Effects of Within-Group Homogeneity on Parameter Estimation of the Multilevel Rasch Model

If a hierarchical structure exists in educational measurement data and examinees within groups are homogeneous, a multilevel item response theory (MLIRT) model may be appropriate. Among the MLIRT models, the multilevel Rasch model is equivalent to a generalized linear mixed model (GLMM) with a logit link where person abilities are considered random effects and item difficulties fixed effects. Then the lme4 package in R can be used to fit the multilevel Rasch model. In this study, it was shown how the multilevel Rasch model can be formulated as a three-level GLMM, followed by a simulation analysis, where intraclass correlation (lCC) of latent abilities as a measure of within-group homogeneity was manipulated from low to high under the conditions of small to large numbers of examinees and items. Item parameter estimates by marginal maximum likelihood estimation (MMLE) were compared with those obtained under the GLMM framework. Biases of item parameter estimates by both methods were not evident in all conditions. However, estimation results by MMLE became proportionally less accurate as the ICC increased when the number of examinees was small. If the number of examinees was large, estimation accuracies of MMLE were acceptable even when a high level of within-group homogeneity existed. GLMM produced stable results at all levels of ICC. In all conditions, the number of examinees was more influential than the number of items.