g02bsf computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting cases with missing values from only those calculations involving the variables for which the values are missing; 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 g02bsf or nagf_correg_coeffs_kspearman_miss_pair.
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 the th observation for the th variable is a missing value, i.e., if a missing value, , has been declared for the th variable, and (see also Section 7); and otherwise, for and .
The observations are first ranked, a pair of variables at a time as follows:
For a given pair of variables, and say, each of the observations for which the product , for , 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 variable for which .
The smallest of these valid observations for variable is assigned to rank , the second smallest valid observation for variable the rank , the third smallest 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 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 variable is then ranked in a similar way. The process is then repeated for all pairs of variables and , for and . Let be the rank assigned to the observation when the th and th variables are being ranked, and be the rank assigned to the observation during the same process, for , and .
The quantities calculated are:
(a)Kendall's tau rank correlation coefficients:
and where is the number of ties of a particular value of variable when the th and th variables are being ranked, and the summation is over all tied values of variable .
(b)Spearman's rank correlation coefficients:
and , where is the number of ties of a particular value of variable when the th and th variables are being ranked, 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 , 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 g02bsf is called.
5: – Integer arrayInput
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.
6: – Real (Kind=nag_wp) arrayInput
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).
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.
, 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 g02bsf is called.
10: – IntegerOutput
On exit: the minimum number of cases used in the calculation of any of the correlation coefficients (when cases involving missing values have been eliminated).
11: – Real (Kind=nag_wp) arrayOutput
On exit: the number of cases,
, actually used in the calculation of the rank correlation coefficient for the th and th variables, for and .
12: – IntegerInput
On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bsf is called.
13: – Integer arrayWorkspace
14: – Integer arrayWorkspace
15: – Integer arrayWorkspace
16: – Integer arrayWorkspace
17: – Real (Kind=nag_wp) arrayWorkspace
18: – Real (Kind=nag_wp) arrayWorkspace
19: – 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 since useful values can be provided in some output arguments even when on exit. 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:
Note: in some cases g02bsf may return useful information.
On entry, .
On entry, .
On entry, and .
On entry, and .
On entry, and .
On entry, .
Constraint: or .
After observations with missing values were omitted, fewer than two cases remained for at least one pair of variables. (The pairs of variables involved can be determined by examination of the contents of the array cnt). All means, standard deviations, sums of squares and cross-products, and correlation-like coefficients based on two or more cases are returned by the routine even if .
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.
You are warned of the need to exercise extreme care in your selection of missing values. g02bsf 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.
8Parallelism and Performance
g02bsf is not threaded in any implementation.
The time taken by g02bsf depends on and , and the occurrence of missing values.
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, second and third variables respectively. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables, omitting cases with missing values from only those calculations involving the variables for which the values are missing. The program, therefore, eliminates cases , , and in calculating and correlation between the first and second variables, cases , and for the first and third variables, and cases , and for the second and third variables.