Estimation for the Rasch Model When Both Ability and Difficulty Parameters Are Random

Abstract

Estimation of the parameters of the Rasch model, a one-parameter item response model, is considered when both the item parameters and the ability parameters are considered random quantities. It is assumed that the item parameters are drawn from a N(γ, τ²) distribution, and the abilities are drawn from a N(0, σ²) distribution. A variation of the EM algorithm is used to find approximate maximum likelihood estimates of γ, τ, and σ. A second approach assumes that the difficulty parameters are drawn from a uniform distribution over part of the real line. Real and simulated data sets are discussed for illustration.