General marginal-free association indices for contingency tables: From the Altham index to the intrinsic association coefficient

Notwithstanding a large body of literature on log-linear models and odds ratios, no general marginal-free index of the association in a contingency table has gained a wide acceptance. Building on a framework developed by L. A. Goodman, we put into light the direct links between odds ratios, the Altham index, the intrinsic association coefficient, and coefficients in log-multiplicative models including Unidiff and row-column association models. We devise a normalized version of the latter coefficient varying between 0 and 1, which offers a simpler interpretation than existing indices similar to the correlation coefficient. We illustrate with the case of educational and socioeconomic homogamy among 149 European regions how this index can be used either alone in a non- or semiparametric approach or combined with models, and how it can protect against incorrect conclusions based on models which rely on strong assumptions to summarize the strength of association as a single parameter