Mixed Compensation Multidimensional Item Response Theory

Computerized Assisted Testing (CAT) has supported the development of numerous adaptive testing approaches. Such approach as Item Response Theory (IRT) estimates a student's competency level by modeling a test as a function of the individual's knowledge ability, and the parameters of the question (i.e. item). Multidimensional Item Response Theory (MIRT) extends IRT so that each item depends on multiple competency areas (i.e., knowledge dimensions). MIRT models consider two opposing types of relationship between knowledge dimensions: compensatory and noncompensatory. In a compensatory model, having a higher competency with one knowledge dimension compensates for having a lower competence in another dimension. Conversely, in a noncompensatory model all the knowledge dimensions are independent and do not compensate for each other. However, using only one type of relationship at a time restricts the use of MIRT in practice. In this work, we generalize MIRT to a mixed-compensation multidimensional item response theory (MCMIRT) model that incorporates both types of relationships. We also relax the MIRT assumption that each item must include every knowledge dimension. Thus, the MCMIRT can better represent real-world curricula. We show that our approach outperforms random item selection with synthetic data.