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AbstractWhen the response pattern in a test item deviates from the deterministic pattern, the percentage of correct answers (p) is shown to be a biased estimator for the latent item difficulty (π). This is specifically true with the items of medium item difficulty. Four elements of impurities in p are formalized in the binary settings and four new estimators of π are proposed and studied. Algebraic reasons and a simulation suggest that, except the case of deterministic item discrimination, the real item difficulty is almost always more extreme than what p indicates. This characteristic of p to be biased toward a medium-leveled item difficulty has a strict consequence to item response theory (IRT) and Rasch modeling. Because the classical estimator of item difficulty p is a biased estimator of the latent difficulty level, the item parameters A and B and the person parameter θ within IRT modeling are, consequently, biased estimators of item discrimination and item difficulty as well as ability levels of the test takers.

The article "The effect of various simultaneous sources of mechanical error in the estimators of correlation causing deflation in reliability: seeking the best options of correlation for deflation-corrected reliability".

AbstractIn general linear modeling (GLM), eta squared (η2) is the dominant statistic for the explaining power of an independent variable. This article discusses a less-studied deficiency in η2: its values are seriously deflated, because the estimates by coefficient eta (η) are seriously deflated. Numerical examples show that the deflation in η may be as high as 0.50–0.60 units of correlation and in η2 as high as 0.70–0.80 units of explaining power. A simple mechanism to evaluate and correct the artificial attenuation is proposed. Because the formulae of η and point-biserial correlation are equal, η can also get negative values. While the traditional formulae give us only the magnitude of nonlinear association, a re-considered formula for η gives estimates with both magnitude and direction in binary cases, and a short-cut option is offered for the polytomous ones. Although the negative values of η are not relevant when η2 is of interest, this may be valuable additional information when η is used with non-nominal variables.

AbstractEstimates 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).

AbstractAlthough 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.