### Comparing k Independent Samples

One goal in comparing k independent samples is to determine whether the location parameters (medians) of the populations are different. Another goal
is to determine whether the scale parameters for the populations are different. For example, suppose new employees are randomly
assigned to one of three training programs. At the end of the program, the employees are given a standard test that provides
a rating score of their job ability. The goal of analysis is to compare the median scores for the three groups and decide
whether the differences are real or due to chance alone.

To compare k independent samples, either the NPAR1WAY or the FREQ procedure provides a Kruskal-Wallis test. PROC NPAR1WAY
also provides the Savage, median, and Van der Waerden (normal scores) tests. In addition, PROC NPAR1WAY produces the following
tests for scale differences: Siegel-Tukey test, Ansari-Bradley test, Klotz test, and Mood test. PROC NPAR1WAY also provides
the Conover test, which can be used to test for differences in both location and scale. Note that you can obtain exact p-values for all of these tests.

Additionally, you can specify the SCORES=DATA option to use the input data observations as scores. This enables you to produce
a very wide variety of tests. You can construct any scores for your data with the DATA step, and then PROC NPAR1WAY computes
the corresponding linear rank and one-way ANOVA tests. You can also analyze the raw data with the SCORES=DATA option; for
two-sample data, this permutation test is known as Pitman’s test.

See Chapter 83: The NPAR1WAY Procedure, for details, formulas, and examples.

To produce a Kruskal-Wallis test in the FREQ procedure,
use SCORES=RANK and the CMH2 option in the TABLES statement. Then, look at the second Cochran-Mantel-Haenszel statistic (labeled
"Row Mean Scores Differ") to obtain the Kruskal-Wallis test. The FREQ procedure also provides the Jonckheere-Terpstra test,
which is more powerful than the Kruskal-Wallis test for comparing k samples against ordered alternatives. The exact test is also available. In addition, you can obtain a ridit analysis, developed
by Bross (1958), by specifying SCORES=RIDIT or SCORES=MODRIDIT in the TABLES statement in the FREQ procedure. See Chapter 40: The FREQ Procedure, for more information.