NAG Library Routine Document

G12ABF

INTEGER	N, IC(N), GRP(N), NGRP, IFREQ(*), DF, ND, DI(LDN), NI(LDN), LDN, IFAIL
REAL (KIND=nag_wp)	T(N), WT(*), TS, P, OBSD(NGRP), EXPT(NGRP)
CHARACTER(1)	FREQ, WEIGHT

3 Description

A survivor function,

S (t)

, is the probability of surviving to at least time

t

. Given a series of

n

failure or right-censored times from

g

groups G12ABF calculates a rank statistic for testing the null hypothesis

$H_{0} : S_{1} (t) = S_{2} (t) = \dots = S_{g} (t), \forall t \leq τ$

where

τ

is the largest observed time, against the alternative hypothesis

$H_{1} :$ at least one of the $S_{i} (t)$ differ, for some $t \leq τ$ .

Let

t_{i}

, for

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

, denote the list of distinct failure times across all

g

groups and

w_{i}

a series of

n_{d}

weights. Let

d_{i j}

denote the number of failures at time

t_{i}

in group

j

and

n_{i j}

denote the number of observations in the group

j

that are known to have not failed prior to time

t_{i}

, i.e., the size of the risk set for group

j

at time

t_{i}

. If a censored observation occurs at time

t_{i}

then that observation is treated as if the censoring had occurred slightly after

t_{i}

and therefore the observation is counted as being part of the risk set at time

t_{i}

. Finally let

d_{i} = \sum_{j = 1}^{g} d_{i j} and n_{i} = \sum_{j = 1}^{g} n_{i j} .

The (weighted) number of observed failures in the

j

th group,

O_{j}

, is therefore given by

O_{j} = \sum_{i = 1}^{n_{d}} w_{i} d_{i j}

and the (weighted) number of expected failures in the

j

th group,

E_{j}

, by

E_{j} = \sum_{i = 1}^{n_{d}} w_{i} \frac{n_{i j} d_{i}}{n_{i}} .

x

denotes the vector of differences

x = (O_{1} - E_{1}, O_{2} - E_{2}, \dots, O_{g} - E_{g})

and

V_{j k} = \sum_{i = 1}^{n_{d}} w_{i}^{2} (\frac{d_{i} (n_{i} - d_{i}) (n_{i} n_{i k} I_{j k} - n_{i j} n_{i k})}{n_{i}^{2} (n_{i} - 1)})

where

I_{j k} = 1

j = k

and

0

otherwise, then the rank statistic,

T

, is calculated as

T = x V^{-} x^{T}

where

V^{-}

denotes a generalized inverse of the matrix

V

. Under the null hypothesis,

T \sim χ_{ν}^{2}

where the degrees of freedom,

ν

, is taken as the rank of the matrix

V

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

Rostomily R C, Duong D, McCormick K, Bland M and Berger M S (1994) Multimodality management of recurrent adult malignant gliomas: results of a phase II multiagent chemotherapy study and analysis of cytoreductive surgery Neurosurgery 35 378

5 Parameters

1: $N$ – INTEGERInput

On entry:

n

, the number of failure and censored times.

Constraint:

N \geq 2

2: $T (N)$ – REAL (KIND=nag_wp) arrayInput

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

Constraint:

T (i) \neq T (j)

for at least one

i \neq j

, for

i = 1, 2, \dots, N

and

j = 1, 2, \dots, N

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.

Constraints:

$IC (i) = 0$ or $1$ , for $i = 1, 2, \dots, N$ ;
$IC (i) = 0$ for at least one $i$ .

4: $GRP (N)$ – INTEGER arrayInput

On entry:

GRP (i)

contains a flag indicating which group the

i

th observation belongs in, for

i = 1, 2, \dots, N

Constraints:

$1 \leq GRP (i) \leq NGRP$ , for $i = 1, 2, \dots, N$ ;
each group must have at least one observation.

5: $NGRP$ – INTEGERInput

On entry:

g

, the number of groups.

Constraint:

2 \leq NGRP \leq N

6: $FREQ$ – 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'

7: $IFREQ (*)$ – INTEGER arrayInput

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

N

FREQ ='F'

On entry: if

FREQ ='F'

IFREQ (i)

must contain the frequency (number of observations) to which each entry in T corresponds.

FREQ ='S'

, each entry in T is assumed to correspond to a single observation, i.e., 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

8: $WEIGHT$ – CHARACTER(1)Input

On entry: indicates if weights are to be used.

