlations within clusters. We employ a Monte Carlo expectation–maximisation algorithm to estimate the model parameters and fixed effects central subspace, and discuss methods for associated uncertainty quantification and prediction. Simulation studies demonstrate that our approach performs strongly against both estimating equation methods for estimating the fixed effects central subspace, and SDR methods which do not account for within‐cluster correlation. Finally, we apply the proposed approach to a data set on air pollutant monitoring across 13 stations in the Eastern United States.

A Festschrift for Geoff McLachlan

“…The leading paper of Tomarchio, Ingrassia & Melnykov (2022) follows naturally from Geoff's interest in the study of mixture of normal distributions for vectorial data (Theme (i)), and extends such an approach to the problem domain of matrix data clustering. The works of Hui & Nghiem (2022) and Scrucca (2022) can also be viewed as contributions to this theme, with particular focus on the use of latent space representations via dimensionality reduction in order to facility better mixture-based clustering outcomes. Other contributions to Theme (i) include the works of Durand et al (2022), Greve et al (2022), andHennig &Coretto (2022), who each provide differing perspectives and solutions to the problem of clustering and mixture model estimation when the underlying number of clusters is unknown.…”