KaplanMeierEstimates Class

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

Inheritance Hierarchy

SystemObject
Imsl.StatKaplanMeierEstimates

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class KaplanMeierEstimates

<SerializableAttribute>
Public Class KaplanMeierEstimates

[SerializableAttribute]
public ref class KaplanMeierEstimates

[<SerializableAttribute>]
type KaplanMeierEstimates =  class end

The KaplanMeierEstimates type exposes the following members.

Constructors

	Name	Description
	KaplanMeierEstimates	Constructor for KaplanMeierEstimates.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetGroupTotal	Returns the total number in the group for the specified group value.
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetLogLikelihood	Returns the Kaplan-Meier log-likelihood of the group with the specified group value.
	GetNumberAtRisk	Returns the number of individuals at risk at each failure point.
	GetNumberOfFailures	Returns the number of failures which occurred at each failure point.
	GetStandardErrors	Returns Greenwood's estimated standard errors.
	GetSurvivalProbabilities	Returns the estimated survival probabilities.
	GetTotalNumberOfFailures	Returns the total number failing in the group for the specified group value.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	CensorColumn	The column index of x containing the optional censoring code for each observation.
	FrequencyColumn	The column index of x containing the frequency of response for each observation.
	NumberOfRowsMissing	The number of rows of data in x that contain missing values in one or more specific columns of x.
	ResponseColumn	The column index of x containing the response time for each observation.
	Sorted	The boolean which indicates that the column of response times in x are already sorted.
	StratumColumn	The column index of x containing the stratum number for each observation.

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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.

Reference

Imsl.Stat Namespace

Other Resources

Example