Evaluation of Covariate Effect Size based on Parameter Variability

Aims: Parameter variability in a model can be described as an explained parameter variability with covariates (EPV) and unexplained parameter variability (UPV) [1,2]. The differences in EPV (dEPV) and UPV (dUPV) between models are indices to assess an impact of covariate effect [2]. In covariate modeling based on the difference in the objective function value (dOFV), dEPV and dUPV can be alternates of dOFV, however, there is no report quantifying impacts of dEPV and dUPV on covariate modeling. The aim of this study was to evaluate how much dEPV or dUPV had an impact on covariate modeling with evaluation of covariate effect size.

Methods: The simulation study was performed using an exponential covariate model given by TVCLi = θ1·exp(θ2·COVRi), where TVCLi is a typical clearance value for the ith subject and COVR1i is a normaly-distributed continuos covariate assumed by N(0, variance(COVR)). The individual clearance for ith subject (CLi) was given by CLi = TVCLi·exp(ηCLi), where ηCLi is normally distributed with N(0, ωCL2). For the log-transformed CLi, dEPV between the covariate model and the base mdoel (i.e., the model with no covariate) can be described by variance(log(TVCLi)), and dUPV can be described by a difference in ωCL2 between the base model and the covariate model. As the crucial part of this study, we used the ratio of 95th percentile for TVCLi relative to 50th percentile as the "inferential index" to be the covariate effect size, where the inferential index of less than 1.25 was considered to describe a slight effect size. As the simulation condition, variance(COVR) of 0.09 and the coefficients for covariate effect (i.e., θ2) of 0.1, 0.5, and 1 were used. The simulation conditions to result in low to high η shrinkage were also considered. Simulations using a power covariate model with a log-normal-distributed covariate were also tested. The stochastic simulation and estimation approach was performed to evaluate the relationship between the inferential index and dEPV or dUPV. NONMEM ver. 7.3 [3] and Perl-speaks NONMEM [4] were used for the evaluation.

Results: dEPV and dUPV were typically consistent with the expected value calculated from each simulation condition for both types of covariate model (exponential model and power model). The relationship between the inferential index and dEPV or dUPV was described with the theoretical equation under the normality for log-trabsformed TVCLi, which was given by exp[qnorm(0.95)·sqrt(dEPV or dUPV)]. Based on the theoretical equation, dEPV or dUPV was was 0.0184, 0.0608 and 0.178 when the inferential index of 1.25, 1.5 and 2, respectively, suggesting less than 0.0184 of dEPV or dUPV idicate slight covariate effect size. In case of the high-η-shrinkage condition, dUPV was smaller than the expected value, whereas dEPV was typically consistent with the expected value.

Conclusion: dEPV and dUPV described the covariate effect size in the theoretical manner. Less than 0.0184 of dEPV or dUPV would indicate slight covariate effect size (inferential index < 1.25) on the covariate modeling. dEPV can be used for evaluating the covariate effect size in case of high η shrinkage.

References: 1. Lagishetty CV, Vajjah P, Duffull SB. A reduction in between subject variability is not mandatory for selecting a new covariate. J Pharmacokinet Pharmacodyn. 2012;39:383-92. 2. Hennig S, Karlsson MO. Concordance between criteria for covariate model building. J Pharmacokinet Pharmacodyn. 2014;41:109-25. 3. Beal SL, Sheiner LB, Boeckmann AJ 1989-2006. NONMEM users guide. Icon Development Solutions, Ellicott City, MD. 4. Lindbom L, Pihlgren P, Jonsson EN. PsN-Toolkit–a collection of computer intensive statistical methods for non-linear mixed effect modeling using NONMEM. Comput. Methods Programs Biomed. 2005;79:241-57.