Improved transformed statistics for the test of independence in r×s contingency tables

We consider a class of statistics Cφ based on φ-divergence for the test of independence in r×s contingency tables. The class of statistics Cφ includes the statistics Ra 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 R2/3 corresponds to the statistic recommended by Cressie and Read [Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. B 46 (1984) 440–464] for the goodness-of-fit test. All members of statistics Cφ have the same chi-square limiting distribution under the hypothesis of independence. In this paper, we show the derivation of an expression of approximation for the distribution of Cφ under the hypothesis of independence. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, we propose a new approximation of the distribution of Cφ. Furthermore, on the basis of the approximation, we obtain transformations that improve the speed of convergence to the chi-square limiting distribution of Cφ. As a competitor of the transformed statistic, we derive a moment-corrected-type statistic. By numerical comparison in the case of Ra, we show that the transformed R1 statistic performs very well.