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 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

nag_surviv_logrank (g12abc) calculates the rank statistics, which can include the logrank test, for comparing survival curves.

2 Specification

#include <nag.h>

#include <nagg12.h>

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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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.
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_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 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.

9 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 Documentnag_surviv_logrank (g12abc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_surviv_logrank (g12abc)