Consistent and Scalable Composite Likelihood Estimation of Probit Models with Crossed Random Effects

TL;DR

This paper introduces a composite likelihood method for probit models with crossed random effects, significantly reducing computational complexity and enabling analysis of very large datasets.

Contribution

It develops a scalable composite likelihood approach that replaces high-dimensional integrals with one-dimensional ones, allowing consistent parameter estimation in large-scale crossed effects models.

Findings

Computation scales linearly with sample size.

Method successfully applied to five million observations.

Achieves accurate estimation without high-dimensional integrals.

Abstract

Estimation of crossed random effects models commonly requires computational costs that grow faster than linearly in the sample size $N$, often as fast as $\Omega(N^{3/2})$, making them unsuitable for large data sets. For non-Gaussian responses, integrating out the random effects to get a marginal likelihood brings significant challenges, especially for high dimensional integrals where the Laplace approximation might not be accurate. We develop a composite likelihood approach to probit models that replaces the crossed random effects model with some hierarchical models that require only one-dimensional integrals. We show how to consistently estimate the crossed effects model parameters from the hierarchical model fits. We find that the computation scales linearly in the sample size. We illustrate the method on about five million observations from Stitch Fix where the crossed effects formulation would require an integral of dimension larger than $700{,}000$.