Conditional Reliability Coefficients for Test Scores
The most widely used, general index of measurement precision for psychological and educational test scores is the reliability coefficient—a ratio of true variance for a test score to the true-plus-error variance of the score. In item response theory (IRT) models for test scores, the information function is the central, conditional index of measurement precision. In this inquiry, conditional reliability coefficients for a variety of score types are derived as simple transformations of information functions. It is shown, for example, that the conditional reliability coefficient for an ordinary, number-correct score, X, is equal to, ρ(X,X′|θ)=I(X,θ)/[I(X,θ)+1] Where: θ is a latent variable measured by an observed test score, X; p(X, X′|θ) is the conditional reliability of X at a fixed value of θ; and I(X, θ) is the score information function. This is a surprisingly simple relationship between the 2, basic indices of measurement precision from IRT and classical test theory (CTT). This relationship holds for item scores as well as test scores based on sums of item scores—and it holds for dichotomous as well as polytomous items, or a mix of both item types. Also, conditional reliabilities are derived for computerized adaptive test scores, and for θ-estimates used as alternatives to number correct scores. These conditional reliabilities are all related to information in a manner similar-or-identical to the 1 given above for the number-correct (NC) score.