2. Measures of Ordinal Association: Goodman and Kruskal’s Gamma

Measures of ordinal association between two variables use the property that the categories of ordinal variables have a natural order; but the way in which these measures use this information may differ considerably. Some measures are based on the intuitive notion that ordinal association should have an interpretation analogous to that of association for metric variables, say, Pearson’s correlation; in this group, we find Spearman’s $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} , Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , polychoric correlation, Somer’s d, and Goodman-Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} . These coefficients allow us to make statements of the general form “if scores on X increase, then most probably, scores on Y will increase” (Kampen and Swyngedouw Reference Kampen and Swyngedouw2000). Other measures of ordinal association have a background in information theory and have interpretations in terms of stochastic entropy (Bryson and Phillips Reference Bryson and Phillips1975; Laird Reference Laird1979; Gilula et al. Reference Gilula, Krieger and Ritov1988). Henceforth, we will focus on the former family of measures.

Let us consider a pair of ordinal variables (X, Y), with H and K ordered categories, respectively, and introduce the concept of concordance/discordance by considering two independent realizations $(x_i,y_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_i,y_i)$$\end{document} and $(x_j,y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_j,y_j)$$\end{document} . $(x_i,y_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_i,y_i)$$\end{document} and $(x_j,y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_j,y_j)$$\end{document} will be said to be concordant if $x_i<x_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i<x_j$$\end{document} and $y_i<y_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i<y_j$$\end{document} , or if $x_i>x_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i>x_j$$\end{document} and $y_i>y_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i>y_j$$\end{document} . Conversely, $(x_i,y_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_i,y_i)$$\end{document} and $(x_j,y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_j,y_j)$$\end{document} will be said to be discordant if $x_i<x_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i<x_j$$\end{document} and $y_i>y_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i>y_j$$\end{document} , or if $x_i>x_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_i>x_j$$\end{document} and $y_i<y_j$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_i<y_j$$\end{document} . Thus, one can define the probability of concordance as

$\begin{array}{c}{\Pi }_c=Pr\left(X_i,<,X_j,,\text{and},,Y_i,<,Y_j\right)+Pr\left(X_i,>,X_j,,\text{and},,Y_i,>,Y_j\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varPi _c = Pr\left\{ X_i<X_j \text { and } Y_i<Y_j \right\} + Pr\left\{ X_i>X_j \text { and } Y_i>Y_j \right\} \end{aligned}$$\end{document}

and similarly the probability of discordance as

$\begin{array}{c}{\Pi }_d=Pr\left(X_i,<,X_j,,\text{and},,Y_i,>,Y_j\right)+Pr\left(X_i,>,X_j,,\text{and},,Y_i,<,Y_j\right).\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varPi _d = Pr\left\{ X_i<X_j \text { and } Y_i>Y_j \right\} + Pr\left\{ X_i>X_j \text { and } Y_i<Y_j \right\} . \end{aligned}$$\end{document}

The gamma coefficient belongs to a larger family of ordinal correlation measures (see, e.g., Woods Reference Woods2009, for an exhaustive account); it is defined as the following ratio (see, e.g., Agresti Reference Agresti2010, pp.186–187):

(1) $\begin{array}{c}\gamma =\frac{{\Pi }_c-{\Pi }_d}{{\Pi }_c+{\Pi }_d}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma =\frac{\varPi _c - \varPi _d}{\varPi _c+\varPi _d} \end{aligned}$$\end{document}

${\Pi }_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _c$$\end{document} and ${\Pi }_d$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _d$$\end{document} can be conveniently expressed in terms of the joint probabilities $p_{\mathrm{ij}}=P(X=x_i,Y=y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}=P(X=x_i,Y=y_j)$$\end{document} :

$\begin{array}{c}{\Pi }_c=2\underset{i<h}{\sum \sum }\underset{j<k}{\sum \sum }{p}_{\mathrm{ij}}{p}_{\mathrm{hk}},{\Pi }_d=2\underset{i<h}{\sum \sum }\underset{j>k}{\sum \sum }{p}_{\mathrm{ij}}{p}_{\mathrm{hk}}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \varPi _c=2\mathop {\sum \sum }_{i<h}\mathop {\sum \sum }_{j<k} p_{ij}p_{hk},\quad \varPi _d=2\mathop {\sum \sum }_{i<h}\mathop {\sum \sum }_{j>k} p_{ij}p_{hk}. \end{aligned}$$\end{document}

By defining the quantities

$\begin{array}{c}{p}_{\mathrm{hk}}^{\left(c\right)}=\sum _{a<h}\sum _{b<k}{p}_{\mathrm{ab}}+\sum _{a>h}\sum _{b>k}{p}_{\mathrm{ab}}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{hk}^{(c)} = \sum _{a<h}\sum _{b<k} p_{ab} + \sum _{a>h}\sum _{b>k} p_{ab} \end{aligned}$$\end{document}

and

$\begin{array}{c}{p}_{\mathrm{hk}}^{\left(d\right)}=\sum _{a<h}\sum _{b>k}{p}_{\mathrm{ab}}+\sum _{a>h}\sum _{b<k}{p}_{\mathrm{ab}}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{hk}^{(d)} = \sum _{a<h}\sum _{b>k} p_{ab} + \sum _{a>h}\sum _{b<k} p_{ab} \end{aligned}$$\end{document}

for each $h=1,\cdots ,H$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h=1,\dots ,H$$\end{document} , $k=1,\cdots ,K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k=1,\dots ,K$$\end{document} , then ${\Pi }_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _c$$\end{document} and ${\Pi }_d$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _d$$\end{document} can be rewritten as

$\begin{array}{c}{\Pi }_c=\sum _{h=1}^H\sum _{k=1}^K{p}_{\mathrm{hk}}{p}_{\mathrm{hk}}^{\left(c\right)}\text{,}{\Pi }_d=\sum _{h=1}^H\sum _{k=1}^K{p}_{\mathrm{hk}}{p}_{\mathrm{hk}}^{\left(d\right)}.\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Pi _c =\sum _{h=1}^H\sum _{k=1}^K p_{hk}p_{hk}^{(c)} \text {, } \Pi _d =\sum _{h=1}^H\sum _{k=1}^K p_{hk}p_{hk}^{(d)}. \end{aligned}$$\end{document}

For a bivariate continuous random variable, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} can be still computed through Eq. (1): in this case, since the probability of concordance and the probability of discordance sum up to 1 (there is no probability of tied values for either X or Y), then $\gamma ={\Pi }_c-{\Pi }_d=2{\Pi }_c-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =\varPi _c-\varPi _d=2\varPi _c-1$$\end{document} , and it coincides with Kendall’s rank correlation $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} (Kendall Reference Kendall1945), simply defined as the difference between the probability of concordance and the probability of discordance. We recall that for two continuous random variables X and Y, Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} depends only on their unique copula C (representing the dependence structure of the bivariate random vector (X, Y); see McNeil et al. , Reference McNeil, Frey and Embrechts2005, chapter 5), specifically,

$\begin{array}{c}\tau (X,Y)=4{\int }_0^1{\int }_0^{1}C(u_1,u_2)\text{d}C(u_1,u_2)-1,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tau (X,Y) = 4\int _0^1\int _0^1 C(u_1, u_2) \text {d}C(u_1, u_2) - 1, \end{aligned}$$\end{document}

and not on the marginal distributions of X and Y. $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ranges between $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} and $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} ; it attains the upper bound if and only if X and Y are comonotonic, whereas it attains the lower bound if and only if X and Y are countermonotonic.

