Time-varying covariates, mediation analysis and overadjustment bias in PK/PD modelling
Sebastiaan Camiel Goulooze (1), Nelleke Snelder (1)
(1) LAP&P Consultants BV, Leiden, The Netherlands
Objectives: Pharmacokinetic-pharmacodynamic (PK/PD) modelling benefits from the inclusion of covariates to enhance predictive capabilities and explain variability [1]. However, the inclusion of covariates may introduce bias for a variety of reasons [2]. This study uses simulations to illustrate overadjustment bias in PK/PD modelling when incorporating time-varying covariates that are an intermediate on the causal pathway of the drug effect on the outcome [3]. We demonstrate the use of mediation analysis to estimate the percentage of the drug effect explained by the intermediate (time-varying covariate).
Methods: Simulations were conducted for three scenarios involving a time-varying covariate influencing a numerical outcome variable. Scenario 1 assumes a direct treatment effect on the outcome, not mediated by the covariate. In Scenario 2, the drug effect on the outcome is entirely mediated by the time-varying covariate: drug treatment lowers covariate, which lowers outcome value. Scenario 3 represents a hybrid scenario with 50% direct and 50% mediated treatment effects. Each scenario was simulated 1000 times, with each dataset containing 100 subjects with six observations for both covariate and outcome variables. In each scenario, the exposure-response (E-R) was linear.
Two models were fitted to each dataset: Model A used the baseline covariate as a predictor of the outcome, while Model B used the time-varying covariate instead. The bias was calculated by comparing the estimated E-R slope with the true value. The level of mediation, indicating the percentage of the total drug effect explained by the intermediate (covariate), was calculated as: 100% * (E-R slope Model A - E-R slope Model B) / E-R slope Model A.
Results: In Scenario 1, where the time-varying covariate is not an intermediate for the treatment effect, both models yield unbiased E-R slope estimates. However, Model B (incorporating the time-varying covariate) exhibits a 2.2-fold reduction in the mean standard error of the E-R slope estimate. In Scenarios 2 and 3, Model A produces unbiased E-R slope estimates (average bias of -0.5% and 0.4%, respectively), while Model B, incorporating the time-varying covariate, results in biased treatment effects (average bias of -100.1% and -50.0%, respectively). Across all scenarios, the average estimated level of mediation closely aligns with the theoretical values of the simulation model (i.e., 0% in Scenario 1, 100% in Scenario 2, and 50% in Scenario 3).
Conclusions: Scenario 1 highlights the potential benefits of incorporating time-varying covariates, not only for enhancing statistical fit but also for improving the statistical power to estimate treatment effects. However, when the time-varying covariate acts as an intermediate on the causal pathway between drug treatment and outcome (Scenarios 2 and 3), its inclusion in the model can lead to substantial underestimation of the treatment effect, reaching up to 100%. Generally speaking, the inclusion of a time-varying covariate that is affected by treatment (either as a mediator or consequence of the treatment effect on the outcome) will cause overadjustment bias [4].
In the presence of mediation, the simplest approach is to model the total drug effect on the outcome directly, using only the baseline value of the intermediate as a covariate in the model (similar to model A) to avoid overadjustment bias. However, a limitation of this approach is that it doesn’t characterize the underlying relationships of treatment, intermediate and outcome. If disentangling these relationships is of interest, one can use more elaborate sequential or joint PK/PD modelling approaches in which the intermediate variable is modelled as an additional outcome variable, rather than as a covariate.
The level of mediation of the drug effect by the intermediate can be calculated using the formula provided above under certain assumptions. High levels of mediation may support the use of the intermediate as a surrogate endpoint or bridging biomarker [5]. This has relevance for personalised medicine and drug development, especially when clinical endpoints are rare or necessitate prolonged follow-up. An example is a PK/PD analysis of finerenone that showed that the drug effect of slowing down kidney function decline (late outcome, requiring long follow-up) could be characterised via its effect on albuminuria (early outcome), supporting albuminuria as a potential surrogate endpoint for future studies [6].
References:
[1] Mould et al. CPT Pharmacometrics Syst Pharmacol (2012) 1(9):e6.
[2] Hernán et al. Causal Inference: What if (2022) Boca Raton: Chapman & Hall/CRC.
[3] Schisterman et al. Epidemiology (2009) 20:488–495.
[4] Lu et al. Epidemiology (2021) 32:e22-e23.
[5] Fleming et al. Ther Innov Regul Sci (2023) 57:109–120.
[6] Eissing et al. Diabetes Obes Metab (2023). doi: 10.1111/dom.15387.