Regularized Reduced Rank Regression for mixed predictor and response variables

TL;DR

This paper introduces GMR4, a regularized reduced rank regression model for mixed predictor and response variables, enhancing high-dimensional data analysis with regularization and optimal scaling.

Contribution

The paper extends the GMR3 model by incorporating regularization techniques and a cross-validation procedure, improving performance and interpretability in high-dimensional mixed data settings.

Findings

Simulation studies demonstrate effective regularization and rank estimation.

Application to healthcare data yields sparse, interpretable predictors.

Model outperforms existing methods in high-dimensional scenarios.

Abstract

In this paper, we introduce the Generalized Mixed Regularized Reduced Rank Regression model (GMR4), an extension of the GMR3 model designed to improve performance in high-dimensional settings. GMR3 is a regression method for a mix of numeric, binary and ordinal response variables, while also allowing for mixed-type predictors through optimal scaling. GMR4 extends this approach by incorporating regularization techniques, such as Ridge, Lasso, Group Lasso, or any combination thereof, making the model suitable for datasets with a large number of predictors or collinearity among them. In addition, we propose a cross-validation procedure that enables the estimation of the rank S and the penalty parameter lambda. Through a simulation study, we evaluate the performance of the model under different scenarios, varying the sample size, the number of non-informative predictors and response dimension. The results of the simulation study guide the choice of the penalty parameter lambda in the empirical application ISSP: Health and Healthcare I-II (2023), which includes mixed-type predictors and ordinal responses. In this application, the model results in a sparse and interpretable solution, with a limited set of influential predictors that provide insights into public attitudes toward healthcare.