Like Pearson’s correlation and other correlation measures such as Spearman’s rho and the aforementioned Kendall’s tau, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} takes values in $[-1,+1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,+1]$$\end{document} . The values $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} , 0, and $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} are attained when ${\Pi }_c=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _c=0$$\end{document} , ${\Pi }_c={\Pi }_d$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _c=\varPi _d$$\end{document} , ${\Pi }_d=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPi _d=0$$\end{document} , respectively. There are however some potential problems with the gamma coefficient, which were immediately recognized by Goodman and Kruskal (Reference Goodman and Kruskal1954):

• • $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is unstable over various “cutting points”, that is to say, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} tends to increase as the categories of a contingency table are collapsed, since $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} gives no consideration to tied pairs, as can be seen from Eq. (1), and the number of tied pairs increases as the table is collapsed;

• • $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} also usually yields greater association values than other measures of ordinal association, as it does not consider any of the tied pairs;

• • Finally, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is a weakly monotonic measure of ordinal association, i.e., it reaches $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} under a variety of cell frequency configurations, not only in case of strict perfect association (Berry et al. Reference Berry, Johnston and Mielke2018). For example, the bivariate distribution of Table 1, where all the joint probabilities are zero except those labelled with $\times$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} , presents a value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} equal to $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} , although the relationship between X and Y is not perfectly monotonic (Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and Spearman’s $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} would be strictly smaller than 1).

With the objective of overcoming these pitfalls, some modifications to the gamma coefficient or alternative measures of ordinal association have been proposed (Rousson Reference Rousson2007; Kvålseth Reference Kvålseth2017, Reference Kvålseth2018).

Table 1. Graphic for a weakly monotonic relationship

A remark on the possible extension of the gamma coefficient to higher dimensions can be made. Let us consider the d-variate ordinal random vector $X={(X_1,X_2,\cdots ,X_d)}^{⊺}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pmb {X}=(X_1, X_2,\dots , X_d)^\intercal $$\end{document} , with $d>2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>2$$\end{document} ; then, a $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} association matrix remains defined as

$\begin{array}{c}\Gamma \left(X\right)={\left[{\gamma }_{\mathrm{ij}}\right]}_{i=1,\cdots ,d;j=1,\cdots ,d}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (\pmb {X})=[\gamma _{ij}]_{i=1,\dots ,d;j=1,\dots ,d} \end{aligned}$$\end{document}

with ${\gamma }_{\mathrm{ij}}=\gamma (X_i,X_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{ij}=\gamma (X_i,X_j)$$\end{document} . This matrix however is not necessarily positive semidefinite, as happens for Spearman’s or Kendall’s rank correlation matrices, which are both valid correlation matrices (see McNeil et al. Reference McNeil, Frey and Embrechts2005, p.207). An easy counterexample is provided as follows. Consider the trivariate distribution of Table 2. The values of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for the bivariate distributions (X, Y), (X, Z), and (Y, Z) are ${\gamma }_{\mathrm{xy}}=1/3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{xy}=1/3$$\end{document} , ${\gamma }_{\mathrm{xz}}=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{xz}=1$$\end{document} , and ${\gamma }_{\mathrm{yz}}=-7/9$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{yz}=-7/9$$\end{document} , respectively. The corresponding $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} association matrix is:

$\begin{array}{c}\Gamma (X,Y,Z)=\left(\begin{array}{ccc}1& 1/3& 1\\ 1/3& 1& -7/9\\ 1& -7/9& 1\end{array}\right);\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Gamma (X,Y,Z)=\begin{bmatrix} 1 &{} 1/3 &{} 1\\ 1/3 &{} 1 &{} -7/9 \\ 1 &{} -7/9 &{} 1 \end{bmatrix}; \end{aligned}$$\end{document}

its three eigenvalues are 2.119, 1.322, and $-0.441$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-0.441$$\end{document} ; since one of them is negative, the matrix is clearly not positive semidefinite.

Table 2. A trivariate probability distribution and its corresponding bivariate marginal distributions

With n pairs of observations constituting a multinomial sample, the sample analog of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is given by $\hat{\gamma }=(C-D)/(C+D)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }} = (C-D)/(C+D)$$\end{document} , with C the number of concordant pairs and D the number of discordant pairs. It is possible to prove that $\sqrt{n}(\hat{\gamma }-\gamma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{n}({\hat{\gamma }}-\gamma )$$\end{document} is asymptotically normal with mean zero and variance given by

$\begin{array}{c}{\sigma }^2=\sum _i\sum _j{p}_{\mathrm{ij}}{\varphi }_{\mathrm{ij}}^2/{({\Pi }_c+{\Pi }_d)}^4=\frac{16}{{({\Pi }_c+{\Pi }_d)}^4}\sum _i\sum _j{p}_{\mathrm{ij}}{\left({\Pi }_c,p_{\mathrm{ij}}^{\left(d\right)},-,{\Pi }_d,p_{\mathrm{ij}}^{\left(d\right)}\right)}^2\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\sigma }^2 = \sum _i\sum _j p_{ij} \phi _{ij}^2/(\Pi _c+\Pi _d)^4 = \frac{16}{(\Pi _c+\Pi _d)^4}\sum _i\sum _j p_{ij}\left( \Pi _cp_{ij}^{(d)} - \Pi _dp_{ij}^{(d)}\right) ^2 \end{aligned}$$\end{document}

where ${\varphi }_{\mathrm{ij}}=4({\Pi }_d{p}_{\mathrm{ij}}^{\left(c\right)}-{\Pi }_c{p}_{\mathrm{ij}}^{\left(d\right)})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{ij}=4(\Pi _d p_{ij}^{(c)} - \Pi _c p_{ij}^{(d)})$$\end{document} (see, e.g., Agresti Reference Agresti2010).

In practice, replacing $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} , ${\Pi }_c$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _c$$\end{document} , and ${\Pi }_d$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _d$$\end{document} with their sample values in ${\sigma }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }^2$$\end{document} yields the ML estimate ${\hat{\sigma }}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\sigma }}^2$$\end{document} of ${\sigma }^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2$$\end{document} . The term $SE=\hat{\sigma }/\sqrt{n}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SE = {\hat{\sigma }}/\sqrt{n}$$\end{document} is an estimated standard error for $\hat{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}$$\end{document} , and Wald confidence interval for $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is $(\hat{\gamma }\pm z_{1-\alpha /2}SE)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\hat{\gamma }} \pm z_{1-\alpha /2}SE)$$\end{document} .

Rosenthal (Reference Rosenthal1966); Gans and Robertson (Reference Gans and Robertson1981) showed that $\hat{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}$$\end{document} has a tendency to converge slowly to normality and to have distributional irregularity, bias, and skewness problems, especially when the true absolute value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is large. O’Gorman and Woolson (Reference O’Gorman and Woolson1988); Carr et al. (Reference Carr, Hafner and Koch1989) pointed out that better convergence occurs using the Fisher-type transform $\hat{\xi }=\frac{1}{2}log[(1+\hat{\gamma })/(1-\hat{\gamma })]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\hat{\xi }} = \frac{1}{2}\log [(1+{\hat{\gamma }})/(1-{\hat{\gamma }})] $$\end{document} , whose asymptotic variance equals the asymptotic variance of $\hat{\gamma }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}$$\end{document} multiplied by ${(1-{\gamma }^2)}^{-2}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 - \gamma ^2)^{-2}$$\end{document} . A confidence interval can be constructed for $\xi$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} and then inverted to one for $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , by using the inverse transformation $\hat{\gamma }=(e^{2\hat{\xi }}-1)/(e^{2\hat{\xi }}+1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\hat{\gamma }} = (e^{2{\hat{\xi }}}-1)/(e^{2{\hat{\xi }}}+1) $$\end{document} .

