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What are the assumptions of the Games-Howell multiple comparisons procedure? Please back up any answers with references.

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Very few, except for the usual assumption of i.i.d. sample (which is common to almost any statistical test) and that it makes sense to compute arithmetic mean. (i.e. it is not ordered or nominal variable). This goes almost without saying, but you will find it at least implicitly in any statistical book mentioning the test.

This test has been shown by the author (Games, P. A., Keselman, H. J., & Clinch, J. J. Tests for homogeneity of variance in factorial designs. Psychological Bulletin, 86, 978-984) to be valid for non uniform sample size (as long as N in each group > 5), and heterogenous variance. All the facts cited here you can find in this article.

  1. Normality: Since it is based on studentized range statistics, the normality of sample if N > 20 is irrelevant. The author says in the article that "Prior studies of the robustness of the t statistic suggest non-normality is not a problem".
  2. Groups don't need to have the same sizes.
  3. In each group should be at least 6 observations.
  4. No assumption of homogenity of variance.
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