Goodman and Kruskal’s Gamma Coefficient for Ordinalized Bivariate Normal Distributions

Abstract

We consider a bivariate normal distribution with linear correlation $$\rho $$ ρ whose random components are discretized according to two assigned sets of thresholds. On the resulting bivariate ordinal random variable, one can compute Goodman and Kruskal’s gamma coefficient, $$\gamma $$ γ , which is a common measure of ordinal association. Given the known analytical monotonic relationship between Pearson’s $$\rho $$ ρ and Kendall’s rank correlation $$\tau $$ τ for the bivariate normal distribution, and since in the continuous case, Kendall’s $$\tau $$ τ coincides with Goodman and Kruskal’s $$\gamma $$ γ , the change of this association measure before and after discretization is worth studying. We consider several experimental settings obtained by varying the two sets of thresholds, or, equivalently, the marginal distributions of the final ordinal variables. This study, confirming previous findings, shows how the gamma coefficient is always larger in absolute value than Kendall’s rank correlation; this discrepancy lessens when the number of categories increases or, given the same number of categories, when using equally probable categories. Based on these results, a proposal is suggested to build a bivariate ordinal variable with assigned margins and Goodman and Kruskal’s $$\gamma $$ γ by ordinalizing a bivariate normal distribution. Illustrative examples employing artificial and real data are provided.

Brief report comparing the Scale of Emotional Development – Short (SED‐S) with other scales for emotional development

“…Mean ED‐level scores were compared using t ‐test to determine possible differences in the outcome of the survey. The per cent agreement per developmental stage was determined, and the Goodman and Kruskal gamma was calculated as a correlation feature for ordinal data (Barbiero & Hitaj 2020). Cohen's weighted kappas investigated the agreement between the different versions (≤0 corresponds to no agreement, 0.01–0.2 no or low agreement, 0.21–0.4 good agreement, 0.41–0.6 moderate agreement, 0.61–0.8 substantial agreement and <0.81 near perfect agreement; McHugh 2012).…”

Link between structural risk factors for adverse impacts of COVID-19 and food insecurity in developed and developing countries

“…In this paper, we use the Gamma coefficient as a measure of association. Also known as Goodman and Kruskal's gamma, it is a nonparametric measure of the strength and direction of association between two variables measured on an ordinal scale (Barbiero & Hitaj, 2020). Its value ranges between -1 and 1, with the sign indicating either a direct or inverse relationship between the pillars studied.…”

Partial credit trees meet the partial gamma coefficient for quantifying DIF and DSF in polytomous items

“…This example illustrates that in larger samples, an additional stopping criterion is needed to stop the tree from growing when DIF and DSF effect sizes are negligible. In this article, we will extend PCtrees by an effect size measure for DIF and DSF in polytomous items, the partial gamma ( p γ ) coefficient (Barbiero & Hitaj, 2020;Bjorner et al, 1998;Kreiner, 2003;Olszak & Ritschard, 1995;Siersma & Kreiner, 2009). p γ is an established descriptive measure for DIF and DSF in polytomous items that comes with a statistical significance test and a classification scheme to categorize DIF and DSF effects, analogously to the Mantel-Haenszel odds ratio for dichotomous items.…”

“…It is a descriptive measure of association between ordinal variables, similar to a correlation coefficient. Analogously, it has a standardized metric ranging from −1 to 1 which facilitates the interpretation of the size of the effect (Barbiero & Hitaj, 2020;Woods, 2007). It also allows for statistical significance testing because its test statistic is asymptotically normally distributed (Agresti, 2002;Goodman & Kruskal, 1963;Olszak & Ritschard, 1995;Woods, 2007).…”

“…The γ coefficient is computed based on the number of concordant and discordant pairs (Barbiero & Hitaj, 2020;Goodman & Kruskal, 1972;Kvålseth, 2017;Van Der Ark & Van Aert, 2015):…”