3. Analysis of the Relationship Between $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for Ordinalized Bivariate Normal Distribution

Discretization of continuous variables is commonly encountered in practice. Based on observed nominal age, income, temperature, and depression score, one can derive ordinal variables such as young-middle-old age, low-medium-high income, cold-cool-average-hot temperature, and no-mild-moderate-severe depression. Discretization is usually avoided by statisticians for intuitive and valid reasons, the most prominent of which is the associated power and information loss. Some problems related to identification of statistical models obtained by discretization have been recently raised by Grønneberg and Foldnes (Reference Grønneberg and Foldnes2019). However, simplicity, better interpretability and comprehension of the effects of interest, and superiority of some categorical data measures such as odds ratio have been argued by proponents of discretization (Liu et al. Reference Liu, Hussain, Tan and Dash2002).

The objective of this section is the determination of association magnitude changes when the univariate components of a bivariate normal distribution are ordinalized, i.e., the range of each continuous component is divided into contiguous intervals, which are assigned an ordered category or a consecutive positive integer. A similar and extensive analysis has been conducted by Demirtas and Vardar-Acar (Reference Demirtas and Vardar-Acar2017) to investigate the effects of discretization of continuous variables on the magnitude of linear correlation. If the underlying continuous distribution is bivariate normal, it can be proved that discretization always preserves the sign of linear correlation and more importantly leads to a reduction in magnitude. This result, which has been empirically observed in many simulation studies (see, e.g., Bollen and Barb Reference Bollen and Barb1981) and was claimed to hold only “in large samples” by Demirtas and Vardar-Acar (Reference Demirtas and Vardar-Acar2017), is just a consequence of a previous, more general theoretical result, named Lancaster’s theorem (Lancaster Reference Lancaster1957), which states that the correlation of a bivariate normal cannot increase whatever transformations are applied to its univariate components (see also Mari and Kotz Reference Mari and Kotz2001, p.155, where it is reported as an “extremal property” of the bivariate normal distribution).

For a bivariate normal distribution with correlation coefficient $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} , the following relationship holds between $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} and Kendall’s rank correlation $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} :

(2) $\begin{array}{c}\tau =\frac{2}{\pi }arcsin\rho .\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tau =\frac{2}{\pi } \arcsin \rho . \end{aligned}$$\end{document}

Equation (2) holds also for most elliptical distributions, e.g., for bivariate Student’s t, and for most bivariate distributions whose dependence structure is described by an elliptical copula (McNeil et al. Reference McNeil, Frey and Embrechts2005). If both univariate margins are discretized/ordinalized through pre-specified thresholds, one can compute some measure of ordinal association based on the resulting bivariate ordinal variable, such as Goodman and Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} : we know (see the previous section) that for continuous bivariate random variables, such as the bivariate normal, the definitions of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} coincide. Thus, it makes sense to analyze the relationship between $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} as a measure of the change in association before and after ordinalization.

Let us start from a very simple case, i.e., dichotomization of the margins of a standard bivariate normal random variable $(Z_1,Z_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Z_1,Z_2)$$\end{document} , with zero marginal means, unit marginal variances, and linear correlation $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} . Consider the bivariate ordinal variable (X, Y) obtained as follows:

(3) $\begin{array}{c}X=\left(\begin{array}{cc}{x}_1& Z_1\le 0\\ x_2& Z_1>0\end{array}\right),Y=\left(\begin{array}{cc}{y}_1& Z_2\le 0\\ y_2& Z_2>0\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X={\left\{ \begin{array}{ll} x_1 &{}\quad Z_1\le 0\\ x_2 &{}\quad Z_1>0 \end{array}\right. },\quad Y={\left\{ \begin{array}{ll} y_1 &{} \quad Z_2\le 0\\ y_2 &{} \quad Z_2>0 \end{array}\right. }, \end{aligned}$$\end{document}

with $x_1<x_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_1<x_2$$\end{document} and $y_1<y_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_1<y_2$$\end{document} being two ordered categories. Therefore, the marginal probabilities for X and Y are $P(X=x_1)=P(X=x_2)=1/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X=x_1)=P(X=x_2)=1/2$$\end{document} and analogously $P(Y=y_1)=P(Y=y_2)=1/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(Y=y_1)=P(Y=y_2)=1/2$$\end{document} . Since we know that for a bivariate normal distribution $P(Z_1\le 0,Z_2\le 0)=P(Z_1>0,Z_2>0)=\frac{1}{4}+\frac{1}{2\pi }arcsin\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(Z_1\le 0,Z_2\le 0)=P(Z_1> 0,Z_2> 0)=\frac{1}{4} + \frac{1}{2\pi }\arcsin \rho $$\end{document} (see, e.g., McNeil et al. Reference McNeil, Frey and Embrechts2005), then the joint probability of (X, Y) is that displayed in the following table:

$\begin{array}{c}\begin{array}{cccc}X,Y& y_1& y_2& total\\ x_1& 1/4+1/\left(2\pi \right)arcsin\rho & 1/4-1/\left(2\pi \right)arcsin\rho & 1/2\\ x_2& 1/4-1/\left(2\pi \right)arcsin\rho & 1/4+1/\left(2\pi \right)arcsin\rho & 1/2\\ total& 1/2& 1/2& 1\end{array}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{cccc} X,Y &{} y_1 &{} y_2 &{} total\\ \hline x_1 &{} 1/4+1/(2\pi )\arcsin \rho &{}\quad \quad 1/4-1/(2\pi )\arcsin \rho &{} 1/2 \\ x_2 &{} 1/4-1/(2\pi )\arcsin \rho &{}\quad \quad 1/4+1/(2\pi )\arcsin \rho &{} 1/2\\ \hline total &{} 1/2 &{} 1/2 &{} 1\\ \end{array} \end{aligned}$$\end{document}

Based on it, by using Eq. (1), one can compute the gamma coefficient as a function of $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} :

(4) $\begin{array}{c}\gamma =\frac{4\pi arcsin\rho }{{\pi }^2+4{(arcsin\rho )}^2},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma = \frac{4\pi \arcsin \rho }{\pi ^2+ 4(\arcsin \rho )^2}, \end{aligned}$$\end{document}

or, recalling (2), in terms of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} :

(5) $\begin{array}{c}\gamma =\frac{2\tau }{1+{\tau }^2},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma = \frac{2\tau }{1+\tau ^2}, \end{aligned}$$\end{document}

whose graph is plotted in Fig. 1. This graph clearly shows how Goodman-Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for the dichotomized random variable is a strictly increasing and odd function of its analogue Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} for the bivariate normal distribution and, in absolute value, is always larger than or equal to $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , the equality holding when $\tau =0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =0$$\end{document} (i.e., when the two normal components are independent), when $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} equals 1 (perfectly positively correlated normal random variables), or when $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} equals $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} (perfectly negatively correlated normal random variables). It is worth noting that in this context, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is equal to $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} ( $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} ) only in case of a perfect monotonic (countermonotonic) relationship between X and Y.

