KAPLAN MEIER PLOT

KAPLAN MEIER PLOT

Graphics Command

Generates a Kaplan Meier plot.

In reliability analysis, many data sets consists of a set of failure times, which may be truncated at some limit value. The cumulative distribution function (CDF) is defined as:
F(t) = prob(T < t)

where T is the lifetime of a randomly selected unit.

Given n units, which are ordered from smallest to largest, t₁, t₂, ... , t_n where t_i can represent either a failure time or a censoring time (i.e., the unit was removed from the test before failing), then the Kaplan Meier estimates are given by:

\( \hat{R}(t_i) = \prod_{\begin{array}{c} j=1 \\ j \in S \end{array}}^{i}{\frac{n-j}{n-j+1}} \)

where S is the set of all subscripts j such that t_j) is a failure time (i.e., S excludes the censoring times). That is, for index i, the product is for all indices less than or equal to i that represent actual failure times.

The Kaplan-Meier plot is a plot of \( \hat{R}(t_i) \) versus the failure time.

Once \( \hat{R} \) is computed, then the CDF estimates are

F(t_i) = 1 - R(t_i)

That is, the Kaplan-Meier estimates are a way to estimate the CDF function when you have censored data.

The Kaplan-Meier estimate of the last failure time is zero, which results in a CDF value of 1. Since the reliability (= 1 - CDF) for standard reliability models asymptotically approaches 1 as time approaches infinity, a modified Kaplan-Meier estimate has been developed:

\( \hat{R}(t_i) = \frac{n+0.7}{n+0.4} \prod_{\begin{array}{c} j=1 \\ j \in S \end{array}}^{i}{\frac{n-j}{n-j+1}} \)

Generally, the modified form of the Kaplan-Meier plot is preferred.

The Kaplan-Meier plot can be thought of as an alternative to the empirical CDF plot that can handle data with both failure and censoring times.

In Dataplot, a tag variable identifies whether the corresponding points in the response variable represent failure times or censoring times. A value of 1 indicates a failure time and a value of 0 indicates a censoring time.

KAPLAN MEIER PLOT <y> <tag>             <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable containing failure times;
            <tag> is a tag variable indicating whether the times in <y> are failure times or censoring times;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax plots the unmodified Kaplan-Meier estimates.

MODIFIED KAPLAN MEIER PLOT <y> <tag>             <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable containing failure times;
            <tag> is a tag variable indicating whether the times in <y> are failure times or censoring times;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax plots the modified Kaplan-Meier estimates.

KAPLAN MEIER PLOT Y1 CENSOR
MODIFIED KAPLAN MEIER PLOT Y1 CENSOR

By default, the vertical axis of the Kaplan-Meier plot represents reliability (or survival). Some analysts prefer to plot the CDF on the vertical axis (i.e., 1 - Rhat).

Enter the following command to plot 1 - Rhat:

SET KAPLAN MEIER CDF

Enter the following command to reset the default (plot Rhat):

SET KAPLAN MEIER RELIABILITY

If you want the numeric values of the Kaplan-Meier estimates, do the following:
MODIFIED KAPLAN-MEIER PLOT Y CENSOR
LET RELI = YPLOT
LET FAILTIME = XPLOT


The variables RELI and FAILTIME can then be used in subsequent analysis and output.

None

None

LINES	= Sets the type for plot lines.
CHARACTERS	= Sets the type for plot characters.
EMPIRICAL CDF PLOT	= Generates an empirical cdf plot.
TAIL AREA PLOT	= Generates a tail area plot.
... HAZARD PLOT	= Generates a hazard plot.
PROBABILITY PLOT	= Generates a probability plot.
PLOT	= Generates a data or function plot.

Reliability

1998/8

SKIP 25
READ HAHN.DAT MILES TAG
TITLE MODIFIED KAPLAN MEIER PLOT OF HAHN.DAT
Y1LABEL SURVIVAL
X1LABEL FAILURE TIME
XLIMITS 0 150000
MODIFIED KAPLAN MEIER PLOT MILES TAG