Bayesian modeling of measurement error in predictor variables using item response theory

This paper focuses on handling measurement error in predictor variables using item response theory (IRT). Measurement error is of great important in assessment of theoretical constructs, such as intelligence or the school climate. Measurement error is modeled by treating the predictors as unobserved latent variables and using the normal ogive model to describe the relations between latent variables and their observed indicator variables. The predictor variables can be defined at any level of a hierarchical regression model. The predictor variables are latent but can be measured indirectly by using tests or questionnaires. The observed responses on these itemized instruments are related to the latent predictors by an IRT model. It is shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. Handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. An example using real data from a mathematics test taken by 3,713 fourth graders is given.