void	nag_surviv_logrank (Integer n, const double t[], const Integer ic[], const Integer grp[], Integer ngrp, const Integer ifreq[], const double wt[], double ts, Integer df, double p, double obsd[], double expt[], Integer nd, Integer di[], Integer ni[], Integer ldn, NagError *fail)

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 nag_surviv_logrank (g12abc) 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

Arguments

1: $n$ – IntegerInput

On entry:

n

, the number of failure and censored times.

Constraint:

n \geq 2

2: $t [n]$ – const doubleInput

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

Constraint:

t [i - 1] \neq t [j - 1]

for at least one

i \neq j

, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, n

3: $ic [n]$ – const IntegerInput

On entry:

ic [i - 1]

contains the censoring code of the

i

th observation, for

i = 1, 2, \dots, n

$ic [i - 1] = 0$: the $i$ th observation is a failure time.
$ic [i - 1] = 1$: the $i$ th observation is right-censored.

Constraints:

$ic [i - 1] = 0$ or $1$ , for $i = 1, 2, \dots, n$ ;
$ic [i - 1] = 0$ for at least one $i$ .

4: $grp [n]$ – const IntegerInput

On entry:

grp [i - 1]

contains a flag indicating which group the

i

th observation belongs in, for

i = 1, 2, \dots, n

Constraints:

$1 \leq grp [i - 1] \leq ngrp$ , 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: $ifreq [\dim]$ – const IntegerInput

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

n

, unless ifreq is NULL.

On entry: optionally, the frequency (number of observations) that each entry in t corresponds to. If

ifreq is NULL

then each entry in t is assumed to correspond to a single observation, i.e., a frequency of

1

is assumed.

Constraint: if

ifreq is not NULL

ifreq [i - 1] \geq 0

, for

i = 1, 2, \dots, n

7: $wt [\dim]$ – const doubleInput

Note: the dimension, dim, of the array wt must be at least

ldn

, unless wt is NULL.

On entry: optionally, the

n_{d}

weights,

w_{i}

, where

n_{d}

is the number of distinct failure times. If

wt is NULL

then

w_{i} = 1

for all

i

Constraint: if

wt is not NULL

wt [i - 1] \geq 0.0

, for

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

8: $ts$ – double *Output

On exit:

T

, the test statistic.

9: $df$ – Integer *Output

On exit:

ν

, the degrees of freedom.

10: $p$ – double *Output

On exit:

P (X \geq T)

, when

X \sim χ_{ν}^{2}

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

11: $obsd [ngrp]$ – doubleOutput

On exit:

O_{i}

, the observed number of failures in each group.

12: $expt [ngrp]$ – doubleOutput

On exit:

E_{i}

, the expected number of failures in each group.

13: $nd$ – Integer *Output

On exit:

n_{d}

, the number of distinct failure times.

14: $di [ldn]$ – IntegerOutput

On exit: the first nd elements of di contain

d_{i}

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

t_{i}

15: $ni [ldn]$ – IntegerOutput

On exit: the first nd elements of ni contain

n_{i}

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

t_{i}

16: $ldn$ – 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.

17: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_GROUP_OBSERV: On entry, group $〈value〉$ has no observations.
NE_INT: On entry, $ldn = 〈value〉$ .
Constraint: $ldn \geq 〈value〉$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .
NE_INT_2: On entry, $ngrp = 〈value〉$ and $n = 〈value〉$ .
Constraint: $2 \leq ngrp \leq n$ .
NE_INT_ARRAY: On entry, $grp [〈value〉] = 〈value〉$ and $ngrp = 〈value〉$ .
Constraint: $1 \leq grp [i - 1] \leq ngrp$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_CENSOR_CODE: On entry, $ic [〈value〉] = 〈value〉$ .
Constraint: $ic [i - 1] = 0$ or $1$ .
NE_INVALID_FREQ: On entry, $ifreq [〈value〉] = 〈value〉$ .
Constraint: $ifreq [i - 1] \geq 0$ .
NE_NEG_WEIGHT: On entry, $wt [〈value〉] = 〈value〉$ .
Constraint: $wt [i - 1] \geq 0.0$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_OBSERVATIONS: On entry, all observations are censored.
NE_TIME_SERIES_IDEN: On entry, all the times in t are the same.
NE_ZERO_DF: The degrees of freedom are zero.

7

Accuracy

Not applicable.

8

Parallelism and Performance

nag_surviv_logrank (g12abc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_surviv_logrank (g12abc) 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 function. 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 nag_surviv_logrank (g12abc) when

wt is NULL

. 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 nag_surviv_logrank (g12abc) 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 Function Document

nag_surviv_logrank (g12abc)

▸▿ 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