CADEM: A conditional augmented data EM algorithm for fitting one parameter probit models

Abstract

In this article we develop an estimation method based on the augmented data scheme and EM/SEM (Stochastic EM) algorithms for fitting one-parameter probit (Rasch) IRT (Item Response Theory) models. Instead of using the S steps of the SEM algorithm, that is, instead of simulating values for the unobserved variables (augmented data and the latent traits), we consider the conditional expectations of a set of unobserved variables on the other set of unobserved variables, the current estimates of the parameters and the observed data, based on the full conditional distributions from the Gibbs sampling algorithm. Our method, named the CADEM algorithm (conditional augmented data EM), presents straightforward E steps, which avoid the need to evaluate the usual integrals, also facilitating the M steps, without the need to use numerical methods of optimization. We use the CADEM algorithm to obtain both maximum likelihood estimates and maximum a posteriori estimates of the difficulty parameters for the one-parameter probit (Rasch) model. Also, we obtain estimates for the latent traits, based on conditional expectations. In addition, we show how to calculate the associated standard errors. Some directions are provided to extend our approach to other IRT models. In this respect, we perform a simulation study to compare the estimation methods. The results indicated that our approach is quite comparable to the usual marginal maximum likelihood (MML) and Gibbs sampling methods (GS) in terms of parameter recovery. However, CADEM is as fast as MML and as flexible as GS.