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 R 0 is the log likelihood ratio statistic and R 1 is Pearson’s X 2 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.