A new solution to deal with eta-shrinkage: the Weighted EBEs!
Kevin Le Boedec, Celine M. Laffont and Didier Concordet.
UMR181 Physiopathologie et Toxicologie Expérimentales, INRA, ENVT, Toulouse, France.
Objectives: Empirical Bayes Estimates (EBEs) of individual PK/PD parameters are given by all population PK/PD software. They are widely used for covariate model building and during model evaluation (via IPRED and IWRES). Unfortunately, they suffer from the shrinkage phenomenon that makes fuzzy the potential relationship between covariates and individual parameters and which may lead to the so-called ‘perfect-fit’ phenomenon [1]. Eta-shrinkages greater than 20-30% have been advocated to be ‘lethal’ for the use of EBEs [2], leaving the population PK analyst quite lonely with his job! However, globally high eta-shrinkage does not imply that all EBEs are shrunk: some individuals may have informative EBEs. We thus propose new weighted EBEs (WEBE) that allow the analyst to evidence the hidden relationship between covariates and individual PK/PD parameters by fully exploiting all information available on each individual.
Methods: The method we propose comes from the seminal work of Lange and Ryan [3] about linear mixed effects models. Currently, all individuals contribute with the same weight to all exploratory (e.g. EBEs vs. covariates) and diagnostic plots (e.g. IWRES vs. time, Q-Q plots of eta/epsilon), irrespective of the uncertainty on EBEs. The main idea of the present method relies on weighting EBE of an individual proportionally to its precision. Weighted plots and statistics (IWRES, ETAS …) are derived from these weighted EBEs. The performance of our method was assessed over a range of population PK and/or PD models.
Results: Our weighted plots showed good performances in detecting hidden relationships between individuals PK/PD parameters and covariates, even when eta-shrinkage was very high, provided rich data were available for some individuals. The corresponding statistics fully exploit these weights. Of course, our method does not create any information: forget it when all individuals are poorly documented!
Conclusions: We have developed a new method to reveal hidden relationships between individual PK parameters and covariates and to help in model validation. Its good theoretical properties were confirmed by several simulation studies using different PK and/or PD models.
References:
[1] Karlsson MO and Savic RM. Diagnosing model diagnostics. Clin. Pharmacol. Ther. 2007; 82: 17-20.
[2] Savic RM and Karlsson MO. Importance of Shrinkage in Empirical Bayes Estimates for Diagnostics: Problems and Solutions. AAPS J. 2009;11:558–69.
[3] Lange N and Ryan L. Assessing normality in random effects models. Ann. Statist. 1989; 17:624–642.