Redefining Item Response Models for Small Samples

Popular item response theory (IRT) models are considered complex, mainly due to the inclusion of a random factor variable (latent variable). The random factor variable represents the incidental parameter problem since the number of parameters increases when including data of new persons. Therefore, IRT models require a specific estimation method and large samples for accurate parameter estimation. The two-parameter IRT model is redefined by analytically integrating out the random person factor in the latent response formulation of the model to make it suitable for small sample applications. This IRT Bayesian covariance structure model (IRT-BCSM) describes the clustering of (latent) responses by persons through a structured covariance matrix in which a common covariance parameter represents the dependence implied by a unidimensional random factor variable. The IRT-BCSM is a latent-variable-free model and consists of (fixed) item parameters and a common covariance parameter, where person parameters can be post-hoc sampled. An efficient Gibbs sampler is proposed for parameter estimation. In simulation studies, the performance of the IRT-BCSM is compared to two-parameter IRT models for small samples, and results show an optimal performance of the IRT-BCSM even in sample sizes as small as 50 to 100 persons and 5 to 10 items. Generalizations of the IRT-BCSM show that a redefinition of more complex IRT models also lead to much more efficient parameterizations, which can broaden the scope of IRT applications.