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 scalemixture distribution under certain relative rates of growth. A simulation study

is used to verify the finite sample performance of our theoretical results.

Key words and phrases: Asymptotic independence, Clustered data, Large sample distribution, Longitudinal data, Prediction

Information

Preprint No.SS-2023-0343
Manuscript IDSS-2023-0343
Complete AuthorsXu Ning, Francis Hui, Alan Welsh
Corresponding AuthorsXu Ning
Emailsnicksonnz@hotmail.com

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Acknowledgments

Xu Ning was supported by the Australian Government Research Training

Program Scholarship. Francis Hui and Alan Welsh were supported by an

Australian Research Council Discovery Project DP230101908.

Supplementary Materials

The online Supplementary Material contains proofs of our theorems and

extra simulation results.

Supplementary materials are available for download.