g02bq 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 method.
- Type: System..::..Int32On entry: , the number of observations or cases.Constraint: .
- Type: System..::..Int32On entry: , the number of variables.Constraint: .
- Type: array<System..::..Double,2>[,](,)[,][,]An array of size [dim1, m]Note: dim1 must satisfy the constraint:On entry: must be set to data value , the value of the th observation on the th variable, for and .
- Type: System..::..Int32On entry: the type of correlation coefficients which are to be calculated.
Constraint: , or .
- Only Kendall's tau coefficients are calculated.
- Both Kendall's tau and Spearman's coefficients are calculated.
- Only Spearman's coefficients are calculated.
- Type: array<System..::..Double,2>[,](,)[,][,]An array of size [dim1, m]Note: dim1 must satisfy the constraint: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.)
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:
|(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
Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (LDX, LDRR) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
On entry, . On entry, . On entry, , or .
The method used is believed to be stable.
The time taken by g02bq 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.