Nonparametric Test Statistics

This section provides formulas for the test statistics used in the Wilcoxon, Median, and van der Waerden tests.

The tests are based on scores and use the following notation.

j = 1,...,n

n1, n2, ..., nk

The numbers of observations in each of the k levels of X.

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.

A function of the midranks used to define scores for the various tests.

α(Rj)

The function α applied to the midrank of the jth observation.

The function α defines scores as follows:

Wilcoxon scores

Median scores

Let nt denote the number of observations tied at the median. Then nt is given by the following:

van der Waerden scores

Two-Sample Normal Approximations

Tests based on the normal approximation are given only when X has exactly two levels. The notation used in this section is defined in Notation. The statistics that appear in the Two-Sample Normal Approximation report are defined below.

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.

The value of Z is given as follows:

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.

E(S)

The expected value of S under the null hypothesis. Denote the number of observations in the smaller level, or in the last level in the value ordering if the two groups have the same number of observations, by nl:

Var(S)

Define ave to be the average score across all observations. Then the variance of S is given as follows:

One-Way ChiSquare Approximations

Note: The ChiSquare test based on the Wilcoxon scores is knows as the Kruskal-Wallis test.

The notation used in this section is defined in Notation. The following quantities are used in calculating the ChiSquare statistic:

The total of the scores for the ith level of X.

E(Ti)

The expected value of the total score for level i under the null hypothesis of no difference in levels, given as follows:

Var(T)

Define ave to be the average score across all observations. Then the variance of T is given as follows:

The value of the test statistic is given below. This statistic is asymptotically ChiSquare on k - 1 degrees of freedom.