# NAG FL Interfaceg12abf (logrank)

## 1Purpose

g12abf calculates the rank statistics, which can include the logrank test, for comparing survival curves.

## 2Specification

Fortran Interface
 Subroutine g12abf ( n, t, ic, grp, ngrp, freq, wt, ts, df, p, obsd, expt, nd, di, ni, ldn,
 Integer, Intent (In) :: n, ic(n), grp(n), ngrp, ifreq(*), ldn Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: df, nd, di(ldn), ni(ldn) Real (Kind=nag_wp), Intent (In) :: t(n), wt(*) Real (Kind=nag_wp), Intent (Out) :: ts, p, obsd(ngrp), expt(ngrp) Character (1), Intent (In) :: freq, weight
#include <nag.h>
 void g12abf_ (const Integer *n, const double t[], const Integer ic[], const Integer grp[], const Integer *ngrp, const char *freq, const Integer ifreq[], const char *weight, const double wt[], double *ts, Integer *df, double *p, double obsd[], double expt[], Integer *nd, Integer di[], Integer ni[], const Integer *ldn, Integer *ifail, const Charlen length_freq, const Charlen length_weight)
The routine may be called by the names g12abf or nagf_surviv_logrank.

## 3Description

A survivor function, $S\left(t\right)$, 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
• ${H}_{0}:{S}_{1}\left(t\right)={S}_{2}\left(t\right)=\cdots ={S}_{g}\left(t\right),\forall t\le \tau$
where $\tau$ is the largest observed time, against the alternative hypothesis
• ${H}_{1}:$ at least one of the ${S}_{i}\left(t\right)$ differ, for some $t\le \tau$.
Let ${t}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{n}_{d}$, denote the list of distinct failure times across all $g$ groups and ${w}_{i}$ a series of ${n}_{d}$ weights. Let ${d}_{ij}$ denote the number of failures at time ${t}_{i}$ in group $j$ and ${n}_{ij}$ 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
 $di = ∑ j=1 g d ij and ni = ∑ j=1 g n ij .$
The (weighted) number of observed failures in the $j$th group, ${O}_{j}$, is therefore given by
 $Oj = ∑ i=1 nd wi d ij$
and the (weighted) number of expected failures in the $j$th group, ${E}_{j}$, by
 $Ej = ∑ i=1 nd wi n ij di ni .$
If $x$ denotes the vector of differences $x=\left({O}_{1}-{E}_{1},{O}_{2}-{E}_{2},\dots ,{O}_{g}-{E}_{g}\right)$ and
 $V jk = ∑ i=1 nd w i 2 di ni - di ni n i k I jk - n ij n ik n i 2 ni - 1$
where ${I}_{jk}=1$ if $j=k$ and $0$ otherwise, then the rank statistic, $T$, is calculated as
 $T = x V- xT$
where ${V}^{-}$ denotes a generalized inverse of the matrix $V$. Under the null hypothesis, $T\sim {\chi }_{\nu }^{2}$ where the degrees of freedom, $\nu$, is taken as the rank of the matrix $V$.

