We use simulated data to examine the consequences of depletion of susceptibles for hazard ratio (HR) estimators based on a propensity score (PS). First, we show that the depletion of susceptibles attenuates marginal HRs toward the null by amounts that increase with the incidence of the outcome, the variance of susceptibility, and the impact of susceptibility on the outcome. If susceptibility is binary then the Bross bias multiplier, originally intended to quantify bias in a risk ratio from a binary confounder, also quantifies the ratio of the instantaneous marginal HR to the conditional HR as susceptibles are depleted differentially. Second, we show how HR estimates that are conditioned on a PS tend to be between the true conditional and marginal HRs, closer to the conditional HR if treatment status is strongly associated with susceptibility and closer to the marginal HR if treatment status is weakly associated with susceptibility. We show that associations of susceptibility with the PS matter to the marginal HR in the treated (ATT) though not to the marginal HR in the entire cohort (ATE). Third, we show how the PS can be updated periodically to reduce depletion-of-susceptibles bias in conditional estimators. Although marginal estimators can hit their ATE or ATT targets consistently without updating the PS, we show how their targets themselves can be misleading as they are attenuated toward the null. Finally, we discuss implications for the interpretation of HRs and their relevance to underlying scientific and clinical questions.
Consequences of depletion of susceptibles for hazard ratio estimators based on propensity scores.