Testing homogeneity of the multinomial proportions

In this article, we investigate some statistical procedures for testing homogeneity in a multinomial population. First we focus on the simpler problem of testing homogeneity in the entire vector of multinomial proportions. When the null hypothesis of uniformity is violated, some of the cell frequencies will tend to be small and some will tend to be large. We propose to investigate the performance of tests based on X(k) = max {X1, ...., Xk} or X(1) = min {X1, ...., Xk}, which are functions of the minimal sufficient statistics for this problem. The performance of the likelihood ratio test implemented by the asymptotic chi-square distribution and the simulated exact distribution will be compared to a test which is based on the maximum, the minimum, the pair maximum and minimum, the range, and the ratio statistic: X(k)/X(1). We also investigate the more difficult problem of testing homogeneity of a subset of the p-vector. For this problem, we focus only on likelihood ratio tests and investigate two implementations. Simulation studies are presented to evaluate the accuracy of achieving the targeted significance level and to compare the powers of the test statistics mentioned above for a range of alternatives. Based on the numerical results, recommendations are given for employing these testing procedures.