g02br computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data, omitting completely any cases with a missing observation for any variable; 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 , the value of the th observation on the th variable, where and
- Type: array<System..::..Int32>()An array of size [m]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.On exit: the array miss is overwritten by the method, and the information it contained on entry is lost.
- Type: array<System..::..Double>()An array of size [m]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 [Accuracy]).On exit: the array xmiss is overwritten by the method, and the information it contained on entry is lost.
- 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.)
- Type: System..::..Int32%On exit: the number of cases, , actually used in the calculations (when cases involving missing values have been eliminated).
- Type: array<System..::..Int32>()An array of size [n]On exit: holds the value if the th case was included in the calculations, and the value if the th case contained a missing value for at least one variable. That is, (see [Description]), for .
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 observation contains a missing value for any of those variables for which missing values have been declared, i.e., if for any for which an has been assigned (see also [Accuracy]); and otherwise, for .
The observations are first ranked as follows.
For a given variable, say, each of the observations for which , () has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitudes of the other observations on that same variable for which .
The smallest of these valid observations 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 such observation is given the rank , where .
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 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 process is repeated for each of the variables.
Let be the rank assigned to the observation when the th variable is being ranked. For those observations, , for which , , for .
The quantities calculated are:
|(a)||Kendall's tau rank correlation coefficients:
and where is the number of ties of a particular value of variable , and the summation is over all tied values of variable .
|(b)||Spearman's rank correlation coefficients:
where and 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 .
- After observations with missing values were omitted, fewer than cases remained.
You are warned of the need to exercise extreme care in your selection of missing values. g02br 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 g02br 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 and third variables respectively; no missing value is specified for the second variable. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables, omitting completely all cases containing missing values; cases , and are therefore eliminated, leaving only six cases in the calculations.