Although Goodman-Kruskal gamma (G) is used relatively rarely it has promising potential as a coefficient of association in educational settings. Characteristics of G are studied in three sub-studies related to educational measurement settings. G appears to be unexpectedly appealing as an estimator of association between an item and a score because it strictly indicates the probability to get a correct answer in the test item given the score, and it accurately produces perfect latent association irrespective of distributions, degrees of freedom, number of tied pairs and tied values in the variables, or the difficulty levels in the items. However, it underestimates the association in an obvious manner when the number of categories in the item is more than four. Towards this, a dimension-corrected G (G2) is proposed and its characteristics are studied. Both G and G2 appear to be promising alternatives in measurement modelling settings, G with binary items and G2 with binary, polytomous and mixed datasets.

Estimates of reliability by traditional estimators are deflated, because the item-total or item-score correlation (Rit) or principal component or factor loading (λi) embedded in the estimators are seriously deflated. Different optional estimators of correlation that can replace Rit and λi are compared in this article. Simulations show that estimators such as polychoric correlation (RPC), gamma (G), dimension-corrected G (G2), and attenuation-corrected Rit (RAC) and eta (EAC) reflect the true correlation without any loss of information with several sources of technical or mechanical error in the estimators of correlation (MEC) including extreme item difficulty and item variance, small number of categories in the item and in the score, and the varying distributions of the latent variable. To obtain deflation-corrected reliability, RPC, G, G2, RAC, and EAC are likely to be the best options closely followed by r-bireg or r-polyreg coefficient (RREG).

Underestimation of reliability is discussed from the viewpoint of deflation in estimates of reliability caused by artificial systematic technical or mechanical error in the estimates of correlation (MEC). Most traditional estimators of reliability embed product–moment correlation coefficient (PMC) in the form of item–score correlation (Rit) or principal component or factor loading (λi). PMC is known to be severely affected by several sources of deflation such as the difficulty level of the item and discrepancy of the scales of the variables of interest and, hence, the estimates by Rit and λi are always deflated in the settings related to estimating reliability. As a short-cut to deflation-corrected estimators of reliability, this article suggests a procedure where Rit and λi in the estimators of reliability are replaced by alternative estimators of correlation that are less deflated. These estimators are called deflation-corrected estimators of correlation (DCER). Several families of DCERs are proposed and their behavior is studied by using polychoric correlation coefficient, Goodman–Kruskal gamma, and Somers delta as examples of MEC-corrected coefficients of correlation.

Objective: To study indoor environmental quality (IEQ) in elementary school buildings and its association with students' learning outcomes. Conclusion: Ventilation is associated with thermal comfort and students' learning outcomes.The ventilation system requires scheduled maintenance or replacement as well as ongoing ventilation adjustment to accommodate the number of students at any one time.

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.