Adjusted Nelson-Aalen Estimators by Inverse Treatment Probability Weighting With an Estimated Propensity Score.

Inverse probability of treatment weighting (IPW) has been well applied in causal inference to estimate population-level estimands from observational studies. For time-to-event outcomes, the failure time distribution can be estimated by estimating the cumulative hazard in the presence of random right censoring. IPW can be performed by weighting the event counting process and at-risk process by the inverse treatment probability, resulting in an adjusted Nelson-Aalen estimator for the population-level counterfactual cumulative incidence function. We consider the adjusted Nelson-Aalen estimator with an estimated propensity score in the competing risks setting. When the estimated propensity score is regular and asymptotically linear, we derive the influence functions for the counterfactual cumulative hazard and cumulative incidence. Then we establish the asymptotic properties for the estimators. We show that the uncertainty in the estimated propensity score contributes to an additional variation in the estimators. However, through simulation and real-data application, we find that such an additional variation is usually small.