Figure 1. Graph of the gamma coefficient for a dichotomized bivariate normal random variable with Kendall’s rank correlation $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , see Eq. (5). The dashed line is the 45 degrees line passing through the origin

By choosing thresholds different from zero for the random components $Z_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_1$$\end{document} and $Z_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_2$$\end{document} , we can numerically obtain the value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} corresponding to any value of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . Similarly, one can discretize either or both continuous random components into more than two categories, applying different sets of thresholds (or, equivalently, assigning different marginal distributions to the final ordinal variables). In general, let us consider a bivariate standard normal distribution $(Z_1,Z_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Z_1,Z_2)$$\end{document} whose two continuous components are discretized into two ordinal variables X and Y according to the following scheme:

$\begin{array}{c}X=\left(\begin{array}{cc}{x}_1& Z_1\le {\theta }_1\\ x_2& {\theta }_1<Z_2\le {\theta }_2\\ \cdots \\ x_H& {\theta }_{H-1}<Z_1\le {\theta }_H=+\infty \end{array}\right),Y=\left(\begin{array}{cc}{y}_1& Z_2\le {\eta }_1\\ y_2& {\eta }_1<Z_2\le {\eta }_2\cdots \\ y_K& {\eta }_{K-1}<Z_2\le {\eta }_K=+\infty \end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} X={\left\{ \begin{array}{ll} x_1 &{} Z_1\le \theta _1\\ x_2 &{} \theta _1< Z_2\le \theta _2\\ \cdots \\ x_H &{} \theta _{H-1}< Z_1\le \theta _H=+\infty \end{array}\right. }, \quad Y={\left\{ \begin{array}{ll} y_1 &{}\quad Z_2\le \eta _1\\ y_2 &{}\quad \eta _1<Z_2\le \eta _2 \cdots \\ y_K &{}\quad \eta _{K-1}< Z_2\le \eta _K=+\infty \end{array}\right. } \end{aligned}$$\end{document}

where ${\theta }_1<{\theta }_2<\cdots <{\theta }_H=\infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _1<\theta _2<\dots <\theta _H=\infty $$\end{document} and ${\eta }_1<{\eta }_2<\cdots <{\eta }_K=\infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _1<\eta _2<\dots <\eta _K=\infty $$\end{document} constitute two sets of thresholds. If we denote with F(i, j) the bivariate joint cumulative probability $P(X\le x_i,Y\le y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(X\le x_i,Y\le y_j)$$\end{document} of the bivariate ordinal variable (X, Y), $i=1,\cdots ,H$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots ,H$$\end{document} , $j=1,\cdots ,K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,\dots ,K$$\end{document} , and let $F_1\left(i\right):=P(X\le x_i)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1(i):=P(X\le x_i)$$\end{document} and $F_2\left(j\right):=P(Y\le y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2(j):=P(Y\le y_j)$$\end{document} be the two marginal cumulative distributions, we have $F_1\left(i\right)=\Phi \left({\theta }_i\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1(i)=\Phi (\theta _i)$$\end{document} and $F_2\left(j\right)=\Phi \left({\eta }_j\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2(j)=\Phi (\eta _j)$$\end{document} , and the joint cumulative distribution function (cdf) of (X, Y) is

(6) $\begin{array}{c}F(i,j)={\Phi }_{\tau }({\theta }_i,{\eta }_j)={\Phi }_{\tau }(F_1^{-1}\left(i\right),F_2^{-1}\left(j\right)),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(i,j)=\Phi _{\tau }(\theta _i,\eta _j)=\Phi _{\tau }(F_1^{-1}(i),F_2^{-1}(j)), \end{aligned}$$\end{document}

where ${\Phi }_{\tau }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{\tau }$$\end{document} is the joint cdf of a bivariate normal with standard components and rank correlation $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , and $F_1^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1^{-1}$$\end{document} ( $F_2^{-1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2^{-1}$$\end{document} ) is the generalized inverse of $F_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document} ( $F_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} ). The joint probabilities $p_{\mathrm{ij}}=P(X=x_i,Y=y_j)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}=P(X=x_i,Y=y_j)$$\end{document} are then obtained as

(7) $\begin{array}{c}{p}_{\mathrm{ij}}=F(i,j)-F(i-1,j)-F(i,j-1)+F(i-1,j-1)=\sum _{t=0}^1\sum _{v=0}^{1}F(i-t,j-v)·{(-1)}^{t+v}\\ \end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} p_{ij} = F(i,j) - F(i-1,j) - F(i,j-1) + F(i-1,j-1) = \sum _{t=0}^1\sum _{v=0}^1 F(i-t,j-v)\cdot (-1)^{t+v}\nonumber \\ \end{aligned}$$\end{document}

and based on these $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} , the gamma coefficient can be computed using Eq. (1).

In the following study, we will consider three different possible types of ordinal marginal distributions, namely, (i) uniform (i.e., equiprobable categories) (ii) symmetrical (non-uniform), based on normal scores, and (iii) asymmetrical (triangular). We will examine what kind of relationship $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} have under these three macro-settings by considering the same distribution for the two margins and varying the number of categories. Relevant code developed in the R environment (R Core Team 2020) is freely available as supplementary material.

Figure 2. Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for an ordinalized bivariate normal random variable with Kendall’s rank correlation $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ; the categories of the two ordinal variables are set both equal to an integer K and have uniform probabilities 1/K. The dashed line is the 45 degrees line passing through the origin

Figure 2 displays the graphs of the $(\tau ,\gamma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ,\gamma )$$\end{document} curve when the two marginal distributions have the same number of equiprobable categories $H=K=3,4,5,10,20$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=K=3,4,5,10,20$$\end{document} . Table 3 reports the values of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ranging from $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} to $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} , with step length of 0.1, and several values of the common number of categories of the two identical margins ( $H=K=2,3,5,7,10,20,50,100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=K=2, 3, 5, 7, 10, 20, 50, 100$$\end{document} ). Note that the $(\tau ,\gamma )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\tau ,\gamma )$$\end{document} curve, for a given H, is strictly increasing and always passes through the points $(-1,-1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1,-1)$$\end{document} , (0, 0), and (1, 1), and, as one can expect, by increasing H (i.e., when the ordinal distributions “resemble” a continuous one), it tends to get close to the 45-degree line passing through the origin. Note that $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is an odd function of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , i.e., $\gamma \left(\tau \right)=-\gamma (-\tau )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\tau )=-\gamma (-\tau )$$\end{document} . For a given value of $\tau >0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is a decreasing function of H; for a given value of $\tau <0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau <0$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is an increasing function of K; as K tends toward $\infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} seems to converge, though slowly, to $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . It is worth observing, however, that even for a moderately large number of categories, there is a substantial difference between the values of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} before and after ordinalization. When $H=100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=100$$\end{document} , there is still a relative difference of up to $2\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\%$$\end{document} between the values of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} .

