On Forward Sufficient Dimension Reduction for Categorical and Ordinal Responses

TL;DR

This paper introduces a forward sufficient dimension reduction method tailored for categorical and ordinal responses within multinomial generalized linear models, improving efficiency over pairwise approaches.

Contribution

It extends existing forward regression techniques to handle multinomial responses directly, providing a consistent estimator with proven convergence rates.

Findings

Estimator shown to be consistent and convergent

Algorithm developed using repeated forward regression applications

Method validated through simulations and real data applications

Abstract

We present a forward sufficient dimension reduction method for categorical or ordinal responses by extending the outer product of gradients and minimum average variance estimator to multinomial generalized linear model. Previous work in this direction extend forward regression to binary responses, and are applied in a pairwise manner to multinomial data, which is less efficient than our approach. Like other forward regression-based sufficient dimension reduction methods, our approach avoids the relatively stringent distributional requirements necessary for inverse regression alternatives. We show consistency of our proposed estimator and derive its convergence rate. We develop an algorithm for our methods based on repeated applications of available algorithms for forward regression. We also propose a clustering-based tuning procedure to estimate the tuning parameters. The effectiveness of our estimator and related algorithms is demonstrated via simulations and applications.