Wilcoxon Scores

Wilcoxon scores are the ranks of the observations, which can be written as

where is the rank of observation , and is the score of observation .

Using Wilcoxon scores in the linear rank statistic for two-sample data produces the rank sum statistic of the Mann-Whitney-Wilcoxon test. Using Wilcoxon scores in the one-way ANOVA statistic produces the Kruskal-Wallis test. Wilcoxon scores are locally most powerful for location shifts of a logistic distribution.

When computing the asymptotic Wilcoxon two-sample test, PROC NPAR1WAY uses a continuity correction by default, as described in the section Continuity Correction. If you specify the CORRECT=NO option in the PROC NPAR1WAY statement, the procedure does not use a continuity correction.

Median Scores

Median scores equal for observations greater than the median, and otherwise. In terms of the observation ranks, median scores are defined as

Using median scores in the linear rank statistic for two-sample data produces the two-sample median test. The one-way ANOVA statistic with median scores is equivalent to the Brown-Mood test. Median scores are particularly powerful for distributions that are symmetric and heavy-tailed.

Van der Waerden (Normal) Scores

Van der Waerden scores are the quantiles of a standard normal distribution and are also known as quantile normal scores. Van der Waerden scores are computed as

where is the cumulative distribution function of a standard normal distribution. These scores are powerful for normal distributions.

Savage Scores

Savage scores are expected values of order statistics from the exponential distribution, with subtracted to center the scores around . Savage scores are computed as

Savage scores are powerful for comparing scale differences in exponential distributions or location shifts in extreme value distributions (Hajek 1969, p. 83).

Siegel-Tukey Scores

Siegel-Tukey scores are defined as

where the score values continue to increase in this pattern toward the middle ranks until all observations have been assigned a score.

When computing the asymptotic Siegel-Tukey two-sample test, PROC NPAR1WAY uses a continuity correction by default, as described in the section Continuity Correction. If you specify the CORRECT=NO option in the PROC NPAR1WAY statement, the procedure does not use a continuity correction.

Ansari-Bradley Scores

Ansari-Bradley scores are similar to Siegel-Tukey scores, but Ansari-Bradley scoring assigns the same score value to corresponding extreme ranks. (Siegel-Tukey scores are a permutation of the ranks .) Ansari-Bradley scores are defined as

Equivalently, Ansari-Bradley scores are equal to

Klotz Scores

Klotz scores are the squares of the Van der Waerden (normal) scores. Klotz scores are computed as

where is the cumulative distribution function of a standard normal distribution.

Mood Scores

Mood scores are computed as the square of the difference between the observation rank and the average rank. Mood scores can be written as

Conover Scores

Conover scores are based on the squared ranks of the absolute deviations from the sample means. For observation the absolute deviation from the mean is computed as

where is the value of observation , observation belongs to sample , and is the mean of sample . The values of are ranked, and the Conover score for observation is computed as

Following Conover (1999), when there are ties among the values of , PROC NPAR1WAY assigns the average rank to each of the tied observations. To compute the average rank, PROC NPAR1WAY ranks the as if there were no ties, and then averages the ranks of the tied observations.

The Conover score test is also known as the squared ranks test for variances. See Conover (1999) for more information.