An alternative measure of ordinal association as a value-validity correction of the Goodman–Kruskal gamma

Even though the Goodman–Kruskal gamma (γ) is perhaps the most popular measure of association between two ordinal categorical variables, it does have some well-known limitations. In particular, it is widely recognized that its sample form γ^$\hat{\gamma }$ takes on values that may be highly inflated, making it incomparable with other measures such as the frequently used Kendall's tau-b (τ^b${\hat{\tau }_b}$). Such overstatement of a characteristic, varying in extent, has important implications for the validity and reliability of results and conclusions. By imposing a value–validity condition on association measures, this paper derives an alternative measure γ^c${\hat{\gamma }_{\rm{c}}}$ as a simple correction of γ^$\hat{\gamma }$. The new measure takes on reasonable values throughout its range from −1 to 1 and is quite comparable with τ^b${\hat{\tau }_b}$, but has advantages over it. Numerical values of γ^c${\hat{\gamma }_{\rm{c}}}$ are compared with those of τ^b${\hat{\tau }_b}$ and other measures for various reported datasets, and a statistical inference procedure is discussed.