Asymptotics of numerical integration for two-level mixed models

TL;DR

This paper analyzes the asymptotic behavior of numerical integration methods in two-level mixed models, demonstrating the importance of adaptive techniques for consistent inference and deriving error rates under regularity conditions.

Contribution

It provides a theoretical analysis of the convergence rates of adaptive versus non-adaptive numerical integration in mixed models, establishing conditions for consistency of maximum likelihood estimators.

Findings

Adaptive numerical integration achieves optimal stochastic error rates.

Non-adaptive methods can lead to inconsistent estimators under weak conditions.

Regularity conditions ensure the applicability of the derived convergence rates.

Abstract

We study mixed models with a single grouping factor, where inference about unknown parameters requires optimizing a marginal likelihood defined by an intractable integral. Low-dimensional numerical integration techniques are regularly used to approximate these integrals, with inferences about parameters based on the resulting approximate marginal likelihood. For a generic class of mixed models that satisfy explicit regularity conditions, we derive the stochastic relative error rate incurred for both the likelihood and maximum likelihood estimator when adaptive numerical integration is used to approximate the marginal likelihood. We then specialize the analysis to well-specified generalized linear mixed models having exponential family response and multivariate Gaussian random effects, verifying that the regularity conditions hold, and hence that the convergence rates apply. We also prove that for models with likelihoods satisfying very weak concentration conditions that the maximum likelihood estimators from non-adaptive numerical integration approximations of the marginal likelihood are not consistent, further motivating adaptive numerical integration as the preferred tool for inference in mixed models. Code to reproduce the simulations in this paper is provided at https://github.com/awstringer1/aq-theory-paper-code.