Rasch measurement: Advanced and specialized applications
Mixed models take the dependency between observations based on the same person into account by introducing one or more random effects. After introducing the mixed model framework, it is explained, by taking the Rasch model as a generic example, how item response models can be conceptualized as generalized linear and nonlinear mixed models. Common estimation methods for generalized linear and nonlinear models are discussed. In a simulation study, the performance of four estimation methods is assessed for the Rasch model under different conditions regarding the number of items and persons, and the degree of interindividual differences. The estimation methods included in the study are: an approximation of the integral over the random effect by means of Gaussian quadrature; direct maximization with a sixth-order Laplace approximation to the integrand; a linearized approximation of the nonlinear model employing PQL2; and finally a Bayesian MCMC method. It is concluded that the estimation methods perform almost equally well, except for a slightly worse recovery of the variance parameter for PQL2 and MCMC.