$WEIGHT ='U'$: All weights are assumed to be $1$ .
$WEIGHT ='W'$: The weights, $w_{i}$ are supplied in WT.

Constraint:

WEIGHT ='U'

'W'

9: $WT (*)$ – REAL (KIND=nag_wp) arrayInput

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

LDN

WEIGHT ='W'

On entry: if

WEIGHT ='W'

, WT must contain the

n_{d}

weights,

w_{i}

, where

n_{d}

is the number of distinct failure times.

WEIGHT ='U'

, WT is not referenced and

w_{i} = 1

for all

i

Constraint: if

WEIGHT ='W'

WT (i) \geq 0.0

, for

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

10: $TS$ – REAL (KIND=nag_wp)Output

On exit:

T

, the test statistic.

11: $DF$ – INTEGEROutput

On exit:

ν

, the degrees of freedom.

12: $P$ – REAL (KIND=nag_wp)Output

On exit:

P (X \geq T)

, when

X \sim χ_{ν}^{2}

, i.e., the probability associated with TS.

13: $OBSD (NGRP)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

O_{i}

, the observed number of failures in each group.

14: $EXPT (NGRP)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

E_{i}

, the expected number of failures in each group.

15: $ND$ – INTEGEROutput

On exit:

n_{d}

, the number of distinct failure times.

16: $DI (LDN)$ – INTEGER arrayOutput

On exit: the first ND elements of DI contain

d_{i}

, the number of failures, across all groups, at time

t_{i}

17: $NI (LDN)$ – INTEGER arrayOutput

On exit: the first ND elements of NI contain

n_{i}

, the size of the risk set, across all groups, at time

t_{i}

18: $LDN$ – INTEGERInput

On entry: the size of arrays DI and NI. As

n_{d} \leq n

, if

n_{d}

is not known a priori then a value of N can safely be used for LDN.

Constraint:

LDN \geq n_{d}

, the number of unique failure times.

19: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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, all the times in T are the same.

$IFAIL = 3$: On entry, $IC (〈value〉) = 〈value〉$ .
Constraint: $IC (i) = 0$ or $1$ .

$IFAIL = 4$: On entry, $GRP (〈value〉) = 〈value〉$ and $NGRP = 〈value〉$ .
Constraint: $1 \leq GRP (i) \leq NGRP$ .

$IFAIL = 5$: On entry, $NGRP = 〈value〉$ and $N = 〈value〉$ .
Constraint: $2 \leq NGRP \leq N$ .

$IFAIL = 6$: On entry, FREQ had an illegal value.

$IFAIL = 7$: On entry, $IFREQ (〈value〉) = 〈value〉$ .
Constraint: $IFREQ (i) \geq 0$ .

$IFAIL = 8$: On entry, WEIGHT had an illegal value.

$IFAIL = 9$: On entry, $WT (〈value〉) = 〈value〉$ .
Constraint: $WT (i) \geq 0.0$ .

$IFAIL = 11$: The degrees of freedom are zero.

$IFAIL = 18$: On entry, $LDN = 〈value〉$ .
Constraint: $LDN \geq 〈value〉$ .

$IFAIL = 31$: On entry, all observations are censored.

$IFAIL = 41$: On entry, group $〈value〉$ has no observations.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

G12ABF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G12ABF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The use of different weights in the formula given in Section 3 leads to different rank statistics being calculated. The logrank test has

w_{i} = 1

, for all

i

, which is the equivalent of calling G12ABF when

WEIGHT ='U'

. Other rank statistics include Wilcoxon (

w_{i} = n_{i}

), Tarone–Ware (

w_{i} = \sqrt{n_{i}}

) and Peto–Peto (

w_{i} = \tilde{S} (t_{i})

where

\tilde{S} (t_{i}) = \prod_{t_{j} \leq t_{i}} \frac{n_{j} - d_{j} + 1}{n_{j} + 1}

) amongst others.

Calculation of any test, other than the logrank test, will probably require G12ABF to be called twice, once to calculate the values of

n_{i}

and

d_{i}

to facilitate in the computation of the required weights, and once to calculate the test statistic itself.

10 Example

This example compares the time to death for

51

adults with two different types of recurrent gliomas (brain tumour), astrocytoma and glioblastoma, using a logrank test. For further details on the data see Rostomily et al. (1994).

NAG Library Routine DocumentG12ABF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

G12ABF