Multiple Comparisons of Polynomial Distributions

Asymptotically correct 90 and 95 percentage points are given for multiple comparisons with control and for all pair comparisons of several independent samples of equal size from polynomial distributions. Test statistics are the maxima of the X²-statistics for single comparisons. For only two categories the asymptotic distributions of these test statistics result from DUNNETT'S many-one tests and TUKEY'S range test (cf. MILLER, 1981). The percentage points for comparisons with control are computed from the limit distribution of the test statistic under the overall hypothesis H₀. To some extent the applicability of these bounds is investigated by simulation. The bounds can also be used to improve Holm's sequentially rejective Bonferroni test procedure (cf. HOLM, 1979). The percentage points for all pair comparisons are obtained by large simulations. Especially for 3×3-tables the limit distribution of the test statistic under H₀ is derived also for samples of unequal size. Also these bounds can improve the corresponding Bonferroni-Holm procedure. Finally from SKIDÁK's probability inequality for normal random vectors (cf. SKIDÁK, 1967) a similar inequality is derived for dependent X²-variables applicable to simultaneous X²-tests.