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

The type double function is imsls_d_kaplan_meier_estimates.

Pointer to an array of length n_observations×2. The first column contains the estimated survival probabilities, and the second column contains Greenwood’s estimate of the standard deviation of these probabilities. If the i-th observation contains censor codes out of range or if a variable is missing, then the corresponding elements of the return value are set to missing (NaN, not a number). Similarly, if an element in the return value is not defined, then it is set to missing.

Function imsls_f_kaplan_meier_estimates computes Kaplan-Meier (or product-limit) estimates of survival probabilities for a sample of failure times that can be right censored or exact times. 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 = xi-1, irt is a failure time if δi is 0, where δI = xi-1, icen. Also, ti is a right 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), and n(j) is the number of observation that have not failed prior to 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 specified). 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

Function imsls_f_kaplan_meier_estimates 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 is not specified, there is no grouping column, and all observations are assumed to come from the same sample.

When failures and right-censored observations are tied and the data are to be sorted by imsls_f_kaplan_meier_estimates (IMSLS_SORTED optional argument is not used), imsls_f_kaplan_meier_estimates assumes that the time of censoring for the tied-censored observations is immediately after the tied failure (within the same sample). When the IMSLS_SORTED optional argument is used, 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 IMSLS_SORTED optional argument is used, 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 IMSLS_PRINT option prints life tables. One table for each stratum is printed. In addition to the survival probabilities at each failure point, the following is also printed: the number of individuals remaining at risk, Greenwood’s estimate of the standard errors for the survival probabilities, and the Kaplan-Meier log-likelihood. The Kaplan-Meier log-likelihood is computed as:

where the sum is with respect to the distinct failure times s(j), d(j).

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 imsls_f_kaplan_meier_estimates.

Function imsls_f_kaplan_meier_estimates could have been called with the IMSLS_SORTED optional argument if the censored observations had been sorted with respect to the failure time variable. IMSLS_PRINT option is used to print the life tables.