Tests for k Samples

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 receive a standard test that gives 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 tests. In addition, PROC NPAR1WAY produces the following tests for scale differences: Siegel-Tukey test, Ansari-Bradley test, Klotz test, and Mood test. Note that you can obtain exact p -values for all of these tests.

In addition, 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 using 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 52, '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 29, 'The FREQ Procedure,' for more information.

Comparing k Dependent Samples

Friedman's test enables you to compare the locations of three or more dependent samples. You can obtain Friedman's Chi-square with the FREQ procedure by using the CMH2 option and SCORES=RANK and looking at the second CMH statistic in the output. For an example, see Chapter 29, 'The FREQ Procedure.' This chapter also contains formulas and other details on the CMH statistics. For a discussion of how to use the RANK and GLM procedures to obtain Friedman's test, refer to Ipe (1987).