Directional nature of Goodman–Kruskal gamma and some consequences: identity of Goodman–Kruskal gamma and Somers delta, and their connection to Jonckheere–Terpstra test statistic

Abstract

Although usually taken as a symmetric measure, G is shown to be a directional coefficient of association. The direction in G is not related to rows or columns of the cross-table nor the identity of the variables to be a predictor or a criterion variable but, instead, to the number of categories in the scales. Under the conditions where there are no tied pairs in the dataset, G equals Somers’ D so directed that the variable with a wider scale (X) explains the response pattern in the variable with a narrower scale (g), that is, D(g│X). Hence, G = G(g│X) = D(g│X) but G ≠ D(X│g) and G ≠ D(symmetric). If there are tied pairs, the estimates by G = G(g│X) are more liberal in comparison with those by D(g│X). Algebraic relation of G and D with Jonckheere–Terpstra test statistic (JT) is derived. Because of the connection to JT, G = G(g│X) and D = D(g│X) indicate the proportion of logically ordered test-takers in the item after they are ordered by the score. It is strongly recommendable that gamma should not be used as a symmetric measure, and it should be used directionally only when willing to explain the behaviour of a variable with a narrower scale by the variable with a wider scale. This fits well with the measurement modelling settings.

Deflation-Corrected Estimators of Reliability

“…D appeared to be less affected by MEC than PMC but not to the extent as RPC and G (see also Figure 1). The reason for the latter is that while RPC and G are not affected by the tied cases, D is, specifically, with short tests (see the differences of D and G also in Metsämuuronen, 2021a).…”

“…The notation in names ρ α_RPCiθ , ρ α_Giθ , and ρ α_Diθ refers to the facts that the base of the estimator is alpha (α), the weight factor is manifested as RPC, G, or D representing different types of correlations between item and the score variable, and the manifestation of the score variable (θ) could be a raw score (θ X ) or factor score variable (θ FA ), as examples. Some of these kinds of estimators are discussed by Metsämuuronen and Ukkola (2019) and Metsämuuronen (2020bMetsämuuronen ( , 2021a. Another type of solution is discussed by Zumbo et al (2007) and Gadermann et al (2012) by replacing the matrix of PMCs by a matrix of RPCs in forming the factor loadings; this leads to a coefficient called ordinal alpha discussed above.…”

Rank–Polyserial Correlation: A Quest for a “Missing” Coefficient of Correlation

“…After all, estimators of the association have a central role to play, for example, as the estimators of the item-score association and embedded to estimators of reliability (see discussion in, e.g., [26,29,30]). This perspective leads us to compare the options of R RP with its traditional alternatives: R PP = ρ gX , often called item-total correlation (Rit), Henrysson's corrected item-total correlation (R PPH ) [31], also known as item-rest correlation (Rir), coefficient eta directed so that X explains the order in g or "g given X" 2 , 2 A specific peculiarity in naming of the directions may be necessary to discuss here (see also [2,29,32]) because the directional estimators are used in what follows. With the truly directional estimators D and eta, in widely used software packages as IBM SPSS, SAS, and R-libraries, this specific direction is traditionally named as "X dependent" (see, e.g., [15,[33][34][35][36][37]).…”

“…(e.g., [32,33,37]) where P is the sum of the concordant pairs of two observations x i and x j , and, correspondingly, y i and y j , and Q is the sum of the discordant pairs. Because of Eq.…”

Improving the Sustainability Effectiveness of Traditional Arts and Crafts Using Supply–Demand and Ordered Logistic Regression Techniques in Taiyuan, China

“…Moreover, the test of a parallel line is applied to the model to verify further whether the influence coefficient of the explanatory variables on the OR logarithm is constant or not when the cutting points of the dependent variables take different values. To check for independence, we used Goodman-Kruskal's gamma test and Pearson's chi-squared test [28].…”