Iterative Chi-Square Test for Equivalence of Multiple Treatment Groups

A measure of closeness among three or more means from normal populations was proposed by Ng (2000). The F-statistic was proposed to test the null hypothesis that this measure of closeness is greater than or equal to some prespecified δ (>0), against the alternative hypothesis that this measure of closeness is smaller than δ. The null distribution of the F-statistic at the boundary has a non-central F-distribution with a noncentral parameter that depends on δ and the assumed common variance (σ) with equal sample sizes. Thus, the critical value for the proposed test depends on σ, which must be estimated, resulting in an inflation of the type I error rate. Ng (2001) described an iterative approach to resolve this problem. Briefly, multiplying the chi-square statistic by σ results in a test statistic, T1, which does not depend on σ. However, its distribution depends on σ. To start the iterative process, determine the critical value for T1 as a function of σ, define a test statistic by subtracting the estimated critical value from T1, and then calculate its critical value as a function of σ. This iterative process stops when the critical value becomes reasonably flat and the inflation of the type I error rate becomes negligible. In this paper, the critical values and the type I error rates at each step will be presented for a specific situation.