Computes Kaplan‑Meier estimates of survival probabilities in stratified samples.

Routine KAPMR computes Kaplan‑Meier (or product‑limit) estimates of survival probabilities for a sample of failure times that possibly contain right censoring. A survival probability S(t) is defined as 1 ‑ F(t), where F(t) is the cumulative distribution function of the failure times (t). Greenwood’s estimate of the standard errors of the survival probability estimates are also computed. (See Kalbfleisch and Prentice, 1980, pages 13 and 14.)

Let (ti, δi), for i = 1, …, n denote the failure/censoring times and the censoring codes for the n observations in a single sample. Here, ti = X(i, IRT) is a failure time if δi is 0, where δi = X(i, ICEN). Also, ti is a censoring time if δi is 1. Rows in X containing values other than 0 or 1 for δi are ignored. Let the number of observations in the sample that have not failed by time s(i) be denoted by n(i), where s(i) is an ordered (from smallest to largest) listing of the distinct failure times (censoring times are omitted). Then the Kaplan‑Meier estimate of the survival probabilities is a step function, which in the interval from s(i) to s(i+1) (including the lower endpoint) is given by

where d(j) denotes the number of failures occurring at time s(j). Note that one row of X may correspond to more than one failed (or censored) observation when the frequency option is in effect (IFRQ is not zero). The Kaplan‑Meier estimate of the survival probability prior to time s(1) is 1.0, while the Kaplan‑Meier estimate of the survival probability after the last failure time is not defined.

Greenwood’s estimate of the variance of

in the interval from s(i) to s(i+1) is given as

Routine KAPMR computes the single sample estimates of the survival probabilities for all samples of data included in X during a single call. This is accomplished through the IGRP column of X, which if present, must contain a distinct code for each sample of observations. If IGRP = 0, there is no grouping column, and all observations are assumed to be from the same sample.

When failures and right‑censored observations are tied and the data are to be sorted by KAPMR (ISRT is not 1), KAPMR assumes that the time of censoring for the tied‑censored observations is immediately after the tied failure (within the same sample). When the ISRT = 1 option is in effect, the data are assumed to be sorted from smallest to largest according to column IRT of X within each stratum. Furthermore, a small increment of time is assumed (theoretically) to elapse between the failed and censored observations that are tied (in the same sample). Thus, when the ISRT = 1 option is in effect, the user must sort all of the data in X from smallest to largest according to column IRT (and column IGRP, if present). By appropriate sorting of the observations, the user can handle censored and failed observations that are tied in any manner desired.

The following example is taken from Kalbfleisch and Prentice (1980, page 1). The first column in X contains the death/censoring times for rats suffering from vaginal cancer. The second column contains information as to which of two forms of treatment were provided, while the third column contains the censoring code. Finally, the fourth column contains the frequency of each observation. The product‑limit estimates of the survival probabilities are computed for both groups with one call to KAPMR. In this example, the output in SPROB has been equivalenced with columns 5 and 6 of X so that the input and output matrices could be printed together. Routine KAPMR could have been called with the ISRT = 1 option in effect if the censored observations had been sorted with respect to the failure time variable.