Measures of Ordinal Association II

Abstract

Chapter 10 is the second of two chapters describing connections, equivalencies, and relationships relating to measures of ordinal association. First, Kendall’s $\tau _{b}$ measure of ordinal association and Stuart’s $\tau _{c}$ measure of ordinal association are described and the connection linking Kendall’s $\tau _{b}$, Stuart’s $\tau _{c}$, and permutation test statistic $\delta$ is established. An example illustrates the connections among $\tau _{a}$, $\tau _{b}$, $\tau _{c}$ and $\delta$. Second, Goodman and Kruskal’s $\gamma$ measure of ordinal association is presented and the connection linking Goodman and Kruskal’s $\gamma$ and permutation test statistic $\delta$ is detailed. Third, Somers’ $d_{yx}$ and $d_{xy}$ measures of ordinal association are described and the connections linking Somers’ $d_{yx}$ and $d_{xy}$ and permutation test statistic $\delta$ are established. Fourth, Wilson’s e measure of ordinal association is described and the connection linking Wilson’s e and test statistic $\delta$ is delineated. Fifth, the connection linking Kendall’s S measure of ordinal association and Mann and Whitney’s U two-sample rank-sum statistic is described. Finally, Cohen’s weighted kappa measure of inter-rater agreement is presented and the connection linking $\kappa _{w}$ and permutation measure $\Re$ is described.

Preview of Chap. 11

Chapter 11 describes connections, equivalencies, and relationships relating to nominal-level (categorical) variables. First, Pearson’s chi-squared test of independence is described and illustrated with an example analysis. Second, Pearson’s $\phi ^{2}$, Pearson’s C, Tschuprov’s$T^{2}$, and Cramér’s$V^{2}$ measures of nominal association are described and the connections linking the measures to Pearson’s $\chi ^{2}$ are established. Third, the connections linking Goodman and Kruskal’s$t_{a}$ and $t_{b}$ measures with $\chi ^{2}$ and permutation test statistic $\delta$ are established. Fourth, McNemar’s$Q_{\text{M}}$ and Cochran’s$Q_{\text{C}}$ measures of change are described and the connections linking $Q_{\text{M}}$, $Q_{\text{C}}$ and permutation test statistic $\delta$ are presented. Fifth, Cohen’s unweighted kappa ($\kappa$) and weighted kappa ($\kappa _{w}$) are described and the connections linking $\kappa$, $\kappa _{w}$, and Mielke and Berry’s permutation-based measure of chance-corrected agreement, $\Re$, are established. Finally, the connections linking Pearson’s$\chi ^{2}$ test of independence and Pearson’s $r_{xy}^{2}$ product-moment correlation coefficient are described and illustrated for $r \times c$ contingency tables.