Table 3. Values of Goodman and Kruskal’s gamma for several combinations of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and number of categories of the two identical marginal triangular distributions

We now move to non-uniform symmetrical ordinal distributions; in particular, we consider a probability mass function that somewhat resembles the probability density function of the normal variable. For this aim, the $(-4,+4)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-4,+4)$$\end{document} interval, whose probability for a standard normal variable is almost equal to 1, is divided into H equal-width adjacent intervals, $\left(-,4,+,8,(,i,-,1,),/,H,,,-,4,+,8,i,/,H\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( -4+8(i-1)/H,-4+8i/H\right) $$\end{document} , $i=1,\cdots ,H$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots ,H$$\end{document} . A similar construction was employed by Becker (Reference Becker1989) (who named it the “uniform cut-points (UCP) method” and employed it to study similarities between uniform association models and ordinalized bivariate normal distribution) and Ferrari and Barbiero (Reference Ferrari and Barbiero2012) (to study the effects of discretization of a bivariate normal distribution on the correlation coefficient). To the i-th interval, the i-th ordered category is associated, whose probability is thus equal to $\Phi (-4+8i/H)-\Phi (-4+8(i-1)/H)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi (-4+8i/H)-\Phi (-4+8(i-1)/H)$$\end{document} , $i=1,\cdots ,H$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots ,H$$\end{document} ; the residual probabilities of $(-\infty ,-4)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\infty ,-4)$$\end{document} and $(4,+\infty )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(4,+\infty )$$\end{document} are assigned to the first and last categories, respectively. Thus, the probability distribution is always symmetrical about the central category (if H is odd) or categories (if H is even). The probabilities obtained according to this scheme are graphically displayed, for $K=5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=5$$\end{document} and $K=10$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=10$$\end{document} , in Fig. 3.

Table 4 reports the values of Goodman and Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ranging from $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} to $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} , with a step length of 0.1, and several values of the common number of categories of the two identical symmetrical margins. Note that as in the case of equiprobable categories, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is an odd function of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . For a given value of $\tau >0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} and for $K>3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K>3$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is a decreasing function of K; for a given value of $\tau <0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau <0$$\end{document} and for $K>3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K>3$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is an increasing function of K (for $K=2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=2$$\end{document} , we obtain again a uniform distribution with two equally probable categories). As K tends toward $\infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} converges to $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , but much more slowly than in the case of uniform margins. When $K=100$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=100$$\end{document} , the relative difference between $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} in is not negligible at all, attaining $4.5\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4.5\%$$\end{document} . This fact can be easily explained by just looking at the barplot in the right side of Fig. 3: among the K categories, only the central ones have appreciable probabilities, so the “actual” number of categories is (much) smaller than K, and then a larger value of K is required to obtain the same difference between $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} that is achieved in case of uniform distributions.

Figure 3. Barplot of two symmetrical non-uniform distributions, with 5 (left) and 10 (right) categories, obtained by mimicking the probability density function of a standard normal variable

Table 4. Values of Goodman and Kruskal’s gamma for several combinations of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and number of categories of the two identical marginal symmetrical distributions (uniform cut-points)

In the end, we consider strongly asymmetrical distributions, namely triangular distributions with H categories, where the probability of the i-th category is proportional to i (through some positive constant $\alpha$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} ). Since under this assumption, ${\sum }_{i=1}^H{p}_i={\sum }_{i=1}^H\alpha i=\alpha H(H+1)/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^H p_i = \sum _{i=1}^H \alpha i=\alpha H(H+1)/2$$\end{document} , then $p_i=2i/\left[H(H+1)\right]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_i=2i/[H(H+1)]$$\end{document} . Choosing this distribution as the marginal distribution of both X and Y, Table 5 reports the corresponding values of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} ranging from $-1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1$$\end{document} to $+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+1$$\end{document} , with a step length of 0.1, and for several values of the common number of categories. One can notice that, differently from the two cases analyzed before, due to the asymmetrical nature of the margins, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is no longer an odd function of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , $\gamma \left(\tau \right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (\tau )$$\end{document} being generally different from $-\gamma (-\tau )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\gamma (-\tau )$$\end{document} .

Table 5. Values of Goodman and Kruskal’s gamma for several combinations of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and number of categories of the two identical marginal triangular distributions

For a given value of $\tau >0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau >0$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is a decreasing function of K; for a given value of $-0.9<\tau <0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-0.9<\tau <0$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is an increasing function of K. When $\tau =-0.9$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =-0.9$$\end{document} , we have a curious behavior of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} as a function of K, which is displayed in Fig. 4.

Figure 4. Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for an ordinalized bivariate normal random variable with Kendall’s rank correlation $\tau =-0.9$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =-0.9$$\end{document} as a function of the common number of categories (from 2 to 10) of the two ordinal triangular variables

It is worth underlining how the values of $\left|\gamma \right|$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\gamma |$$\end{document} are significantly larger than the corresponding $\left|\tau \right|$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\tau |$$\end{document} , especially when the number of categories is small. This feature is more apparent for negative values of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . For example, when $\tau =-0.5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =-0.5$$\end{document} and $H=5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=5$$\end{document} , the value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is $-0.69721$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-0.69721$$\end{document} . Even when discretizing the two continuous components through an extremely large number of ordered categories, such as 100, the relative difference between the values of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is still non-negligible (not smaller than $2.6\%$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.6\%$$\end{document} when $-0.9\le \tau \le 0.9$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-0.9\le \tau \le 0.9$$\end{document} ).

4. Building a Bivariate Ordinal Variable with Prescribed Margins and Gamma Coefficient

Researchers can be interested in building and simulating samples from a bivariate (multivariate) ordinal distribution with prescribed marginal distributions and (pairwise) levels of association, expressed in terms of the gamma coefficient. Such a concern may arise when one has to show the appropriateness and study the performance or robustness of a novel statistical technique, since it is not always possible to do so by using analytic arguments, except in elementary cases; on the contrary, generating data replicates that mimic the real data’s characteristics of interest allows one to study the performance of the statistical method in any given setting. For example, researchers may be interested in assigning and preserving in their simulation study the marginal structures as well as ordinal associations (Demirtas Reference Demirtas2006). In other circumstances, their concern is matching marginal means, variances, skewnesses and kurtoses, and intercorrelations (Vale and Maurelli Reference Vale and Maurelli1983). van der Ark and van Aert (Reference van der Ark and van Aert2015), with the aim of investigating the performance of different types of confidence intervals for $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , constructed several bivariate ordinal distributions by assigning each a fixed value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and imposing constraints on the margins, considering uniform or skewed distributions.

However, when assigning the margins and association (correlation) value for two random variables, a uniqueness issue arises for the corresponding bivariate distribution. In fact, one of the fallacies of Pearson’s correlation is that if we consider a bivariate random vector (X, Y) with assigned marginal distributions $F_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document} and $F_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} and a feasible value of $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} , then $F_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document} , $F_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} , and $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} do not in general determine the joint distribution F univocally; on the contrary, there may exist several (even infinite) joint distributions whose margins are $F_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document} and $F_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} and whose linear correlation is $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} (see, e.g., McNeil et al. Reference McNeil, Frey and Embrechts2005, p. 202). This issue is not overcome even by other dependence measures such as Spearman’s $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} and Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , nor most likely by Kruskal and Goodman’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} . Nevertheless, if we restrict our focus to the family of joint distributions obtained by discretizing a specific class of continuous random vectors, we can find that there is a unique joint distribution satisfying the match of margins and association value.

