Consistent Estimation of Rasch Item Parameters and Their Standard Errors Under Complex Sample Designs

U.S. state educational testing programs administer tests to track student progress and hold schools accountable for educational outcomes. Methods from item response theory, especially Rasch models, are usually used to equate different forms of a test. The most popular method for estimating Rasch models yields inconsistent estimates and relies on ad hoc adjustments to obtain good approximations. Furthermore, psychometricians have paid little attention to the estimation of effective standard errors for Rasch models, especially under complex sample designs. This article presents a computationally efficient, statistically consistent estimator for Rasch models, based on a nonparametric marginal maximum likelihood approach, along with complete, design-consistent estimators of the standard error, based on the full information matrix and including covariance terms among items, covariances between items, and parameters of the distribution of the latent trait. Simulations support the consistency of the estimators in both simple random samples and more realistic multistage samples.