## 4References

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

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of failure and censored times.
Constraint: ${\mathbf{n}}\ge 2$.
2: $\mathbf{t}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the observed failure and censored times; these need not be ordered.
Constraint: ${\mathbf{t}}\left(\mathit{i}\right)\ne {\mathbf{t}}\left(\mathit{j}\right)$ for at least one $\mathit{i}\ne \mathit{j}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
3: $\mathbf{ic}\left({\mathbf{n}}\right)$Integer array Input
On entry: ${\mathbf{ic}}\left(\mathit{i}\right)$ contains the censoring code of the $\mathit{i}$th observation, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
${\mathbf{ic}}\left(i\right)=0$
the $i$th observation is a failure time.
${\mathbf{ic}}\left(i\right)=1$
the $i$th observation is right-censored.
Constraints:
• ${\mathbf{ic}}\left(\mathit{i}\right)=0$ or $1$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$;
• ${\mathbf{ic}}\left(i\right)=0$ for at least one $i$.
4: $\mathbf{grp}\left({\mathbf{n}}\right)$Integer array Input
On entry: ${\mathbf{grp}}\left(\mathit{i}\right)$ contains a flag indicating which group the $\mathit{i}$th observation belongs in, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
Constraints:
• $1\le {\mathbf{grp}}\left(\mathit{i}\right)\le {\mathbf{ngrp}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$;
• each group must have at least one observation.
5: $\mathbf{ngrp}$Integer Input
On entry: $g$, the number of groups.
Constraint: $2\le {\mathbf{ngrp}}\le {\mathbf{n}}$.
6: $\mathbf{freq}$Character(1) Input
On entry: indicates whether frequencies are provided for each time point.
${\mathbf{freq}}=\text{'F'}$
Frequencies are provided for each failure and censored time.
${\mathbf{freq}}=\text{'S'}$
The failure and censored times are considered as single observations, i.e., a frequency of $1$ is assumed.
Constraint: ${\mathbf{freq}}=\text{'F'}$ or $\text{'S'}$.
7: $\mathbf{ifreq}\left(*\right)$Integer array Input
Note: the dimension of the array ifreq must be at least ${\mathbf{n}}$ if ${\mathbf{freq}}=\text{'F'}$.
On entry: if ${\mathbf{freq}}=\text{'F'}$, ${\mathbf{ifreq}}\left(i\right)$ must contain the frequency (number of observations) to which each entry in t corresponds.
If ${\mathbf{freq}}=\text{'S'}$, 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 ${\mathbf{freq}}=\text{'F'}$, ${\mathbf{ifreq}}\left(\mathit{i}\right)\ge 0$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
8: $\mathbf{weight}$Character(1) Input
On entry: indicates if weights are to be used.
${\mathbf{weight}}=\text{'U'}$
All weights are assumed to be $1$.
${\mathbf{weight}}=\text{'W'}$
The weights, ${w}_{i}$ are supplied in wt.
Constraint: ${\mathbf{weight}}=\text{'U'}$ or $\text{'W'}$.
9: $\mathbf{wt}\left(*\right)$Real (Kind=nag_wp) array Input
Note: the dimension of the array wt must be at least ${\mathbf{ldn}}$ if ${\mathbf{weight}}=\text{'W'}$.
On entry: if ${\mathbf{weight}}=\text{'W'}$, wt must contain the ${n}_{d}$ weights, ${w}_{i}$, where ${n}_{d}$ is the number of distinct failure times.
If ${\mathbf{weight}}=\text{'U'}$, wt is not referenced and ${w}_{i}=1$ for all $i$.
Constraint: if ${\mathbf{weight}}=\text{'W'}$, ${\mathbf{wt}}\left(\mathit{i}\right)\ge 0.0$, for $\mathit{i}=1,2,\dots ,{n}_{d}$.
10: $\mathbf{ts}$Real (Kind=nag_wp) Output
On exit: $T$, the test statistic.
11: $\mathbf{df}$Integer Output
On exit: $\nu$, the degrees of freedom.
12: $\mathbf{p}$Real (Kind=nag_wp) Output
On exit: $P\left(X\ge T\right)$, when $X\sim {\chi }_{\nu }^{2}$, i.e., the probability associated with ts.
13: $\mathbf{obsd}\left({\mathbf{ngrp}}\right)$Real (Kind=nag_wp) array Output
On exit: ${O}_{i}$, the observed number of failures in each group.
14: $\mathbf{expt}\left({\mathbf{ngrp}}\right)$Real (Kind=nag_wp) array Output
On exit: ${E}_{i}$, the expected number of failures in each group.
15: $\mathbf{nd}$Integer Output
On exit: ${n}_{d}$, the number of distinct failure times.
16: $\mathbf{di}\left({\mathbf{ldn}}\right)$Integer array Output
On exit: the first nd elements of di contain ${d}_{i}$, the number of failures, across all groups, at time ${t}_{i}$.
17: $\mathbf{ni}\left({\mathbf{ldn}}\right)$Integer array Output
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: $\mathbf{ldn}$Integer Input
On entry: the size of arrays di and ni. As ${n}_{d}\le n$, if ${n}_{d}$ is not known a priori then a value of n can safely be used for ldn.
Constraint: ${\mathbf{ldn}}\ge {n}_{d}$, the number of unique failure times.
19: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, all the times in t are the same.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ic}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ic}}\left(i\right)=0$ or $1$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{grp}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{ngrp}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{grp}}\left(i\right)\le {\mathbf{ngrp}}$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{ngrp}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $2\le {\mathbf{ngrp}}\le {\mathbf{n}}$.
${\mathbf{ifail}}=6$
On entry, freq had an illegal value.
${\mathbf{ifail}}=7$
On entry, ${\mathbf{ifreq}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ifreq}}\left(i\right)\ge 0$.
${\mathbf{ifail}}=8$
On entry, weight had an illegal value.
${\mathbf{ifail}}=9$
On entry, ${\mathbf{wt}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{wt}}\left(i\right)\ge 0.0$.
${\mathbf{ifail}}=11$
The degrees of freedom are zero.
${\mathbf{ifail}}=18$
On entry, ${\mathbf{ldn}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldn}}\ge 〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=31$
On entry, all observations are censored.
${\mathbf{ifail}}=41$
On entry, group $〈\mathit{\text{value}}〉$ has no observations.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism 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.

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 ${\mathbf{weight}}=\text{'U'}$. Other rank statistics include Wilcoxon (${w}_{i}={n}_{i}$), Tarone–Ware (${w}_{i}=\sqrt{{n}_{i}}$) and Peto–Peto (${w}_{i}=\stackrel{~}{S}\left({t}_{i}\right)$ where $\stackrel{~}{S}\left({t}_{i}\right)=\prod _{{t}_{j}\le {t}_{i}}\phantom{\rule{0.25em}{0ex}}\frac{{n}_{j}-{d}_{j}+1}{{n}_{j}+1}$) 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.

## 10Example

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).

### 10.1Program Text

Program Text (g12abfe.f90)

### 10.2Program Data

Program Data (g12abfe.d)

### 10.3Program Results

Program Results (g12abfe.r)