Estimation of a Ramsay-Curve Item Response Theory Model by the Metropolis–Hastings Robbins–Monro Algorithm

Scott Monroe and Li Cai

Abstract
In Ramsay curve item response theory (RC-IRT) modeling, the shape of the latent trait distribution is estimated simultaneously with the item parameters. In its original implementation, RC-IRT is estimated via Bock and Aitkin’s EM algorithm, which yields maximum marginal likelihood estimates. This method, however, does not produce the parameter covariance matrix as an automatic byproduct on convergence. In turn, researchers are limited in when they can employ RC-IRT, as the covariance matrix is needed for many statistical inference procedures. The present research remedies this problem by estimating the RC-IRT model parameters by the Metropolis–Hastings Robbins–Monro (MH-RM) algorithm. An attractive feature of MH-RM is that the structure of the algorithm makes estimation of the covariance matrix convenient. Additionally, MH-RM is ideally suited for multidimensional IRT, whereas EM is limited by the ‘‘curse of dimensionality.’’ Based on the current research, when RC-IRTor similar IRT models are eventually generalized to include multiple latent dimensions, MHRM would appear to be the logical choice for estimation.

Keywords
Ramsay curve, item response theory, EM algorithm, MH-RM, density estimation