Let us consider a bivariate ordinal variable with joint probabilities $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} , $i=1,\cdots ,H$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\dots ,H$$\end{document} , $j=1,\cdots ,K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,\dots ,K$$\end{document} , and marginal probailities $p_{i·}={\sum }_{j=1}^K{p}_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{i\cdot }=\sum _{j=1}^K p_{ij}$$\end{document} and $p_{·j}={\sum }_{i=1}^H{p}_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\cdot j}=\sum _{i=1}^H p_{ij}$$\end{document} . Let us start with the simplest case: for a bivariate ordinal distribution with dichotomous margins, $H=K=2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=K=2$$\end{document} , the problem consists of finding the solution to the following equation system:

$\begin{array}{c}\left(\begin{array}{cc}{p}_{11}{p}_{22}(1-\gamma )& =p_{12}{p}_{21}(1+\gamma )\\ p_{12}& =1-p_{·1}-p_{22}\\ p_{21}& =1-p_{1·}-p_{22}\\ p_{11}& =p_{1·}-p_{12}\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} p_{11}p_{22}(1-\gamma )&{}=p_{12}p_{21}(1+\gamma )\\ p_{12}&{}=1-p_{\cdot 1}-p_{22}\\ p_{21}&{}=1-p_{1\cdot }-p_{22}\\ p_{11}&{}=p_{1\cdot }-p_{12} \end{array}\right. } \end{aligned}$$\end{document}

which becomes

$\begin{array}{c}\left(\begin{array}{cc}(p_{1·}+p_{·1}+p_{22}-1)p_{22}(1-\gamma )& =(1-p_{·1}-p_{22})(1-p_{1·}-p_{22})(1+\gamma )\\ p_{12}& =1-p_{·1}-p_{22}\\ p_{21}& =1-p_{1·}-p_{22}\\ p_{11}& =p_{1·}-p_{12}\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} (p_{1\cdot }+p_{\cdot 1}+p_{22}-1)p_{22}(1-\gamma ) &{}=(1-p_{\cdot 1}-p_{22})(1-p_{1\cdot }-p_{22})(1+\gamma )\\ p_{12}&{}=1-p_{\cdot 1}-p_{22}\\ p_{21}&{}=1-p_{1\cdot }-p_{22}\\ p_{11}&{}=p_{1\cdot }-p_{12} \end{array}\right. } \end{aligned}$$\end{document}

and then

$\begin{array}{c}\left(\begin{array}{cc}2\gamma p_{22}^2-p_{22}(1+3\gamma -2\gamma (p_{1·}+p_{·1}))+(1+\gamma )·(1-p_{1·})(1-p_{·1})& =0\\ p_{12}& =1-p_{·1}-p_{22}\\ p_{21}& =1-p_{1·}-p_{22}\\ p_{11}& =p_{1·}-p_{12}\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} 2\gamma p_{22}^2 - p_{22} (1+3\gamma -2\gamma (p_{1\cdot }+p_{\cdot 1})) +(1+\gamma )\cdot (1-p_{1\cdot })(1-p_{\cdot 1}) &{}=0\\ p_{12}&{}=1-p_{\cdot 1}-p_{22}\\ p_{21}&{}=1-p_{1\cdot }-p_{22}\\ p_{11}&{}=p_{1\cdot }-p_{12} \end{array}\right. } \end{aligned}$$\end{document}

From the first equation of the system above, we derive that $p_{22}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{22}$$\end{document} is the unique real root (the one with minus sign) of the second-order equation $a{x}^2+bx+c=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ax^2+bx+c=0$$\end{document} with

$\begin{array}{c}\left(\begin{array}{cc}a& =2\gamma \\ b& =-(1+3\gamma -2\gamma ·p_{1·}-2\gamma ·p_{·1})\\ c& =(1+\gamma )·(1-p_{1·})·(1-p_{·1})\end{array}\right);\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} a&{}=2\gamma \\ b&{}=-(1+3\gamma -2\gamma \cdot p_{1\cdot }-2\gamma \cdot p_{\cdot 1})\\ c&{}=(1+\gamma )\cdot (1-p_{1\cdot })\cdot (1-p_{\cdot 1}) \end{array}\right. }; \end{aligned}$$\end{document}

the other joint probabilities can then be easily computed in cascade from the remaining equations. Thus, the problem has a unique feasible solution. For example, if we assign the margins $p_{1·}=0.5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{1\cdot }=0.5$$\end{document} , $p_{·1}=0.5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\cdot 1}=0.5$$\end{document} , and set $\gamma =3/5$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =3/5$$\end{document} , we obtain the following second-degree equation: $3x^2-4x+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3 x^2 - 4 x + 1$$\end{document} , whose roots are $p_{22,1}=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{22,1}=1$$\end{document} and $p_{22,2}=1/3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{22,2}=1/3$$\end{document} ; taking the latter leads to the solution:

$\begin{array}{c}\left(\begin{array}{cc}{p}_{22}& =1/3\\ p_{12}& =1/6\\ p_{11}& =1/3\\ p_{12}& =1/6\end{array}\right)\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} p_{22}&{}=1/3\\ p_{12}&{}=1/6\\ p_{11}&{}=1/3\\ p_{12}&{}=1/6 \end{array}\right. } \end{aligned}$$\end{document}

Note that this set of probabilities corresponds to the joint distribution of the bivariate ordinal variable obtained by discretizing a bivariate normal (3) with $\rho =1/2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho =1/2$$\end{document} (or, equivalently, $\tau =1/3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =1/3$$\end{document} ; see Eq. (4)).

For larger numbers of categories, there will be infinite solutions, which are not easy to derive, due to the nonlinear nature of the problem: the gamma coefficient is not linear in the $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} ’s, so the equation system is nonlinear, as we have already experienced with a $2\times 2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} contingency table. This means that there are several bivariate ordinal distributions $p_{\mathrm{ij}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}$$\end{document} whose margins are the assigned $p_{i·}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{i\cdot }$$\end{document} and $p_{·j}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{\cdot j}$$\end{document} and whose gamma coefficient is equal to a prespecified value $\gamma \in [-1,+1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in [-1,+1]$$\end{document} . If one is interested in building just one bivariate ordinal variable with assigned margins and gamma, he/she can restrict his/her attention to the class of bivariate ordinal variables with the assigned margins derived by ordinalization of a standard bivariate normal distribution; then he/she can exploit the results of the previous section, by accommodating the value of the parameter $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} in order to match the assigned value $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} between the assigned ordinal margins. Due to the monotonic relationship between $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , whose form depends on the selected margins, and the fact that $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} can span its entire natural range $[-1,+1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,+1]$$\end{document} , which we proved empirically, we are confident that there will be a unique bivariate ordinalized distribution satisfying the requested requisites. This means that among all the bivariate distributions with assigned margins and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , we select the unique one whose underlying continuous latent model is the bivariate normal.

