NAG Library Routine Document
G02BQF
1 Purpose
G02BQF computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is preserved, and the ranks of the observations are not available on exit from the routine.
2 Specification
SUBROUTINE G02BQF ( |
N, M, X, LDX, ITYPE, RR, LDRR, KWORKA, KWORKB, WORK1, WORK2, IFAIL) |
INTEGER |
N, M, LDX, ITYPE, LDRR, KWORKA(N), KWORKB(N), IFAIL |
REAL (KIND=nag_wp) |
X(LDX,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.
The observations are first ranked, as follows.
For a given variable, say, each of the observations, , has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitude of the other observations on that same variable.
The smallest observation 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 observation for variable is given the rank .
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 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.
The quantities calculated are:
(a) |
Kendall's tau rank correlation coefficients:
and |
if |
|
if |
|
if |
and , being the number of ties of a particular value of variable , and the summation being over all tied values of variable . |
(b) |
Spearman's rank correlation coefficients:
where 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 data value , the value of the th observation on the th variable, for and .
- 4: – INTEGERInput
-
On entry: the first dimension of the array
X as declared in the (sub)program from which G02BQF is called.
Constraint:
.
- 5: – 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 .
- 6: – 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.)
- 7: – INTEGERInput
-
On entry: the first dimension of the array
RR as declared in the (sub)program from which G02BQF is called.
Constraint:
.
- 8: – INTEGER arrayWorkspace
- 9: – INTEGER arrayWorkspace
- 10: – REAL (KIND=nag_wp) arrayWorkspace
- 11: – REAL (KIND=nag_wp) arrayWorkspace
-
- 12: – 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 | . |
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
The method used is believed to be stable.
8 Parallelism and Performance
G02BQF 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 G02BQF depends on and .
10 Example
This example reads in a set of data consisting of nine observations on each of three variables. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables.
10.1 Program Text
Program Text (g02bqfe.f90)
10.2 Program Data
Program Data (g02bqfe.d)
10.3 Program Results
Program Results (g02bqfe.r)