Stochastic EM for estimating the parameters of a multilevel IRT model

An item response theory (IRT) model is used as a measurement error model for the dependent variable of a multilevel model where tests or questionnaires consisting of separate items are used to perform a measurement error analysis. The advantage of using latent scores as dependent variables of a multilevel model is that it offers the possibility of modeling response variation and measurement error and separating the influence of item difficulty and ability level. The two-parameter normal ogive model is used for the IRT model. It is shown that the stochastic EM (expectation-maximization) (SEM) algorithm can be used to estimate the parameters that are close to the maximum likelihood estimated. It turns out that this algorithm is easily implemented. This estimation procedure is compared to an implementation of the Gibbs sample in a Bayesian framework. Examples using real data from a Dutch primary school language test are given.