Casey Davis1, Arya Pourzanjani1
1Daiichi Sankyo, Inc.
Introduction We introduce the Regularized Horseshoe (RHS) in the context of covariate selection for population PK/PD models. Covariate selection for these models is important for understanding when dose adjustments may be needed and what groups of patients may benefit most for a particular therapy. Fitting a single full model, such as FFEM or FREM [1], that includes all such combinations is often difficult in practice, as maximum-likelihood estimates of such models often have large variance and numerical issues. Several approaches to covariate selection exist in the pharmacometrics literature [2]. Two of the most common stepwise approaches are p-value or Akaike Information Criteria(AIC)-based forward and/or backward selection and Conditional Sampling used for Stepwise Approach based on Correlation (COSSAC) [3]. Unlike these stepwise approaches which require several model fits, the RHS can simultaneously assess all possible parameter-covariate relationships in a single model fit by leveraging the fact that such relationships are usually sparse in practice. Furthermore, the RHS avoids the over-estimation of effect sizes that commonly occurs with stepwise approaches[4] and avoids overfitting by averaging over the posterior uncertainty of possible parameter-covariate relationships. This leads to improved predictive performance on held-out data. Objectives •To introduce and adapt RHS in the context of population PK/PD models. •To evaluate the applicability and effectiveness of RHS in covariate selection for population PK/PD models. •To compare RHS with traditional population PK/PD stepwise covariate modeling (SCM) and COSSAC. Methods We compare the performance of RHS with SCM and COSSAC in both simulated and real-world data examples. In the simulated data examples, data are simulated from a two-compartment IV infusion model with 4 parameters per subject and 10 possible covariates, and thus 40 possible parameter-covariate relationships, We simulate across 4 ground-truth levels of sparsity scenarios (0%, 10%, 50%, 70%) of these 40 possible parameter-covariate relationships. We consider dataset sizes of 12, 40, and 300 subjects to get a realistic range across the various stages of drug development. Lastly, to assess the statistical properties of the methods over repeated draws of data, these simulation settings are replicated and fit 100 times each. Non-zero covariates were also given varying levels of realistic effect sizes. Several metrics of the estimators were assessed including bias, mean-square-error (MSE), type 1 error, type 2 error, type S error, and type M error [5]. In the real-world data examples, we evaluate the predictive performance on 4 publicly available population PK/PD datasets using the log pointwise predictive density (lppd) [6]. This was accomplished by using leave-subject-out cross-validation (LSO-CV). We use MCMC sampling in Stan [6] to fit RHS for each model, and we use Monolix to implement the SCM and COSSAC approaches. Results On the simulated data examples RHS outperforms SCM and COSSAC in terms of predictive accuracy on held-out data. Furthermore, RHS avoids the over-estimation of effect sizes displayed by the stepwise methods. In scenarios with multiple non-zero covariates that are correlated, RHS takes a hedged approach returning a posterior of several plausible parameter estimates, as opposed to a single point-estimate. As expected, the differences across methods are mitigated as the number of subjects in the model increases. In the real-world data examples, the held-out likelihood for RHS is consistently higher than the models selected by the stepwise approaches. The methods also returned similar results in terms of the parameter-covariate relationships selected. Conclusion A fully Bayesian approach to covariate selection offers several advantages to traditional approaches. First, because of the regularization provided by RHS priors and the accounting of all posterior uncertainty, all covariate relationships of interest can be included in a single model fit. Second, marginal posterior probabilities of the importance of individual covariates can be directly obtained and used to identify important covariates, while accounting for uncertainty inherent in the data. Third, rather than having to make discrete decision rules on which covariates are included in a model, the posterior distribution provides a natural continuous compromise between all such models, weighted by their ability to explain the data. This approach has several practical advantages that merit broader use in the pharmacometrics community.
[1] H Yun, R Svensson, EM Niebecker, and MO Karlsson. Evaluation of FREM and FFEM including use of model linearization. PAGE Abstr 2900. PAGE 22, 2013. [2] Sanghavi, K. e. Covariate modeling in pharmacometrics: General points for consideration. CPT: Pharmacometrics & Systems Pharmacology. 2024 [3] Ayral G, et al. A novel method based on unbiased correlations tests for covariate selection in nonlinear mixed effects models: The COSSAC approach. CPT: pharmacometrics & systems pharmacology, 318-329. 2021. [4] Harrell F, et al. Regression modeling strategies: with applications to linear models, logistic regression, and survival analysis, volume 608. Springer, 2001 [5] Gelman A and Carlin J. Beyond power calculations: Assessing type s (sign) and type m (magnitude) errors. Perspectives on psychological science, 9(6):641–651, 2014. [6] Gelman A, Carlin J, Stern H, Dunson D, Vehtari A, and Rubin D. Bayesian Data Analysis, Third Edition. Chapman & Hall/CRC Texts in Statistical Science. Taylor & Francis, 2013. [7] Carpenter B, et al. Stan: A probabilistic programming language. Journal of statistical software. 2017.
Reference: PAGE 33 (2025) Abstr 11393 [www.page-meeting.org/?abstract=11393]
Poster: Methodology - Covariate/Variability Models