Computes Kaplan-Meier (or product-limit) estimates of survival probabilities for a sample of failure times that possibly contain right consoring.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute] public class KaplanMeierEstimates

Visual Basic (Declaration)
<SerializableAttribute> _ Public Class KaplanMeierEstimates

Visual C++
[SerializableAttribute] public ref class KaplanMeierEstimates

Remarks

Class KaplanMeierEstimates 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 (, $\delta_i$ ), for i = 1,..., n denote the failure censoring times and the censoring codes for the n observations in a single sample. Here, $t_i = x_{i-l, responseIndex}$ is a failure time if $\delta_i$ is 0, where $\delta_i = x_{i-l, censorIndex}$ . Also, is a right censoring time if $\delta_i$ is 1. Rows in x containing values other than 0 or 1 for $\delta_i$ are ignored. Let the number of observations in the sample that have not failed by time $s_{(t)}$ be denoted by $n_{(t)}$ , where $s_{(t)}$ 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

$\hat{S}(t)=\prod_{j=1}^{i}\left ( \frac{n_{(j)}-d_{(j)}}{n_{(j)}} \right )$

where $d_{(j)}$ denotes the number of failures occurring at time $s_{(j)}$ , and $n_{(j)}$ is the number of observations 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 (see FrequencyColumn). 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

$\hat{S}(t)$

in the interval from $s_{(i)}$ to $s_{(i+1)}$ is given as

$\textup{est.var}(\hat{S}(t))=\hat{S}^2(t)\sum_{j=1}^{i}\frac{d_{(j)}}{n_{(j)}(n_{(j)}-d_{(j)})}$

KaplanMeierEstimates 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 stratum column of x, which if present, must contain a distinct code for each sample of observations (see StratumColumn). If a stratum column is not specified, there is no grouping , 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 KaplanMeierEstimates (Sorted=true is not used), KaplanMeierEstimates assumes that the time of censoring for the tied-censored observations is immediately after the tied failure (within the same sample). When Sorted=true is used, the data are assumed to be sorted from smallest to largest according to the response time column of x within each stratum (see ResponseColumn). 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 Sorted=true is used, the user must sort all of the data in x from smallest to largest according to the response time column (and the stratum column, if set). By appropriate sorting of the observations, the user can handle censored and failed observations that are tied in any manner desired.

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.KaplanMeierEstimates

Syntax

Remarks

Inheritance Hierarchy

See Also