Nonparametric Confidence Intervals for the Median in the Presence of Right Censoring

Abstract

A method is proposed for constructing nonpararnetric confidence intervals for the median of a survival distribution. The method applies when survival data are subject to random right censoring. The confidence interval is constructed by inverting a hypothesis test, the test used being one that generalizes the binomial test to accommodate censoring. A nonparametric estimate of the eventual failure time for each censored observation modifies the usual binomial test statistic. The estimates are computed using the Kaplan-Meier product-limit estimate of the survival function. A large simulation study indicates that the nonparametric method performs well for various survival distributions of different shapes. Two existing techniques that assume exponentiality perform well for exponential data; they give good coverage probabilities with intervals that are shorter than those provided by the nonparametric approach. The nonparametric intervals exhibit superior coverage probabilities when data are generated from nonexponential distributions. This is the case for Weibull data with both increasing and decreasing hazard functions. The nonparametric construction is appropriate for samples of size 10 or greater with up to 50% censoring. A Monte Carlo study supports the use of a binomial distribution to approximate the exact discrete distribution of the test statistic. Median survival time is often used for evaluation of the efficacy of a treatment for a chronic disease. One would like to provide a confidence interval for the median, as well as a point estimate. When little is known about the shape of the survival distribution, an interval is needed for which no particular parametric form for the distribution is assumed. A nonparametric confidence interval for the median is easily formed when all observations are complete. The problem increases in subtlety and complexity when incomplete or censored observations exist. In this paper, a method is proposed for constructing nonparametric confidence intervals for the median when observations may be right-censored. The distribution of survival time may be either discrete or continuous, and it need not have finite moments. The method does assume, however, that the censoring mechanism is noninformative, in that censoring acts independently of survival time; see Lagakos (1979). In the absence of censoring, the usual construction of a nonparametric confidence interval for the median inverts the sign test. This test is based upon the number of failures that occur beyond the hypothesized value of the median. The present paper generalizes the procedure to allow for right censoring, by using the Kaplan-Meier estimate of survival to approximate the number of failures beyond the hypothesized median. Confidence interval estimation when data are censored has received recent attention in