Standard errors of prediction in generalized linear mixed models

The unconditional mean squared error of prediction (UMSEP) is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model. But the UMSEP is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data. When the random effects describe variation between independent small domains and domain-specific prediction is of interest, we propose a conditional mean squared error of prediction (CMSEP) as a general measure of prediction variance. The CMSEP is shown to be the sum of the conditional variance and a positive correction that accounts for the sampling variability of parameter estimates. We derive a second-order-correct estimate of the CMSEP that consists of three components: (a) a plug-in estimate of the conditional variance, (b) a plug-in estimate of a Taylor series approximation to the correction term, and (c) a bootstrap estimate of the bias incurred in (a). In the normal case our formulas based on the CMSEP provide a conditional alternative to the unconditional expansions of Fuller and Harter, Kackar and Harville, and Prasad and Rao. In addition, we show that the prediction variance formula obtained by Wolfinger and O'Connell and suggested by Breslow and Clayton is in fact Laplace's approximation to the CMSEP based on the assumption that the variance components are known and ignoring the bias-correction term. Thus this formula has a conditional interpretation in the small-domain setting and should not be interpreted unconditionally. Finally, although use of the CMSEP is motivated using entirely frequentist arguments, our second-order approximation to the CMSEP closely resembles a corresponding expansion for the Bayesian posterior variance.