Writing the relationship between the two association measures as $\gamma =g(\tau ;F_1,F_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma = g(\tau ;F_1,F_2)$$\end{document} , we just need to find the (unique) root $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} of the equation $\gamma -g(\tau ;F_1,F_2)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma -g(\tau ;F_1,F_2)=0$$\end{document} , $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} being an assigned number in $[-1,+1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,+1]$$\end{document} and $F_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document} , $F_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} being assigned ordinal distributions. Recovering the correct $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is a task that can be carried out by using an iterative procedure, which requires setting an initial value. Since we have empirically found that $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} after discretization is – in absolute value – larger than the rank correlation $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} of the bivariate normal random variable, one can use ${\tau }^{\left(0\right)}=\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{(0)}=\gamma $$\end{document} as a starting value for the unknown $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} . One can then determine the corresponding value of the bivariate ordinal random variable ${\gamma }^{\left(0\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^{(0)}$$\end{document} and then iteratively adjust the value of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} according to some updating rule till ${\gamma }^{\left(t\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^{(t)}$$\end{document} converges to the assigned value. The updating process shall take into account that when $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is zero, $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is zero, too, and should prevent the updated value of ${\tau }^{\left(t\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{(t)}$$\end{document} from escaping the interval $[-1,+1]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[-1,+1]$$\end{document} ; a proposal is suggested in the following algorithm:

1. 1. Set ${\tau }^{\left(0\right)}\leftarrow 0,{\gamma }^{\left(0\right)}\leftarrow 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{(0)}\leftarrow 0,\gamma ^{(0)}\leftarrow 0$$\end{document} ; let $ϵ>0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon >0$$\end{document} be an arbitrarily small number

2. 2. Set $t\leftarrow 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\leftarrow 1$$\end{document} and ${\tau }^{\left(t\right)}\leftarrow \gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{(t)}\leftarrow \gamma $$\end{document}

3. 3. Compute $F(i,j;{\tau }^{\left(t\right)})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(i,j;\tau ^{(t)})$$\end{document} by using (6)

4. 4. Compute $p(i,j;{\tau }^{\left(t\right)})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(i,j;\tau ^{(t)})$$\end{document} by using (7)

5. 5. Compute ${\gamma }^{\left(t\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ^{(t)}$$\end{document} for $p(i,j;{\tau }^{\left(t\right)})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(i,j;\tau ^{(t)})$$\end{document} by using (1)

6. 6. If $|{\gamma }^{\left(t\right)}-\gamma |<ϵ$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\gamma ^{(t)} - \gamma |<\epsilon $$\end{document} stop; else

   set $t\leftarrow t+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\leftarrow t+1$$\end{document} ,

   ${\tau }^{\left(t\right)}\leftarrow {\tau }^{(t-1)}+m^{\left(t\right)}(\gamma -{\gamma }^{(t-1)})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{(t)}\leftarrow \tau ^{(t-1)}+m^{(t)}(\gamma -\gamma ^{(t-1)})$$\end{document} , with $m^{\left(t\right)}=\frac{{\tau }^{(t-1)}-{\tau }^{(t-2)}}{{\gamma }^{(t-1)}-{\gamma }^{(t-2)}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m^{(t)}=\dfrac{\tau ^{(t-1)}-\tau ^{(t-2)}}{\gamma ^{(t-1)}-\gamma ^{(t-2)}}$$\end{document} ;

   ${\tau }^{\left(t\right)}\leftarrow \left(\begin{array}{cc}min({\tau }^{\left(t\right)},1)& \text{if}{\tau }^{\left(t\right)}\ge 0\\ max({\tau }^{\left(t\right)},-1)& \text{if}{\tau }^{\left(t\right)}<0\end{array}\right);$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau ^{(t)}\leftarrow {\left\{ \begin{array}{ll} \min (\tau ^{(t)},1)&{} \text { if } \tau ^{(t)}\ge 0\\ \max (\tau ^{(t)},-1) &{} \text { if } \tau ^{(t)}< 0 \end{array}\right. };$$\end{document}

   go back to 3.

Given the monotonic relationship between $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} for the bivariate normal model and its ordinalized counterpart, the above algorithm should be able to recover the value of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} inducing the target $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} in a few steps, for any choice of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} and of $F_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_1$$\end{document} and $F_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_2$$\end{document} . The algorithm is implemented in the R environment and is freely available as supplementary material.

5. Application to Inference

If a bivariate ordinal sample of size n is available, one can assume it is an i.i.d. sample from an ordinalized bivariate normal distribution, or, more generally, from an ordinalized bivariate continuous distribution whose unique copula is the Gaussian one (see, in this sense, Grønneberg and Foldnes Reference Grønneberg and Foldnes2019; Foldnes and Grønneberg Reference Foldnes and Grønneberg2019, for a more detailed account on identifiability issues). One has then to estimate the value of the unknown $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} (or $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} ) and those of the unknown thresholds that define the marginal distributions of X and Y; this can be done by numerically maximizing the joint log-likelihood of the observed sample with respect to all the unknown parameters simultaneously (see Olsson Reference Olsson1979, for details).

As an alternative to this full maximum likelihood approach, one can resort to the following method of moments for estimating the thresholds and $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} :

1. 1. Compute the empirical cumulative distribution functions ${\hat{F}}_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{F}}_1$$\end{document} and ${\hat{F}}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{F}}_2$$\end{document} based on the bivariate sample, and the thresholds by using the inverse cdf (see Sect. 3). Compute the sample value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , ${\hat{\gamma }}_M$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}_M$$\end{document} , based on the bivariate sample.

2. 2. Compute the value of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} , ${\hat{\tau }}_M$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\tau }}_M$$\end{document} , of the underlying bivariate normal distribution, inducing ${\hat{\gamma }}_M$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\gamma }}_M$$\end{document} given ${\hat{F}}_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{F}}_1$$\end{document} and ${\hat{F}}_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{F}}_2$$\end{document} , by resorting to the iterative procedure of Sect. 4. That is, since we can write $\gamma =g(\tau ;F_1,F_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =g(\tau ;F_1,F_2)$$\end{document} and $\tau =g^{-1}(\gamma ;F_1,F_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau =g^{-1}(\gamma ;F_1,F_2)$$\end{document} , ${\hat{\tau }}_M=g^{-1}({\hat{\gamma }}_M;{\hat{F}}_1,{\hat{F}}_2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\tau }}_M = g^{-1}({\hat{\gamma }}_M;{\hat{F}}_1,{\hat{F}}_2)$$\end{document} .

Given the way the estimates are derived, this method can be classified as a two-stage method of moments. The unknown parameters are in fact estimated in two sequential steps: first, the thresholds for $Z_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_1$$\end{document} and $Z_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_2$$\end{document} are estimated independently based on the empirical cdf of X and Y, respectively, and the sample value of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is calculated as well; then, the dependence parameter $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is estimated based on the quantities computed at the first stage.

A numerical example with artificial data is provided in the R code provided as supplementary material, where the two methods are implemented and compared.

6. Real Data

In a now classic study of mental health in Manhattan, New York, Srole and Fischer (Reference Srole and Fischer1978) explore the relationship, among others, between mental impairment (Y) and parents’ socioeconomic status (X). Table 8, from that study, has been used extensively to illustrate the utility and application of models for ordered categorical data.

Table 8. The Midtown Manhattan Study: Mental Health and Parents’ Socioeconomic Status

The sample value of Goodman-Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} is 0.15429. By assuming a bivariate standard normal distribution underlying the bivariate ordinal data, we can derive the maximum likelihood estimates (MLEs) for Kendall’s tau and the thresholds for the two margins (see Table 9). Note that the MLE of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is 0.10762, quite a bit smaller than the corresponding estimate of $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} , confirming the empirical results of the study in Sect. 3. However, the estimate is significantly greater than zero, with an associated standard error of 0.01718, which provides strong evidence that mental health status and parents’ socioeconomic status are positively associated (i.e., higher socioeconomic status is associated with better mental health), even if the strength of that association is quite small. Note also that all the MLEs of the thresholds are highly significant, except for ${\theta }_3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _3$$\end{document} , which is the central threshold for the variable parents’ socioeconomic status.

