Some links between classical and modern test theory via the two-level hierarchical generalized linear model

This article considers some links between classical test theory (CTT) and modern test theory (MTT) such as item response theory (IRT) and the Rasch model in the context of the two-level hierarchical generalized linear model (HGLM). Conceptualizing items as nested within subjects, both the CTT model and the MTT model can be reformulated as an HGLM where item difficulty parameters are represented by fixed effects and subjects' abilities are represented by random effects. In this HGLM framework, the CTT and MTT models differ only in the level 1 sampling model and the associated link function. This article also contrasts the Rasch and two-parameter IRT models by considering the property of specific objectivity in the context of CTT. It is found that the essentially tau-equivalent model exhibits specific objectivity if the data fit the model, but the congeneric measures model does not. Data from English composition scores on essay writing used by Jöreskog (1971) are reanalyzed for illustration.