Estimation of the average causal effect among subgroups defined by post-treatment variables

Background: In clinical trials, when comparing treatments in a subgroup of patients defined by an event that occurred after randomization is required, the standard estimator that adjusts for the post-treatment variable does not have a causal interpretation. Purpose: To address this problem, we formulate clinically relevant causal estimands using the principal stratification framework developed by Frangakis and Rubin [3], and propose a new estimation method for the principal causal effect. Methods: We consider the comparison of the duration of response among patients who responded to chemotherapy in a cancer clinical trial. Our goal is to estimate the local average treatment effect, that is, the treatment difference among patients who would have responded to either treatment. In order to identify this estimand, we make the assumption that the value of the counterfactual indicator of response is independent of both the actual response status and the outcome variable of interest conditional on the covariates. The proposed estimator is a weighted average of the standard estimators for responders where weights are the probability that the response would have occurred had the patient received the other treatment. Results: The proposed method is applied to data from a randomized phase III clinical trial in patients with advanced non-small-cell lung cancer. The average difference for the duration of response among responders estimated by the proposed method and the standard one was 16.1 (days) and 9.5 (days), respectively. We also evaluate the performance of the proposed method through simulation studies, which showed that the proposed estimator was unbiased, while the standard one was largely biased. Conclusions: We have developed an estimation method for the local average treatment effect. For any type of outcome variables, our estimator can be easily constructed and can be interpreted as the treatment effect among patients who would have had the event in either treatment group.