The function may be called by the names: g12abc or nag_surviv_logrank.
3Description
A survivor function, , is the probability of surviving to at least time . Given a series of failure or right-censored times from groups g12abc calculates a rank statistic for testing the null hypothesis
where is the largest observed time, against the alternative hypothesis
at least one of the
differ, for some
.
Let
, for , denote the list of distinct failure times across all groups and a series of weights. Let denote the number of failures at time in group and denote the number of observations in the group that are known to have not failed prior to time , i.e., the size of the risk set for group at time . If a censored observation occurs at time then that observation is treated as if the censoring had occurred slightly after and, therefore, the observation is counted as being part of the risk set at time . Finally let
The (weighted) number of observed failures in the th group, , is, therefore, given by
and the (weighted) number of expected failures in the th group, , by
If denotes the vector of differences
and
where
if and otherwise, then the rank statistic, , is calculated as
where denotes a generalized inverse of the matrix . Under the null hypothesis,
where the degrees of freedom, , is taken as the rank of the matrix .
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 Neurosurgery35 378
5Arguments
1: – IntegerInput
On entry: , the number of failure and censored times.
Constraint:
.
2: – const doubleInput
On entry: the observed failure and censored times; these need not be ordered.
Constraint:
for at least one , for and .
3: – const IntegerInput
On entry: contains the censoring code of the th observation, for .
the th observation is a failure time.
the th observation is right-censored.
Constraints:
or , for ;
for at least one .
4: – const IntegerInput
On entry: contains a flag indicating which group the th observation belongs in, for .
Constraints:
, for ;
each group must have at least one observation.
5: – IntegerInput
On entry: , the number of groups.
Constraint:
.
6: – const IntegerInput
Note: the dimension, dim, of the array ifreq
must be at least
, when .
On entry: optionally, the frequency (number of observations) that each entry in t corresponds to. If then each entry in t is assumed to correspond to a single observation, i.e., a frequency of is assumed.
Constraint:
if ,
, for .
7: – const doubleInput
Note: the dimension, dim, of the array wt
must be at least
, when .
On entry: optionally, the weights, , where is the number of distinct failure times. If then for all .
Constraint:
if ,
, for .
8: – double *Output
On exit: , the test statistic.
9: – Integer *Output
On exit: , the degrees of freedom.
10: – double *Output
On exit: , when , i.e., the probability associated with ts.
11: – doubleOutput
On exit: , the observed number of failures in each group.
12: – doubleOutput
On exit: , the expected number of failures in each group.
13: – Integer *Output
On exit: , the number of distinct failure times.
14: – IntegerOutput
On exit: the first nd elements of di contain , the number of failures, across all groups, at time .
15: – IntegerOutput
On exit: the first nd elements of ni contain , the size of the risk set, across all groups, at time .
16: – IntegerInput
On entry: the size of arrays di and ni. As , if is not known a priori then a value of n can safely be used for ldn.
Constraint:
, the number of unique failure times.
17: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument had an illegal value.
NE_GROUP_OBSERV
On entry, group has no observations.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
NE_INT_2
On entry, and .
Constraint: .
NE_INT_ARRAY
On entry, and .
Constraint: .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_CENSOR_CODE
On entry, .
Constraint: or .
NE_INVALID_FREQ
On entry, .
Constraint: .
NE_NEG_WEIGHT
On entry, .
Constraint: .
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Background information to multithreading can be found in the Multithreading documentation.
g12abc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.
9Further Comments
The use of different weights in the formula given in Section 3 leads to different rank statistics being calculated. The logrank test has , for all , which is the equivalent of calling g12abc when . Other rank statistics include Wilcoxon (), Tarone–Ware () and Peto–Peto ( where ) amongst others.
Calculation of any test, other than the logrank test, will probably require g12abc to be called twice, once to calculate the values of and 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 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).