Improved Transformed Statistics for the Test of One Factor Independence from the Other Two in an r × s × t Contingency Table

Abstract

We consider φ-divergence statistics C_φ for the test of one factor independence from the other two in an r × s × t contingency table. Statistics C_φ include the statistics R^a based on the power divergence as a special case. Statistic R⁰ is the log likelihood ratio statistic and R¹ is Pearson's X² statistic. Statistic R^2/3 corresponds to the statistic for the goodness-of-fit test recommended by Cressie and Read (1984). Statistics C_φ have the same chi-square limiting distribution under the hypothesis that one factor and the other two are independent. In this paper, when we assume that the distribution of C_φ is continuous, we show the derivation of an expression of approximation based on a multivariate Edgeworth expansion for the distribution of C_φ under the hypothesis that one factor and the other two are independent. Using the expression, we propose a new approximation of the distribution of C_φ. In addition, on the basis of the approximation, we obtain transformed statistics that improve the speed of convergence to a chi-square limiting distribution of C_φ. By numerical comparison in the case of R^a, we show that the transformed statistics perform well for a small sample.