Random Effects Model-Based Sufficient Dimension Reduction for Independent Clustered Data

Sufficient dimension reduction (SDR) is a popular class of regression methods which aim to find a small number of linear combinations of covariates that capture all the information of the responses, that is, a central subspace. The majority of current methods for SDR focus on the setting of independent observations, while the few techniques that have been developed for clustered data assume the linear transformation is identical across clusters. In this article, we introduce random effects SDR, where cluster-specific random effect central subspaces are assumed to follow a distribution on the Grassmann manifold, and the random effects distribution is characterized by a covariance matrix that captures the heterogeneity between clusters in the SDR process itself. We incorporate random effect SDR within a model-based inverse regression framework. Specifically, we propose a random effects principal fitted components model, where a two-stage algorithm is used to estimate the overall fixed effect central subspace, and predict the cluster-specific random effect central subspaces. We demonstrate the consistency of the proposed estimators, while simulation studies demonstrate the superior performance of the proposed approach compared to global and cluster-specific SDR approaches. We also present extensions of the above model to handle mixed predictors, demonstrating how random effects SDR can be achieved in the case of mixed continuous and binary covariates. Applying the proposed methods to study the longitudinal association between the life expectancy of women and socioeconomic variables across 117 countries, we find log income per capita, infant mortality, and income inequality are the main drivers of a two-dimensional fixed effect central subspace, although there is considerable heterogeneity in how the country-specific central subspaces are driven by the predictors. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.