Table 9. MLEs for the Midtown Manhattan Mental study data, assuming an ordinalized bivariate normal model

Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1 (maximized value of the log-likelihood function: $-5171.345$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-5171.345$$\end{document} ). The results are obtained by using the package maxLik

If one applies the two-stage method of moments in Sect. 5, the estimate of $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} is equal to 0.10665, just slightly different from the maximum likelihood estimate.

Table 10 displays the expected joint frequencies $n_{\mathrm{ij}}^{\left(o\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{ij}^{(o)}$$\end{document} under the bivariate ordinalized distribution whose parameters are set equal to the corresponding MLEs. Note that under this model, the marginal frequency distributions are not equal to the analogous ones of Table 8 (even if they are very close to each other). This is because the full MLEs of the thresholds are not equal to the corresponding marginal MLEs (which are used as starting values for the optimization routine). Comparing the observed and theoretical contingency tables, one can notice some slight discrepancies between homologous frequencies. In order to (approximately) evaluate the goodness-of-fit of the suggested ordinalized bivariate normal model to the data at issue, we computed the usual chi-squared statistic based on the theoretical and observed frequencies (all theoretical joint frequencies are greater than 5, so there is no need for pooling cells); its value is 8.84. Under the null hypothesis that the data come from the ordinalized bivariate normal distribution with parameters equal to their MLEs, this statistic approximately follows a chi-square distribution with a number of degrees of freedom equal to $24-9-1=14$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$24-9-1=14$$\end{document} , where 24 is the number of pooled frequencies and 9 is the number of estimated parameters. The corresponding p-value is 0.841, leading us to comfortably accept the null hypothesis. The value of the log-likelihood ratio statistic, given by $-2{\sum }_{i=1}^H{\sum }_{j=1}^K{n}_{\mathrm{ij}}log(n_{\mathrm{ij}}^{\left(o\right)}/n_{\mathrm{ij}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-2\sum _{i=1}^H\sum _{j=1}^Kn_{ij}\log (n_{ij}^{(o)}/n_{ij})$$\end{document} , is equal to 8.959 (p-value 0.834) and thus leads to the same conclusion.

Table 10. Expected joint frequencies under the ordinalized bivariate normal model whose parameters are obtained by applying the maximum likelihood method to the data of Table 8

Other more sophisticated models may fit the sample data better. For some examples of alternatives (namely, association models) fitting the Midtown Manhattan Mental study data, see for example Becker (Reference Becker2014). In particular, a uniform association model can be used: the expected cell frequencies can be calculated by using the vcdExtraR package (Friendly Reference Friendly2017) (see the supplementary material). Note that these expected frequencies are quite close to their analogs under the ordinalized bivariate normal model, confirming the findings of Becker (Reference Becker1989) and as stressed by Kateri (Reference Kateri2014). To prove this, one can compute the Kullback-Leibler distances between the two theoretical joint distributions ( $p_{\mathrm{ij}}^{\left(o\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}^{(o)}$$\end{document} and $p_{\mathrm{ij}}^{\left(u\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{ij}^{(u)}$$\end{document} , for the ordinalized normal and uniform association models, respectively): $K_1=\sum \sum p_{\mathrm{ij}}^{\left(o\right)}log(p_{\mathrm{ij}}^{\left(o\right)}/p_{\mathrm{ij}}^{\left(u\right)})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_1=\sum \sum p_{ij}^{(o)} \log (p_{ij}^{(o)}/p_{ij}^{(u)})$$\end{document} and $K_2=\sum \sum p_{\mathrm{ij}}^{\left(u\right)}log\left(p_{\mathrm{ij}}^{\left(u\right)}\right)/p_{\mathrm{ij}}^{\left(o\right)}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2=\sum \sum p_{ij}^{(u)} \log (p_{ij}^{(u)})/p_{ij}^{(o)}$$\end{document} . The values are computed as $K_1=0.0001721$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_1=0.0001721$$\end{document} and $K_2=0.0001724$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_2=0.0001724$$\end{document} , which are very small, proving that the two distributions are very close to each other.

7. Conclusions and Further Research

We focused on the bivariate standard normal variable and studied the change in association before and after ordinalization, measured by Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and Goodman and Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} . This analysis has been facilitated (i) by the wide availability of software implementations of the bivariate normal cdf (despite its non-closed expression), which is required when computing the joint probabilities of the bivariate ordinal distribution resulting from discretization, and (ii) by the analytic relationship between Pearson’s correlation $\rho$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} and Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} for the bivariate normal distribution. We empirically investigated the relationship between $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} by considering several specific configurations for the two final marginal distributions (uniform, unimodal or bimodal symmetric, triangular). The study confirmed a somewhat expected result, i.e., the association measure tends to inflate after discretization, and to a larger extent when the number of ordered categories is small. For a same number of categories, we also highlighted the effect of the marginal probabilities in the change of association. Based on these results, we also elaborated a scheme for building and simulating samples from a bivariate ordinal distribution with assigned margins and value of association.

Other bivariate continuous distributions or bivariate copulas can be explored when the interest is in analyzing the effects of ordinalization on Goodman and Kruskal’s $\gamma$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} or in constructing a bivariate ordinal variable with assigned marginal distributions and association; one should prefer the parametric copula families satisfying conditions (i) and (ii) mentioned above. Moreover, one should focus on so-called “comprehensive” bivariate copulas (Nelsen Reference Nelsen2006, p.15), i.e., copulas able to model continuously the whole range of dependence from the lower to the upper Fréchet bounds passing through the product copula. As for (i), there is a wide variety of parametric copulas with a closed-form expression of their joint cdf. As for (ii), an analytic expression linking the copula parameter to its Kendall’s correlation is not always available. For example, for the Frank family of copulas,

$\begin{array}{c}C(u_1,u_2;\theta )=-\frac{1}{\theta }ln\left(1,+,\frac{(e^{-\theta u_1}-1)(e^{-\theta u_2}-1)}{e^{-\theta }-1}\right),\theta \in R\backslash \left(0\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(u_1,u_2;\theta )=-\frac{1}{\theta } \ln \left[ 1+\frac{(e^{-\theta u_1}-1)(e^{-\theta u_2}-1)}{e^{-\theta }-1}\right] ,\theta \in {\mathbb {R}}\setminus \left\{ 0 \right\} , \end{aligned}$$\end{document}

we have the following relationship between Kendall’s $\tau$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document} and $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} : $\tau \left(\theta \right)=1-4[1-D_1\left(\theta \right)]/\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (\theta ) = 1 - 4[1-D_1(\theta )]/\theta $$\end{document} , with $D_1\left(x\right)=\frac{1}{x}{\int }_0^x\frac{t}{e^t-1}\text{d}t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1(x)=\frac{1}{x}\int _0^x \frac{t}{e^t-1}\text {d}t$$\end{document} , which can be computed numerically. Note that letting the parameter $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} go to 0, the Frank copula boils down to the product copula, with $\tau \left(0\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (0)=0$$\end{document} ; when $\theta \to -\infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \rightarrow -\infty $$\end{document} , C reduces to the countermonotonicity copula; when $\theta \to +\infty$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta \rightarrow +\infty $$\end{document} , C reduces to the comonotonicity copula. The Plackett family of copulas has the following form: