ANOVA and rank tests when the number of treatments is large

In this paper we consider the analysis of variance (ANOVA) F-tests, and rank statistic analogs, for testing equality of treatment means in the one-way and two-way experimental layouts. The rank-based procedures include the Kruskal-Wallis and Friedman statistics with chi-squared critical values, and the "ANOVA on ranks" or F-versions of these procedures. We provide proofs of asymptotic normality for these statistics under the nonstandard assumption that the number of treatments converges to infinity while the number of replications per treatment remains finite. These results confirm the robustness of F-distribution critical values for nonnormal data in situations which have a large number of treatments.Kruskal-Wallis test Friedman test Central limit theorem Type I error robustness Nonnormality