A family of association measures for 2×2 contingency tables based on the ϕ-divergence

The odds ratio is the predominant measure of association in $2\times 2$ contingency tables, which, for inferential purposes, is usually considered on the log-scale. Under an information theoretic set-up, it is connected to the Kullback–Leibler divergence. Considering a generalized family of divergences, the $\varphi$ divergence, alternative association measures are derived for $2\times 2$ contingency tables. Their properties are studied and asymptotic inference is developed. For some members of this family, the estimated association measures remain finite in the presence of a sampling zero while for a subset of these members the estimators of these measures have finite variance as well. Special attention is given to the power divergence, which is a parametric family. The role of its parameter $\lambda$, in terms of the asymptotic confidence intervals’ coverage probability and average relative length, is further discussed. In special probability table structures, for which the performance of the asymptotic confidence intervals for the classical log odds ratio is poor, the measure corresponding to $\lambda =1/3$ is suggested as an alternative.