j = 1,...,n
n1, n2, ..., nk
Rj
The midrank of the jth observation. The midrank is the observation’s rank if it is not tied and its average rank if it is tied.
α(Rj)
The function α applied to the midrank of the jth observation.
The function α defines scores as follows:
Let nt denote the number of observations tied at the median. Then nt is given by the following:
The statistic S is the sum of the values α(Rj) for the observations in the smaller group. If the two levels of X have the same numbers of observations, then the value of S corresponds to the last level of X in the value ordering.
Note: The Wilcoxon test adds a continuity correction. If (S - E(S)) is greater than zero, then 0.5 is subtracted from the numerator. If (S - E(S)) is less than zero, then 0.5 is added to the numerator.
Define ave to be the average score across all observations. Then the variance of S is given as follows:
The notation used in this section is defined in Notation. The following quantities are used in calculating the ChiSquare statistic:
Ti
E(Ti)
The expected value of the total score for level i under the null hypothesis of no difference in levels, given as follows:
Define ave to be the average score across all observations. Then the variance of T is given as follows:

Help created on 9/19/2017