Mixed Models, Random Effects Misspecification and Prediction

Abstract

Linear and generalized linear mixed models are widely used in applied sciences because of their capability to model clustered and spatial/spatio-temporal data. One important aspect when dealing with mixed models is prediction of random effects. It is important to note that when fitting a mixed model, some assumption is often made on the random effects, which can potentially affect prediction. In this talk, we will discuss the impact of misspecification of the random effects on prediction in two different situations: cluster-independent GLMMs and spatio-temporal LMMs, with simulated and real-world data sets.

First, we investigate prediction in cluster-independent GLMMs under the normality assumption of the random effects (when the underlying random effects might be nonnormal). We examined whether this assumption significantly impacts the quality of prediction by comparing the mean squared prediction errors (MSEPs) under a misspecified normal distribution against those under a correctly specified distribution, which was assumed to be a mixture of normal distributions. We found that the unconditional MSEPs are generally higher under the incorrectly assumed normal distribution, especially when the cluster size is small. The MSEPs conditional on the random effects are also generally higher under the misspecified distribution, especially at the region closer to the mean of each component of the underlying mixture distribution (given this distribution is skewed or multimodal

Second, we investigate prediction in spatio-temporal LMMs where the spatio-temporal random effects are normally distributed with a misspecified covariance function. More specifically, we compare the predictive performance of nonstationary-covariance models against that of stationary-covariance models, where the underlying spatiotemporal random processes are nonstationary. The nonstationary covariance is built using a warping function approach, where spatio-temporal coordinates are mapped onto a new deformed space through a warping function, and a stationary covariance is assumed on this new deformed space. The results show a difference in prediction variances (equivalent to conditional MSEPs) from the two models: If the observation locations are evenly distributed on the spatial domain, the prediction variances from the stationary model are homogeneous across the domain, while those from the nonstationary model are related to the warping: in regions of expansion in the deformed space, the prediction variances are larger.

This is joint work with Alan Welsh, Francis Hui, Samuel Muller, Andrew Zammit-Mangion, and Stephen Chuter.