Measures of Fourfold Association I

Chapter 13 is the first of two chapters describing connections, equivalencies, and relationships relating to fourfold contingency tables. First, Pearson’s mean-square contingency coefficient, ϕ 2 $$\phi ^{2}$$ , and tetrachoric correlation coefficient, r tet $$r_{\text{tet}}$$ , are described. Second, Yule’s Q and Y measures are described, and the connections linking Q and Y are established. Third, the odds ratio is described, and the connections linking the odds ratio with Yule’s Q and Yule’s Y statistics are detailed. Fourth, Goodman and Kruskal’s t a $$t_{a}$$ and t b $$t_{b}$$ asymmetric measures of association are presented, and the connections linking t a $$t_{a}$$ , t b $$t_{b}$$ , χ 2 $$\chi ^{2}$$ , Pearson’s ϕ 2 $$\phi ^{2}$$ , and Pearson’s r x y 2 $$r_{xy}^{2}$$ are described. Fifth, Somers’ d y x $$d_{yx}$$ and d x y $$d_{xy}$$ asymmetric measures are described, the connections linking d y x $$d_{yx}$$ and d x y $$d_{xy}$$ with simple percentage differences are established, and the connections with the corresponding unstandardized regression coefficients are detailed. Sixth, Kendall’s τ b $$\tau _{b}$$ measure of ordinal association is described, and the connections among τ b $$\tau _{b}$$ , χ 2 $$\chi ^{2}$$ , Pearson’s ϕ $$\phi $$ , and Pearson’s r x y $$r_{xy}$$ are delineated. Finally, the connections linking Pearson’s ϕ 2 $$\phi ^{2}$$ and Cohen’s κ $$\kappa $$ are established.