NAG Library Routine Document

g12abf (logrank)


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


Fortran Interface
Subroutine g12abf ( n, t, ic, grp, ngrp, freq, ifreq, weight, wt, ts, df, p, obsd, expt, nd, di, ni, ldn, ifail)
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
C Header Interface
#include <nagmk26.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)


A survivor function, St, 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 where τ is the largest observed time, against the alternative hypothesis
Let t i , for i=1,2,,nd, denote the list of distinct failure times across all g groups and wi a series of nd weights. Let dij denote the number of failures at time ti in group j and nij denote the number of observations in the group j that are known to have not failed prior to time ti, i.e., the size of the risk set for group j at time ti. If a censored observation occurs at time ti then that observation is treated as if the censoring had occurred slightly after ti and therefore the observation is counted as being part of the risk set at time ti. Finally let
di = j=1 g d ij   and   ni = j=1 g n ij .  
The (weighted) number of observed failures in the jth group, Oj, is therefore given by
Oj = i=1 nd wi d ij  
and the (weighted) number of expected failures in the jth group, Ej, by
Ej = i=1 nd wi n ij di ni .  
If x denotes the vector of differences x = O1 - E1 , O2 - E2 , , Og - Eg  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 χ ν 2  where the degrees of freedom, ν, is taken as the rank of the matrix V.


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


1:     n – IntegerInput
On entry: n, the number of failure and censored times.
Constraint: n2.
2:     tn – Real (Kind=nag_wp) arrayInput
On entry: the observed failure and censored times; these need not be ordered.
Constraint: titj for at least one ij, for i=1,2,,n and j=1,2,,n.
3:     icn – Integer arrayInput
On entry: ici contains the censoring code of the ith observation, for i=1,2,,n.
the ith observation is a failure time.
the ith observation is right-censored.
  • ici=0 or 1, for i=1,2,,n;
  • ici=0 for at least one i.
4:     grpn – Integer arrayInput
On entry: grpi contains a flag indicating which group the ith observation belongs in, for i=1,2,,n.
  • 1grpingrp, for i=1,2,,n;
  • each group must have at least one observation.
5:     ngrp – IntegerInput
On entry: g, the number of groups.
Constraint: 2ngrpn.
6:     freq – Character(1)Input
On entry: indicates whether frequencies are provided for each time point.
Frequencies are provided for each failure and censored time.
The failure and censored times are considered as single observations, i.e., a frequency of 1 is assumed.
Constraint: freq='F' or 'S'.
7:     ifreq* – Integer arrayInput
Note: the dimension of the array ifreq must be at least n if freq='F'.
On entry: if freq='F', ifreqi must contain the frequency (number of observations) to which each entry in t corresponds.
If 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', ifreqi0, for i=1,2,,n.
8:     weight – Character(1)Input
On entry: indicates if weights are to be used.
All weights are assumed to be 1.
The weights, wi are supplied in wt.
Constraint: weight='U' or 'W'.
9:     wt* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array wt must be at least ldn if weight='W'.
On entry: if weight='W', wt must contain the nd weights, wi, where nd is the number of distinct failure times.
If weight='U', wt is not referenced and wi=1 for all i.
Constraint: if weight='W', wti0.0, for i=1,2,,nd.
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: PXT, when Xχν2, i.e., the probability associated with ts.
13:   obsdngrp – Real (Kind=nag_wp) arrayOutput
On exit: Oi, the observed number of failures in each group.
14:   exptngrp – Real (Kind=nag_wp) arrayOutput
On exit: Ei, the expected number of failures in each group.
15:   nd – IntegerOutput
On exit: nd, the number of distinct failure times.
16:   dildn – Integer arrayOutput
On exit: the first nd elements of di contain di, the number of failures, across all groups, at time ti.
17:   nildn – Integer arrayOutput
On exit: the first nd elements of ni contain ni, the size of the risk set, across all groups, at time ti.
18:   ldn – IntegerInput
On entry: the size of arrays di and ni. As ndn, if nd is not known a priori then a value of n can safely be used for ldn.
Constraint: ldnnd, 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 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).

Error Indicators and Warnings

If on entry 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:
On entry, n=value.
Constraint: n2.
On entry, all the times in t are the same.
On entry, icvalue=value.
Constraint: ici=0 or 1.
On entry, grpvalue=value and ngrp=value.
Constraint: 1grpingrp.
On entry, ngrp=value and n=value.
Constraint: 2ngrpn.
On entry, freq had an illegal value.
On entry, ifreqvalue=value.
Constraint: ifreqi0.
On entry, weight had an illegal value.
On entry, wtvalue=value.
Constraint: wti0.0.
The degrees of freedom are zero.
On entry, ldn=value.
Constraint: ldnvalue.
On entry, all observations are censored.
On entry, group value has no observations.
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.
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.
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.


Not applicable.

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.

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 wi=1, for all i, which is the equivalent of calling g12abf when weight='U' . Other rank statistics include Wilcoxon (wi=ni), Tarone–Ware (wi=ni) and Peto–Peto ( wi = S~ ti  where S~ ti = tj ti nj - dj + 1 nj+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 ni and di to facilitate in the computation of the required weights, and once to calculate the test statistic itself.


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).

Program Text

Program Text (g12abfe.f90)

Program Data

Program Data (g12abfe.d)

Program Results

Program Results (g12abfe.r)