Inference on Generalized Latent Variable Models with High-Dimensional Responses and Covariates

TL;DR

This paper introduces a high-dimensional generalized latent variable model that handles mixed-type responses and complex covariate dependence, with an efficient algorithm and theoretical guarantees for inference.

Contribution

It develops a novel alternating algorithm for inference in complex latent variable models with high-dimensional data, providing statistical guarantees and a debiased estimator.

Findings

Algorithm achieves statistical consistency and error bounds.

Debiased estimator is asymptotically normal.

Method effectively applied to PISA fairness assessment.

Abstract

Regression models with both high-dimensional responses and covariates have attracted growing attention. Standard multivariate regression models become inadequate when the response variables depend not only on observed covariates but also on latent variables that capture key unobserved characteristics. To draw statistical inferences on covariate effects while accounting for latent variables, we consider a high-dimensional generalized latent variable model that accommodates mixed-type responses and allows for flexible dependence between covariates and latent variables, which is more suitable for many real-world applications than existing methods that either rely on a linear regression form or restricted assumptions on the dependence between covariates and latent variables. We develop an alternating algorithm that iteratively updates the regression parameters and the latent variables, transforming an intractable nonconvex problem into a sequence of tractable convex subproblems. Theoretically, we provide algorithmic guarantees by establishing statistical consistency of the resulting estimator and deriving an error bound for it. Further, building on this estimator, we construct a debiased estimator for the covariate effect and establish its asymptotic normality. The effectiveness of the proposed method is demonstrated through an application to evaluating the fairness of the Programme for International Student Assessment (PISA).