A Rasch hierarchical measurement model

Abstract

In this article, a hierarchical measurement model is developed that enables researchers to measure a latent trait variable and model the error variance corresponding to multiple levels. The Rasch hierarchical measurement model (HMM) results when a Rasch IRT model and a one-way ANOVA with random effects are combined (Bryk & Raudenbush, 1992; Goldstein, 1987; Rasch, 1960). This model is appropriate for modeling dichotomous response strings nested within a contextual level. Examples of this type of structure include responses from students nested within schools and multiple response strings nested within people. Model parameter estimates of the Rasch HMM were obtained using the Bayesian data analysis methods of Gibbs sampling and the Metropolis-Hastings algorithm (Gelfand, Hills, Racine-Poon, & Smith, 1990; Hastings, 1970; Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953). The model is illustrated with two simulated data sets and data from the Sloan Study of Youth and Social Development. The results are discussed and parameter estimates for the simulated data sets are compared to parameter estimates obtained using a two-step estimation approach.