Multidimensional IRT Models
R. Darrell Bock, Robert D. Gibbons · 2021
Item factor analysis plays an essential role in the development of tests or scales to measure behavioral tendencies that are considered to be a matter of degree but are observed only as discrete responses. Multiple factor analysis as formulated by Thurstone assumes that the test scores are continuous measurements standardized to mean zero and standard deviation one in the sample. Gibbons and Hedeker showed how parameters of the item bifactor model for binary responses can be estimated by maximum marginal likelihood using a variation of the EM algorithm described by Bock and Aitkin. The first formulation of item response theory estimation of test scores was the maximum likelihood estimator (MLE) derived by Lord from Lazarsfeld’s principle of conditional (or local) independence of item responses. Because of the simple structure of the item-group bifactor loadings, it is possible to perform MLE with two-dimensional quadrature with many quadrature points without excessive computational burden.