Measures of Interval Association

Abstract

Chapter 8 describes connections, equivalencies, and relationships relating to bivariate linear correlation and regression. First, Pearson’s product-moment correlation coefficient is described. Second, a permutation alternative to Pearson’s correlation coefficient is presented for bivariate data and the connection linking the two measures is described. Third, an example analysis illustrates the differences and similarities of the two measures and the asymptotic and Monte Carlo probability values are calculated and compared. Fourth, the point-biserial correlation coefficient is described, an example analysis illustrates the point-biserial correlation coefficient, and a Monte Carlo probability value for the point-biserial correlation coefficient is generated. The connection linking the point-biserial correlation coefficient and Pearson’s product-moment correlation coefficient is established. Fifth, the connection linking Spearman’s rank-order correlation coefficient and Pearson’s product-moment correlation coefficient is detailed and an example analysis illustrates the connection. Sixth, Jaspen’s multi-serial correlation coefficient for one ordinal-level variable and one interval-level variable is described and the connection linking Jaspen’s coefficient and Pearson’s product-moment correlation coefficient is established. Finally, the biserial correlation coefficient is described and the connections linking biserial correlation, point-biserial correlation, Jaspen’s multi-serial correlation, and Pearson’s product-moment correlation are delineated and illustrated.

Preview of Chap. 9

Chapter 9 is the first of two chapters describing connections, equivalencies, and relationships relating to measures of ordinal association. First, Spearman’s rank-order correlation coefficient and Spearman’s footrule measure are presented. Second, permutation test statistic $\delta$ is described and the connections linking Spearman’s two measures with test statistic $\delta$ are established. Third, Kendall’s S test statistic is described and the connections linking Kendall’s S and test statistic $\delta$ are detailed. Fourth, Kendall’s $\tau _{a}$ and $\tau _{b}$ measures of ordinal association are described and the connections linking Kendall’s $\tau _{a}$ and $\tau _{b}$ to test statistic $\delta$ are defined. Fifth, Kendall and Babington Smith’s u measure of pairwise ordinal association is described and its connections with Kendall’s $\tau _{a}$ measure are demonstrated.

Sixth, Stuart’s $\tau _{c}$ measure of ordinal association is described and the connections linking Stuart’s $\tau _{c}$ and test statistic $\delta$ are established. An example illustrates the connections among S, $\tau _{a}$, $\tau _{b}$, $\tau _{c}$ and $\delta$. Seventh, Goodman and Kruskal’s $\gamma$ measure of ordinal association is presented and the connections linking Goodman and Kruskal’s $\gamma$ and test statistic $\delta$ are detailed. Eighth, Somers’ $d_{yx}$ and $d_{xy}$ measures of ordinal association are described and the connections linking Somers’ $d_{yx}$ and $d_{xy}$ and test statistic $\delta$ are established. Ninth, Wilson’s e measure of ordinal association is described and the connections linking Wilson’s e and test statistic $\delta$ are delineated. Finally, the connections linking Kendall’s S measure of ordinal association and Mann and Whitney’s U two-sample rank-sum test statistic are described and illustrated with an example analysis.