Prediction in multilevel generalized linear models

• Published in: Journal of the Royal Statistical Society. Series A, Statistics in society Statistics in society, 2009-06, Vol.172 (3), p.659-687
• Main Authors: Skrondal, Anders, Rabe-Hesketh, Sophia
• Format: Article
• Language: English
• Subjects: Adaptive quadrature; Approximation; Bayes estimators; Best linear unbiased predictor (BLUP); Bootstrap method; Comparative standard error; Covariance matrices; Diagnostic standard error; Empirical Bayes; Error; Errors; Forecasting techniques; Forecasts; Generalized linear mixed model; Generalized linear model; Generalized linear models; gllamm; Linear models; Maximum likelihood estimation; Mean-squared error of prediction; Modeling; Multilevel model; Multilevel models; Multivariate analysis; Parameter estimation; Posterior; Prediction; Random effects; Scoring; Standard deviation; Standard error; Statistical methods; Statistical models
• ISSN: 0964-1998; 1467-985X
• DOI: 10.1111/j.1467-985X.2009.00587.x

We discuss prediction of random effects and of expected responses in multilevel generalized linear models. Prediction of random effects is useful for instance in small area estimation and disease mapping, effectiveness studies and model diagnostics. Prediction of expected responses is useful for planning, model interpretation and diagnostics. For prediction of random effects, we concentrate on empirical Bayes prediction and discuss three different kinds of standard errors; the posterior standard deviation and the marginal prediction error standard deviation (comparative standard errors) and the marginal sampling standard deviation (diagnostic standard error). Analytical expressions are available only for linear models and are provided in an appendix . For other multilevel generalized linear models we present approximations and suggest using parametric bootstrapping to obtain standard errors. We also discuss prediction of expectations of responses or probabilities for a new unit in a hypothetical cluster, or in a new (randomly sampled) cluster or in an existing cluster. The methods are implemented in gllamm and illustrated by applying them to survey data on reading proficiency of children nested in schools. Simulations are used to assess the performance of various predictions and associated standard errors for logistic random-intercept models under a range of conditions.