NAG Library Routine Document

Integer, Intent (In)	::	n, ic(n), ifreq(*)
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	nd, iwk(n)
Real (Kind=nag_wp), Intent (In)	::	t(n)
Real (Kind=nag_wp), Intent (Out)	::	tp(n), p(n), psig(n)
Character (1), Intent (In)	::	freq

C Header Interface

#include <nagmk26.h>

void	g12aaf_ (const Integer n, const double t[], const Integer ic[], const char freq, const Integer ifreq[], Integer nd, double tp[], double p[], double psig[], Integer iwk[], Integer ifail, const Charlen length_freq)

3

Description

A survivor function,

S (t)

, is the probability of surviving to at least time

t

with

S (t) = 1 - F (t)

, where

F (t)

is the cumulative distribution function of the failure times. The Kaplan–Meier or product limit estimator provides an estimate of

S (t)

\hat{S} (t)

, from sample of failure times which may be progressively right-censored.

Let

t_{i}

i = 1, 2, \dots, n_{d}

, be the ordered distinct failure times for the sample of observed failure/censored times, and let the number of observations in the sample that have not failed by time

t_{i}

n_{i}

. If a failure and a loss (censored observation) occur at the same time

t_{i}

, then the failure is treated as if it had occurred slightly before time

t_{i}

and the loss as if it had occurred slightly after

t_{i}

The Kaplan–Meier estimate of the survival probabilities is a step function which in the interval

t_{i}

t_{i + 1}

is given by

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

where

d_{j}

is the number of failures occurring at time

t_{j}

g12aaf computes the Kaplan–Meier estimates and the corresponding estimates of the variances,

\hat{var} (\hat{S} (t))

, using Greenwood's formula,

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

4

References

Gross A J and Clark V A (1975) Survival Distributions: Reliability Applications in the Biomedical Sciences Wiley

Kalbfleisch J D and Prentice R L (1980) The Statistical Analysis of Failure Time Data Wiley

5

Arguments

1: $n$ – IntegerInput

On entry: the number of failure and censored times given in t.

Constraint:

n \geq 2

2: $t (n)$ – Real (Kind=nag_wp) arrayInput

On entry: the failure and censored times; these need not be ordered.

3: $ic (n)$ – Integer arrayInput

On entry:

ic (i)

contains the censoring code of the

i

th observation, for

i = 1, 2, \dots, n

$ic (i) = 0$: The $i$ th observation is a failure time.
$ic (i) = 1$: The $i$ th observation is right-censored.

Constraint:

ic (i) = 0

1

, for

i = 1, 2, \dots, n

4: $freq$ – Character(1)Input

On entry: indicates whether frequencies are provided for each time point.

$freq ='F'$: Frequencies are provided for each failure and censored time.
$freq ='S'$: The failure and censored times are considered as single observations, i.e., a frequency of $1$ is assumed.

Constraint:

freq ='F'

'S'

5: $ifreq (*)$ – Integer arrayInput

Note: the dimension of the array ifreq must be at least

n

freq ='F'

and at least

1

freq ='S'

On entry: if

freq ='F'

ifreq (i)

must contain the frequency of the

i

th observation.

ifreq ='S'

, a frequency of

1

is assumed and ifreq is not referenced.

Constraint: if

freq ='F'

ifreq (i) \geq 0

, for

i = 1, 2, \dots, n

6: $nd$ – IntegerOutput

On exit: the number of distinct failure times,

n_{d}

7: $tp (n)$ – Real (Kind=nag_wp) arrayOutput

On exit:

tp (i)

contains the

i

th ordered distinct failure time,

t_{i}

, for

i = 1, 2, \dots, n_{d}

8: $p (n)$ – Real (Kind=nag_wp) arrayOutput

On exit:

p (i)

contains the Kaplan–Meier estimate of the survival probability,

\hat{S} (t)

, for time

tp (i)

, for

i = 1, 2, \dots, n_{d}

9: $psig (n)$ – Real (Kind=nag_wp) arrayOutput

On exit:

psig (i)

contains an estimate of the standard deviation of

p (i)

, for

i = 1, 2, \dots, n_{d}

10: $iwk (n)$ – Integer arrayWorkspace

11: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this argument, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $freq = 〈value〉$ .
Constraint: $freq ='F'$ or $'S'$ .

$ifail = 3$: On entry, $i = 〈value〉$ and $ic (i) = 〈value〉$ .
Constraint: $ic (i) = 0$ or $1$ .

$ifail = 4$: On entry, $i = 〈value〉$ and $ifreq (i) = 〈value〉$ .
Constraint: $ifreq (i) \geq 0$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

The computations are believed to be stable.

8

Parallelism and Performance

g12aaf is not threaded in any implementation.

9

Further Comments

If there are no censored observations,

\hat{S} (t)

reduces to the ordinary binomial estimate of the probability of survival at time

t

10

Example

The remission times for a set of

21

leukaemia patients at

18

distinct time points are read in and the Kaplan–Meier estimate computed and printed. For further details see page 242 of Gross and Clark (1975).

NAG Library Routine Document

g12aaf (kaplanmeier)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results