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

抄録

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 χ² 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 R⁰, R^2/3, and R¹.