A modified Bayesian information criterion (mBIC) with multiple testing correction for population pharmacokinetic model building
Authors: Hooker Andrew , Chen Xiaomei , Shi Keqi , Nordgren Rikard , Belin Stella , Borg Johan , Schindler Emilie , Karlsson Mats 
Published: February 2025

Objectives: Multiple testing (or multiplicity) is a well-recognized problem in statistics, which occurs when multiple hypothesis tests are performed simultaneously on a single dataset. This problem is especially prominent in the case of evaluating a large pool of predictors or models. The process of pharmacometric model building involves estimations of various models and multiple comparisons among the estimated models. Without correction for multiple testing, a more complicated model tends to be selected due to the inflation of type I error. However, few studies have been done to address this problem. In the field of quantitative trait loci mapping, a modified Bayesian information criterion (mBIC) was previously proposed for linear multiple regression considering multiple testing correction [1,2]. In this work, we expanded and further modified the published mBIC for nonlinear mixed-effect modeling and model development of population pharmacokinetic models. Methods: In the current work, two mBIC equations were proposed for selecting the pharmacokinetic structural model and random effect model, respectively. Briefly, an additional penalty term for multiple testing (2k_M×log⁡(p/E)) is added to the mixed-effect [3] or random-effect BIC [4]. The size of the penalty depends on the size of the search space (p), the a priori value related to the expected model complexity (E), and the actual model elements in the evaluated model (k_M). The impact of E value was evaluated on the size of the penalty in the mBIC and the p-value corresponding to the likelihood ratio test. Moreover, the proposed mBIC was applied to an automatic model development (AMD) procedure using the AMD tool [5] developed by our group on a simulated pharmacokinetic dataset. Results: The additional penalty for multiple testing increases with the decrease in E value. During a structural model search with the default search space of the AMD tool (p=10), the multiple-testing penalty was 4.6, 3.2, and 2.4 for E=1,2,3, respectively, for an additional THETA. With a total of 300 observations, the mBIC corresponded to p values of 0.0013, 0.0028, and 0.0044, respectively. During a search for inter-individual variation models with four pharmacokinetic parameters, the multiple-testing penalty was 3.6, 2.2, and 1.4 for E=1,2,3, respectively, for an additional off-diagonal OMEGA. With a total of 30 subjects, the mBIC corresponded to p values of 0.0082, 0.018, and 0.029, respectively. Through the AMD procedure, the mBIC gave a more parsimonious model compared to the BIC without the multiple testing penalty (a transit compartmental model without versus with a depot compartment). Conclusions: The proposed mBIC is a promising selection criterion that accounts for multiple testing in the case of extensive pharmacokinetic model building and can avoid an overly complicated final model.Citations: [1] Bogdan M, et al . Genetics 2004;167:989–999.[2] Cui X, et al. Handbook of Multiple Comparisons 2021.[3] Delattre M, et al. Electronic Journal of Statistics 2014;8:456–475.[4] Delattre M, Poursat MA. International Journal of Biostatistics 2016;16:75–83.[5] Chen X, et al. PAGE 30 (2022) Abstr 10091.