A multiple-choice SDT model for cognitive diagnosis models

A model for multiple-choice (MC) items based on signal detection theory (SDT), the MC-SDT model (DeCarlo, 2021a), follows from assumptions about perceptual and decision processes involved when examinees choose alternatives for MC items. The model can be expressed as a hierarchical model with an 'item-level', Level 1, and an 'examinee-level', Level 2. Here it is shown that cognitive diagnosis models (CDMs) can also be viewed as consisting of two levels, with the MC-SDT model serving as the first-level model, whereas the second-level model determines the type of CDM. Thus, the theory about how examinees make choices for MC items is unified across different CDMs. The resulting MC-SDT-CDM models are shown to be parsimonious sub-models of MC-CDMs. The models have straightforward interpretations (MC-SDT at Level 1), avoid estimation problems, and are useful for small sample sizes. The models are illustrated with an application to MC items from the TIMSS 2007 4th grade exam.