Closed Testing Procedures Based on x̅ 2-Statistics in Multi-Sample Models with Bernoulli Responses under Simple Ordered Restrictions

Multiple comparison procedures provide differences among the groups that is of interest. The procedures are used in clinical trials and agricultural fields experiments. We consider multi-sample model with Bernoulli responses under simple ordered restrictions of proportions. Shiraishi (2014b) proposed closed testing procedures based on maximum values of two-sample test statistics for all pairwise comparisons. The equality of sample sizes is needed in the asymptotic theory of Shiraishi’s procedures. We propose closed testing procedures based on statistics having asymptotically a χ̄-distribution which is appeared in Chernoff (1954). The proposed procedures are applicable for the models with unequal sample sizes. Although single-step multiple comparison procedures are utilized in general, the power of these procedures is low for a large number of groups. The closed testing procedures stated in the present paper are more powerful than the single-step procedures. Simulation studies are performed under the null hypothesis and some alternative hypotheses. In this studies, the proposed procedures show a good performance. We also illustrate applying to a dose-finding trial data with unequal sample sizes.