The cross-classified multilevel measurement model: an explanation and demonstration

The link between the hierarchical generalized linear model (HGLM) and the Rasch model's parameterization has already been demonstrated by several researchers. Extensions have been described that include higher clustering levels to model more appropriately the contextual effects that are frequently encountered in educational research. However, pure hierarchies are relatively rare and instead cross-classified data structures are more frequently encountered. Cross-classified random effect modeling (CCREM) is still not commonly used. Use of CCREM in combination with the multilevel measurement model (MMM) has been recently introduced and is described further in the current study. Specifically, the link between the MMM and the CCREM MMM (termed "CCMMM" model) is provided. A dataset was simulated to demonstrate interpretation of the CCMMM model's parameters and to compare results under a CCMMM versus HGLM analysis. An Appendix is provided to demonstrate SAS GLIMMIX code used to estimate HGLM and CCMMM models' parameters.