Asymptotic Results for Penalized Quasi-Likelihood Estimation in Generalized Linear Mixed Models

TL;DR

This paper derives large sample asymptotic distributions for penalized quasi-likelihood estimators in generalized linear mixed models, considering both fixed and random effects, with simulations confirming the theoretical findings.

Contribution

It provides the first asymptotic distribution results for PQL estimators in GLMMs with increasing clusters and sizes, under different regimes.

Findings

PQL estimators are asymptotically normal for fixed effects.

The distribution of random effects prediction gap can be a normal scale-mixture.

Simulation results support theoretical asymptotic distributions.

Abstract

Generalized Linear Mixed Models (GLMMs) are widely used for analysing clustered data. One well-established method of overcoming the integral in the marginal likelihood function for GLMMs is penalized quasi-likelihood (PQL) estimation, although to date there are few asymptotic distribution results relating to PQL estimation for GLMMs in the literature. In this paper, we establish large sample results for PQL estimators of the parameters and random effects in independent-cluster GLMMs, when both the number of clusters and the cluster sizes go to infinity. This is done under two distinct regimes: conditional on the random effects (essentially treating them as fixed effects) and unconditionally (treating the random effects as random). Under the conditional regime, we show the PQL estimators are asymptotically normal around the true fixed and random effects. Unconditionally, we prove that while the estimator of the fixed effects is asymptotically normally distributed, the correct asymptotic distribution of the so-called prediction gap of the random effects may in fact be a normal scale-mixture distribution under certain relative rates of growth. A simulation study is used to verify the finite sample performance of our theoretical results.