Fused SDT/IRT Models for Mixed-Format Exams

A psychological framework for different types of items commonly used with mixed-format exams is proposed. A choice model based on signal detection theory (SDT) is used for multiple-choice (MC) items, whereas an item response theory (IRT) model is used for open-ended (OE) items. The SDT and IRT models are shown to share a common conceptualization in terms of latent states of "know/don't know" at the examinee level. This in turn suggests a way to join or "fuse" the models-through the probability of knowing. A general model that fuses the SDT choice model, for MC items, with a generalized sequential logit model, for OE items, is introduced. Fitting SDT and IRT models simultaneously allows one to examine possible differences in psychological processes across the different types of items, to examine the effects of covariates in both models simultaneously, to allow for relations among the model parameters, and likely offers potential estimation benefits. The utility of the approach is illustrated with MC and OE items from large-scale international exams.