The AAPS Journal

, Volume 11, Issue 3, pp 558–569 | Cite as

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

  • Radojka M. Savic
  • Mats O. Karlsson
Research Article


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(η EBE)/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.

Key words

empirical Bayes estimate model building model evaluation NONMEM shrinkage 



The authors would like to thank Dr. Siv Jönsson for valuable comments on the manuscript, Paul Baverel for his help with reviewing shrinkage in real models and datasets, and the anonymous reviewers for their valuable comments on the manuscript.


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Copyright information

© American Association of Pharmaceutical Scientists 2009

Authors and Affiliations

  1. 1.Division of Pharmacokinetics and Drug Therapy, Department of Pharmaceutical Biosciences, Faculty of PharmacyUppsala UniversityUppsalaSweden

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