Simultaneous Confidence Intervals for Contrasts Among Multinomial Populations

Abstract

In the present article, we shall present two different methods for obtaining simultaneous confidence intervals for judging contrasts among multinomial populations, and we shall compare the intervals obtained by these methods with the simultaneous confidence intervals obtained earlier by Gold [5] for all linear functions of multinomial probabilities. One of the methods presented herein is particularly suited to the situation where all contrasts among, say, $I$ multinomial populations may be of interest, where each population consists of, say, $J$ classes. The other method presented herein is suited to certain situations where a specific set of contrasts among these populations is of interest; e.g., where, for each of the $\frac{1}{2}I(I - 1)$ pairs of populations, the $J$ contrasts between the corresponding probabilities associated with the twopopulations in the pair are of interest. For judging all contrasts among the $I$ multinomial populations, the confidence intervals obtained with the first method presented herein have the desirable property that they are shorter than the corresponding intervals obtained with the method presented by Gold [5]. For judging the $\frac{1}{2}I(I - 1)J$ pair-wise contrasts between the multinomial populations, the confidence intervals obtained with the second method presented herein have the desirable property that they are shorter than the corresponding intervals obtained with the first method, for the usual probability levels. In the present paper we shall also solve a problem first mentioned by Gold [5] but left unsolved in the earlier article. Gold took note of the fact that, in the usual analysis of variance context, the simultaneous confidence intervals obtained by Scheffe [14] and by Turkey [15] for judging contrasts among the parameters have the desirable property that rejection of the homogeneity hypothesis by the usual $F$ or Studentized range test implies the existence of at least one relevant contrast for which the corresponding confidence interval does not cover zero (see, for example, [14], pp. 66-77). She also noted that a result analogous to the Scheffe-Tukey result had not yet been obtained for her simultaneous confidence intervals, and she stated that the difficulty seemed to be that the homogeneity test is based on a $\chi^2$ statistic with $(I - 1)(J - 1)$ degrees of freedom in the case where $I$ multinomial populations, each consisting of $J$ classes, are tested for homogeneity, whereas her confidence intervals were based upon the $\chi^2$ distribution with $I(J - 1)$ degrees of freedom. In the present article, one of the methods we shall present for obtaining simultaneous confidence intervals for the contrasts among the $I$ multinomial populations will be based upon the $\chi^2$ distribution with $(I - 1)(J - 1)$ degrees of freedom, and these intervals will have desirable properties somewhat analogous to those enjoyed in the analysis of variance by the Scheffe confidence intervals and by the Tukey confidence intervals. A modification of the usual test of the null hypothesis that the $I$ multinomial populations are homogeneous will be presented herein, and we shall show that this modified test will lead to rejection of the null hypothesis if and only if there is at least one contrast, of the kind presented herein, for which the relevant confidence interval does not cover zero.