Kaplan-Meier curves for survivor causal effects with time-to-event outcomes

In clinical trials, an outcome of interest may be undefined for individuals who die before the outcome is evaluated. One approach to deal with such issues is to consider the survivor causal effect (SCE), which is defined as the effect of treatment on the outcome among the subpopulation that would have survived under either treatment arm. Although several methods have been presented to estimate the SCE with time-to-event outcomes, they are difficult to implement in practice.

We present a simple method to create Kaplan-Meier curves and to estimate the hazard ratio (HR) for the SCE with time-to-event outcomes.

To develop such a method, we applied the weighted average method presented for the SCE to outcomes with no censoring, where weights are calculated using the probability that a patient would have survived had the patient been in the other treatment arm. By multiplying the weight to each patient, Kaplan-Meier curves can be created for the SCE to outcomes with censoring. The HR is then calculated using a weighted proportional hazard model. For this method, two assumptions need to be introduced to achieve unbiasedness.

The proposed method is illustrated using data from a randomized Phase II clinical trial, comparing two chemotherapy treatments with radiotherapy in patients with esophageal cancer. Here, we focus on the loco-regional control rate, which is calculated from the time after randomization until recurrence in the radiation field. The duration is undefined for patients who died without recurrence. The proposed method yielded a HR of 1.026 (95% confidence interval (CI): 0.627, 1.677). The standard method, where data of patients who died without progression were regarded as censored at the time of death, yielded a HR of 1.121 (95% CI: 0.688, 1.827).

The proposed method requires two assumptions. As a general problem, unfortunately, whether these assumptions hold cannot be confirmed from the observed data. Thus, we cannot confirm whether the Kaplan-Meier curves and the HR are unbiased.

We have proposed a simple method for the SCE with time-to-event outcomes, which is easy to implement in practice. The proposed method is a potentially valuable supplement to the standard method.