NAG Library Routine Document
G02BRF
1 Purpose
G02BRF computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data, omitting completely any cases with a missing observation for any variable; the data array is preserved, and the ranks of the observations are not available on exit from the routine.
2 Specification
SUBROUTINE G02BRF ( |
N, M, X, LDX, MISS, XMISS, ITYPE, RR, LDRR, NCASES, INCASE, KWORKA, KWORKB, KWORKC, WORK1, WORK2, IFAIL) |
INTEGER |
N, M, LDX, MISS(M), ITYPE, LDRR, NCASES, INCASE(N), KWORKA(N), KWORKB(N), KWORKC(N), IFAIL |
REAL (KIND=nag_wp) |
X(LDX,M), XMISS(M), RR(LDRR,M), WORK1(N), WORK2(N) |
|
3 Description
The input data consists of
observations for each of
variables, given as an array
where
is the
th observation on the
th variable. In addition, each of the
variables may optionally have associated with it a value which is to be considered as representing a missing observation for that variable; the missing value for the
th variable is denoted by
. Missing values need not be specified for all variables.
Let
if observation
contains a missing value for any of those variables for which missing values have been declared, i.e., if
for any
for which an
has been assigned (see also
Section 7); and
otherwise, for
.
The observations are first ranked as follows.
For a given variable, say, each of the observations for which , () has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitudes of the other observations on that same variable for which .
The smallest of these valid observations for variable is assigned the rank , the second smallest observation for variable the rank , the third smallest the rank , and so on until the largest such observation is given the rank , where .
If a number of cases all have the same value for the given variable,
, then they are each given an ‘average’ rank, e.g., if in attempting to assign the rank
,
observations for which
were found to have the same value, then instead of giving them the ranks
all
observations would be assigned the rank
and the next value in ascending order would be assigned the rank
The process is repeated for each of the
variables.
Let be the rank assigned to the observation when the th variable is being ranked. For those observations, , for which , , for .
The quantities calculated are:
(a) |
Kendall's tau rank correlation coefficients:
where |
|
and |
if |
|
if |
|
if |
and where is the number of ties of a particular value of variable , and the summation is over all tied values of variable . |
(b) |
Spearman's rank correlation coefficients:
where and where is the number of ties of a particular value of variable , and the summation is over all tied values of variable . |
4 References
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
5 Parameters
- 1: – INTEGERInput
-
On entry: , the number of observations or cases.
Constraint:
.
- 2: – INTEGERInput
-
On entry: , the number of variables.
Constraint:
.
- 3: – REAL (KIND=nag_wp) arrayInput
-
On entry: must be set to , the value of the th observation on the th variable, where and
- 4: – INTEGERInput
-
On entry: the first dimension of the array
X as declared in the (sub)program from which G02BRF is called.
Constraint:
.
- 5: – INTEGER arrayInput/Output
-
On entry:
must be set equal to
if a missing value,
, is to be specified for the
th variable in the array
X, or set equal to
otherwise. Values of
MISS must be given for all
variables in the array
X.
On exit: the array
MISS is overwritten by the routine, and the information it contained on entry is lost.
- 6: – REAL (KIND=nag_wp) arrayInput/Output
-
On entry:
must be set to the missing value,
, to be associated with the
th variable in the array
X, for those variables for which missing values are specified by means of the array
MISS (see
Section 7).
On exit: the array
XMISS is overwritten by the routine, and the information it contained on entry is lost.
- 7: – INTEGERInput
-
On entry: the type of correlation coefficients which are to be calculated.
- Only Kendall's tau coefficients are calculated.
- Both Kendall's tau and Spearman's coefficients are calculated.
- Only Spearman's coefficients are calculated.
Constraint:
, or .
- 8: – REAL (KIND=nag_wp) arrayOutput
-
On exit: the requested correlation coefficients.
If only Kendall's tau coefficients are requested (), contains Kendall's tau for the th and th variables.
If only Spearman's coefficients are requested (), contains Spearman's rank correlation coefficient for the th and th variables.
If both Kendall's tau and Spearman's coefficients are requested (
), the upper triangle of
RR contains the Spearman coefficients and the lower triangle the Kendall coefficients. That is, for the
th and
th variables, where
is less than
,
contains the Spearman rank correlation coefficient, and
contains Kendall's tau, for
and
.
(Diagonal terms,
, are unity for all three values of
ITYPE.)
- 9: – INTEGERInput
-
On entry: the first dimension of the array
RR as declared in the (sub)program from which G02BRF is called.
Constraint:
.
- 10: – INTEGEROutput
-
On exit: the number of cases, , actually used in the calculations (when cases involving missing values have been eliminated).
- 11: – INTEGER arrayOutput
-
On exit:
holds the value
if the
th case was included in the calculations, and the value
if the
th case contained a missing value for at least one variable. That is,
(see
Section 3), for
.
- 12: – INTEGER arrayWorkspace
- 13: – INTEGER arrayWorkspace
- 14: – INTEGER arrayWorkspace
- 15: – REAL (KIND=nag_wp) arrayWorkspace
- 16: – REAL (KIND=nag_wp) arrayWorkspace
-
- 17: – INTEGERInput/Output
-
On entry:
IFAIL must be set to
,
. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
.
When the value is used it is essential to test the value of IFAIL on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
-
-
-
-
On entry, | , |
or | . |
-
After observations with missing values were omitted, fewer than cases remained.
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
You are warned of the need to exercise extreme care in your selection of missing values. G02BRF treats all values in the inclusive range
, where
is the missing value for variable
specified in
XMISS.
You must therefore ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.
8 Parallelism and Performance
G02BRF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 time taken by G02BRF depends on and , and the occurrence of missing values.
10 Example
This example reads in a set of data consisting of nine observations on each of three variables. Missing values of and are declared for the first and third variables respectively; no missing value is specified for the second variable. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables, omitting completely all cases containing missing values; cases , and are therefore eliminated, leaving only six cases in the calculations.
10.1 Program Text
Program Text (g02brfe.f90)
10.2 Program Data
Program Data (g02brfe.d)
10.3 Program Results
Program Results (g02brfe.r)