Friedman-Type Statistics and Consistent Multiple Comparisons for Unbalanced Designs with Missing Data

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Abstract
A generalization of the Friedman test using the marginal likelihood principle (Kalbfleisch and Prentice 1973) is presented and its asymptotic power given. It allows use of a variety of score functions and handles ties and unbalanced designs. Some well-known statistics [including the tests of Kruskal and Wallis (1952), Prentice (1979), and Rinaman (1983)] are proven to be special cases, whereas others (e.g., Klotz 1980; Groggel and Skillings 1986; Rai 1987; Skillings and Mack 1981) are shown to be less appropriate. Multiple comparisons are considered under both the global hypothesis and alternatives. Evaluating non-centrality parameters under local shift alternatives, the procedures of Klotz (1980) and (for a special case) Skillings and Mack (1981) can be either anticonservative (showing differences that do not exist) or insensitive (ignoring differences that do exist), depending on the distribution of missing data. A new Scheffé-type procedure for arbitrarily missing data is presented and recommended as consistent and more powerful.

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