Many multilevel linear and item response theory models have been developed to account for multilevel data structures. However, most existing cognitive diagnostic models (CDMs) are unilevel in nature and become inapplicable when data have a multilevel structure. In this study, using the log-linear CDM as the item-level model, multilevel CDMs were developed based on the latent continuous variable approach and the multivariate Bernoulli distribution approach. In a series of simulations, the newly developed multilevel deterministic input, noisy, and gate (DINA) model was used as an example to evaluate the parameter recovery and consequences of ignoring the multilevel structures. The results indicated that all parameters in the new multilevel DINA were recovered fairly well by using the freeware Just Another Gibbs Sampler (JAGS) and that ignoring multilevel structures by fitting the standard unilevel DINA model resulted in poor estimates for the student-level covariates and underestimated standard errors, as well as led to poor recovery for the latent attribute profiles for individuals. An empirical example using the 2003 Trends in International Mathematics and Science Study eighth-grade mathematical test was provided.
Continuous-variable approach to the spectral properties and quantum states of the two-component Bose-Hubbard dimer
