Computes Turnbull’s generalized Kaplan‑Meier estimates of survival probabilities in samples with interval censoring.

Routine TRNBL computes nonparametric maximum likelihood estimates of a survival distribution based upon a random sample of data containing exact failure, right‑censored, leftcensored (interval censored with a left endpoint of zero), or interval‑censored observations. The computational method of Turnbull (1976) is used in computing the probability estimates. The model used is also discussed by Peto (1973).

Routine TRNBL begins by finding a set of regions or “failure intervals” (to distinguish them from “observation failure intervals”) on the positive real axis in which a change in the survival probability occurs. The survival probability is constant outside of these regions, and undefined within them. Each region (failure interval) is composed of a single left and a single right endpoint obtained from the left and right endpoints of the observation failure intervals (for exact failure times, the left and right endpoints are equal). The regions are defined by the fact that no observation interval endpoints are allowed within a region, except at its endpoints. Note that the endpoints of the intervals need not correspond to a single observation. Regions defined by endpoints from two distinct observations are often obtained.

Let pi, i = 1, …, NINTVL denote the change in the survival probability within the i‑th region, and let the region be denoted by ci. Let n = NOBS and suppose that the observation failure interval for observation j is denoted by Ij. The EM (expectation, maximization) algorithm of Dempster, Laird and Rubin (1977) is used to find the optimal

The algorithm is defined as follows:

For given

compute the expected contribution of the j‑th observation to the i‑th change interval as

where δij = 1 if ci ⊆ Ij and δij = 0 otherwise, and fj is the observation frequency.

For given expectations

compute the new probability estimate as

Iterate in this manner until convergence. Convergence is assumed when the relative change in the log‑likelihood

is small (less than EPS). Because the algorithm is slow to converge, 5 expectation‑maximization cycles are considered to be one iteration of the algorithm. The initial estimate for all the

is taken to be one divided by the number of regions (failure intervals).

The following example contains exact failure, right-, left-, and interval‑censored observations. The 20 observations yield 15 change intervals. The last interval is from 192 to ∞, and corresponds to a right‑censored observation. When the last interval is infinite, as is the case here, the second column of SPROB contains +∞ in the NINTVL‑th position. Left‑or right‑censored observations input in X are arbitrarily assigned the value 0.0 for the non‑specified endpoint.