Tests for Patterned Alternatives ink-Sample Problems

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Abstract
This article suggests nonparametric methods of testing for a patterned alternative. As an example, suppose that we suspect that with increasing age, one's ability to perform a certain job tends to improve, but that after some point advancing age tends to mean diminishing performance. Suppose that N employees are grouped by age into k groups and employees' job performances are ranked from 1 to N. Viewing performance as the dependent variable, the particular pattern of increasing group locations followed by decreasing group locations is called an umbrella (pattern). The tests proposed herein allow the experimenter to test for this or any pattern of his choice. Given k groups, let denote the mean rank of the data in group j. Considering only statistics of the form , where Σ aj , = 0, we find a statistic having maximum asymptotic local power and greatest efficiency when the alternative hypothesis consists of a specific pattern. The case in which the researcher specifies a vague pattern is handled by letting the alternative consist of a collection of appropriate specific patterns. As a special case, the alternative “not all medians are equal” produces a test equivalent to the Kruskal—Wallis rank test. The approach extends to the additive two-way layout with b blocks and k treatments. In the “not all medians are equal” case the standardized test statistic equals the square root of the extended Friedman test statistic. Other special cases also lead to tests equivalent to known tests. The tests are reformulated using ranks within pairs of samples. How to adjust when covariates are present is discussed. Also discussed is the particular case of testing for umbrella alternatives, as are inherent differences with a competing procedure of Mack and Wolfe (1981).

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