Importance of Shrinkage in Empirical Bayes Estimates for Diagnostics: Problems and Solutions

Empirical Bayes (“post hoc”) estimates (EBEs) of ηs provide modelers with diagnostics: the EBEs themselves, individual prediction (IPRED), and residual errors (individual weighted residual (IWRES)). When data are uninformative at the individual level, the EBE distribution will shrink towards zero (η-shrinkage, quantified as 1-SD(ηEBE)/ω), IPREDs towards the corresponding observations, and IWRES towards zero (ε-shrinkage, quantified as 1-SD(IWRES)). These diagnostics are widely used in pharmacokinetic (PK) pharmacodynamic (PD) modeling; we investigate here their usefulness in the presence of shrinkage. Datasets were simulated from a range of PK PD models, EBEs estimated in non-linear mixed effects modeling based on the true or a misspecified model, and desired diagnostics evaluated both qualitatively and quantitatively. Identified consequences of η-shrinkage on EBE-based model diagnostics include non-normal and/or asymmetric distribution of EBEs with their mean values (“ETABAR”) significantly different from zero, even for a correctly specified model; EBE–EBE correlations and covariate relationships may be masked, falsely induced, or the shape of the true relationship distorted. Consequences of ε-shrinkage included low power of IPRED and IWRES to diagnose structural and residual error model misspecification, respectively. EBE-based diagnostics should be interpreted with caution whenever substantial η- or ε-shrinkage exists (usually greater than 20% to 30%). Reporting the magnitude of η- and ε-shrinkage will facilitate the informed use and interpretation of EBE-based diagnostics.