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.
The routine may be called by the names g02bqf or nagf_correg_coeffs_kspearman.
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 , 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 .
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
1: – IntegerInput
On entry: , the number of observations or cases.
2: – IntegerInput
On entry: , the number of variables.
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.
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.
, 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.
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 , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or 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).
6Error 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, .
On entry, .
On entry, and .
On entry, and .
On entry, .
Constraint: or .
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The method used is believed to be stable.
8Parallelism 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 .
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.