Improved prediction and flagging of extreme random effects for non-Gaussian outcomes using weighted methods

Investigators often focus on predicting extreme random effects from mixed effects models fitted to longitudinal or clustered data, and on identifying or "flagging" outliers such as poorly performing hospitals or rapidly deteriorating patients. Our recent work with Gaussian outcomes showed that weighted prediction methods can substantially reduce mean square error of prediction for extremes and substantially increase correct flagging rates compared to previous methods, while controlling the incorrect flagging rates. This paper extends the weighted prediction methods to non-Gaussian outcomes such as binary and count data. Closed-form expressions for predicted random effects and probabilities of correct and incorrect flagging are not available for the usual non-Gaussian outcomes, and the computational challenges are substantial. Therefore, our results include the development of theory to support algorithms that tune predictors that we call "self-calibrated" (which control the incorrect flagging rate using very simple flagging rules) and innovative numerical methods to calculate weighted predictors as well as to evaluate their performance. Comprehensive numerical evaluations show that the novel weighted predictors for non-Gaussian outcomes have substantially lower mean square error of prediction at the extremes and considerably higher correct flagging rates than previously proposed methods, while controlling the incorrect flagging rates. We illustrate our new methods using data on emergency room readmissions for children with asthma.