Random Effects Misspecification and its Consequences for Prediction in Generalized Linear Mixed Models

TL;DR

This paper examines how misspecifying the random effects distribution in GLMMs affects prediction accuracy, showing that using a normal assumption when the true distribution is a mixture can lead to larger prediction errors, especially with small clusters.

Contribution

It provides a comprehensive analysis of the impact of random effects distribution misspecification on prediction in GLMMs using theory, simulation, and real data.

Findings

Unconditional MSEPs are larger under misspecified normal random effects.

Optimal shrinkage differs between true and assumed distributions.

Prediction interval coverage remains relatively stable despite misspecification.

Abstract

When fitting generalized linear mixed models (GLMMs), one important decision to make relates to the choice of the random effects distribution. As the random effects are unobserved, misspecification of this distribution is a real possibility. In this article, we investigate the consequences of random effects misspecification for point prediction and prediction inference in GLMMs, a topic on which there is considerably less research compared to consequences for parameter estimation and inference. We use theory, simulation, and a real application to explore the effect of using the common normality assumption for the random effects distribution when the correct specification is a mixture of normal distributions, focusing on the impacts on point prediction, mean squared prediction errors (MSEPs), and prediction intervals. We found that the optimal shrinkage is different under the two random effect distributions, so is impacted by misspecification. The unconditional MSEPs for the random effects are almost always larger under the misspecified normal random effects distribution, especially when cluster sizes are small. Results for the MSEPs conditional on the random effects are more complicated, but they remain generally larger under the misspecified distribution when the true random effect is close to the mean of one of the component distributions in the true mixture distribution. Results for prediction intervals indicate that overall coverage probability is not greatly impacted by misspecification.