On the P-value for Members of the Cressie-Read Family of Divergence Statistics

A few years ago a paper appeared in this Journal that proposed a closed-form approximation of the p-value for Pearson's chi-squared statistic. Extensive empirical and simulation studies were performed and it was shown that the approximation provides very accurate p-values when compared with what we considered to be the ``true'' p-value (obtained using a base R function). It is important to note, however, that Pearson's chi-squared statistic is a special case of the Cressie-Read family of divergence statistics as is, for example, the log-likelihood ratio statistic, the Freeman-Tukey statistic and the Cressie-Read statistic. Therefore, this paper adapts the previously published closed form approximation of the p-value by demonstrating its applicability to any member of the Cressie-Read family of divergence statistics. We also give two further closed form approximations and assess their accuracy by analysing three contingency tables of varying sample size, degrees of freedom and statistical significance of the association.