g02bs 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 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>[,](,)[,][,]Note: dim1 must satisfy the constraint:On entry: must be set to , 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>[,](,)[,][,]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.)
- Type: System..::..Int32%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).
- Type: array<System..::..Double,2>[,](,)[,][,]Note: dim1 must satisfy the constraint:On exit: the number of cases, , actually used in the calculation of the rank correlation coefficient for the th and th variables, for and .
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 [Accuracy]); 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:
|(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
Note: g02bs may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (LDX, LDRR, LDCNT) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
On entry, . On entry, . On entry, , 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 correlation coefficients based on two or more cases are returned by the method even if .
You are warned of the need to exercise extreme care in your selection of missing values. g02bs 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.
The time taken by g02bs 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.