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I am trying to understand the bootstrapping procedure for right-censored failure time data. One of the bootstrapping methods is based on sampling pairs of failure/censoring times from the non-parametric Kaplan-Meier estimators for the failure time distribution and the censoring time distribution. The bootstrap sample is then constructed by only recording the minimums of the failure/censoring times and their associated minimum indicators.

Question: Given a Kaplan-Meier curve (not necessarily a proper CDF due to right-censoring), how do you draw random times according to this estimator? If it were a proper CDF I would sample the failure times according to their respective masses but I am unsure how to handle the right-tail if it does not equal 0.

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As I understand the question, you aren't supposed to sample from the Kaplan-Meier estimator. You're supposed to sample from the data that went into producing that estimator. You don't sample random times; you sample random observations.

That's what bootstrapping is. You sample with replacement from an empirical distribution. Even if the probability distribution associated with the Kaplan-Meier estimator is improper, the empirical distributions of event and censoring times that went into it are proper.

Implementing the proposed scheme is then straightforward. You first identify, for each case contributing to the Kaplan-Meier curve, the actual event or censoring time. You separate out the censored and uncensored groups. Then, at each iteration, sample one from the censored group and one from the uncensored. Choose the lowest time of the pair and mark it with its censored/event status. Repeat until you have as many samples as there were cases in the original data.

Sampling cases instead of sampling from the CDF of times in this context makes further sense, given that much inference on Kaplan-Meier non-parametric and Cox semi-parametric models doesn't depend on the time scale. What matters when comparing such curves is the order of the observations, not the actual times of the observations. You can stretch or shrink the time scale between observation times and still get the same results of comparing 2 curves, although things like median survival-time estimates would change.

That's unlike working with a truly parametric survival model. If you had a truly parametric model you could then sample from the full CDF, a proper distribution, in time as you wish. But that wouldn't be bootstrapping.

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