Two-Sample Inference for Median Survival Times Based on One-Sample Procedures for Censored Survival Data

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Abstract
Confidence intervals for median survival times are derived for censored survival data. The intervals are obtained by using the quantiles of the Kaplan-Meier product-limit estimator and have the same Pitman efficiency as the intervals found by inverting the sign test. Two-sample tests and confidence intervals for the difference in median survival times are then developed based on the comparison of the one-sample confidence intervals. Several methods for choosing the confidence coefficients of the corresponding one-sample confidence intervals are developed. The Pitman efficiencies of these two-sample tests are the same as that of the median test proposed by Brookmeyer and Crowley (1982a). The procedures can also be used for the accelerated failure time model, proportional hazard model, and the Behrens-Fisher problem. Nonparametric two-sample inference procedures are useful in comparing the responses of treatment and control groups. In medical follow-up studies the data are usually subjected to censoring. Although many of the two-sample procedures have been extended to accommodate censored data, we show in this article how to construct two-sample tests and confidence intervals based on one-sample confidence intervals. This method was first discussed in Hettmansperger (1984a) for uncensored data where a confidence interval for the difference in population medians is constructed by subtracting the endpoints of one (one-sample) interval from the opposite endpoints of the other. The test then rejects the null hypothesis of equal medians if the one-sample intervals are disjoint. Hettmansperger used the sign interval as the one-sample interval, which is obtained by inverting the sign test. In the presence of censoring we modify the sign interval to the so-called quantile interval, whose endpoints are the quantiles of the product-limit estimator of Kaplan and Meier (1958). After deriving the asymptotic properties of the quantile interval, we show, for specified overall level α, three ways to select the confidence coefficients for the one-sample quantile intervals. For α = .05 and equal confidence coefficients for both samples, the one sample confidence coefficients are in the neighborhood of .85 under the Koziol-Green model, compared with .975 had Bonferroni inequality or independence of the two samples been used. The procedures are then applied to data from a colorectal cancer clinical trial to compare four treatments, and they appear to be quite robust.

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