A POWER APPROXIMATION OF THE TEST OF INDEPENDENCE IN s×r CONTINGENCY TABLES BASED ON A NORMALIZING TRANSFORMATION

Read and Cressie (1988) introduced a class of the power-divergence statistics R^a for the test of independence in s×r contingency tables. This class includes Pearson's X^2 statistic (when a = 1) and the loglikelihood ratio statistic (when a = 0). All R^a have the same chi-squared limiting null distribution. All R^a have the same noncentral chi-squared limiting distribution under local alternatives, whence the power of the class is the same for all a asymptotically. Applying the power approximation methods for the multinomial goodness-of-fit test developed by Broffitt and Randles (1977) and Drost et al. (1989), Taneichi and Sekiya (1995) proposed three approximations to the power of R^a that vary with the statistic chosen. In this paper we propose a new approximation to the power of R^a. The new approximation is a normal approximation based on normalizing transformations of the statistics. The proposed approximation and the other approximations are compared numerically. As a result of comparison, we find that the proposed approximation is very effective for R^<-1> and R^<-2> when all marginal probabilities are equal. We also find that the approximation is effective for the statistics R^0, R^<2/3>, and R^1.