Multidimensional Item Response Theory

They are closely connected: MIRT is essentially a nonlinear factor analysis for categorical item responses. The item discrimination vectors play the role of factor loadings, and the latent ability dimensions correspond to factors. Full-information item factor analysis and MIRT are two descriptions of the same family of models. The main practical difference is emphasis — MIRT foregrounds item-level response functions and measurement (difficulty, information), while factor analysis foregrounds the covariance structure — but the underlying mathematics overlaps heavily. See the related Factor Analysis entry.

In a compensatory model, abilities combine additively, so being high on one dimension can compensate for being low on another in raising the probability of a correct response. In a noncompensatory (partially compensatory) model, an item requires adequate standing on each needed dimension, so a deficiency on one cannot be fully offset by strength on another — closer in spirit to conjunctive cognitive models. Compensatory MIRT is far more common and easier to estimate; noncompensatory models are used when the task genuinely requires multiple skills jointly.

Through a combination of theory, dimensionality assessment, and model comparison. Substantive understanding of the construct should propose candidate dimensions; empirical tools — parallel analysis, fit indices, information criteria, and inspection of residual dependence — help judge how many are supported. Confirmatory MIRT tests a theory-specified structure, while exploratory MIRT searches for one, but adding dimensions purely to improve fit risks uninterpretable factors. The goal is the smallest number of dimensions that are both well-supported and substantively meaningful.