Item Q-matrix G and S parameters φ r1 r2 r3 1 1 0 0 0.9 0.2 - - 2 1 0 0 0.9 0.2 - - 3 1 0 0 0.9 0.2 - - 4 0 1 0 0.9 - 0.3 5 0 1 0 0.9 - 0.3 6 0 1 0 0.9 - 0.3 7 0 0 1 0.9 - - 0.4 8 0 0 1 0.9 - - 0.4 9 0 0 1 0.9 - - 0.4 10 1 1 0 0.9 0.5 0.7 - 11 1 1 0 0.9 0.5 0.7 - 12 1 0 1 0.9 0.6 - 0.4 13 1 0 1 0.9 0.6 - 0.4 14 0 1 1 0.9 - 0.5 0.5 15 0 1 1 0.9 - 0.5 0.5 Table 5: Q-matrix and parameters setting for reduced NC-RUM model simulation 5.2 Reduced NC-RUM model simualtion In our second simulation setting, we consider a three-attribute NIDA model. The response proba- bility is The Q -matrix and parameters φ$_{j}$, r$_{jk}$ are listed in Table 5. We consider sample size 2000. The latent class of attribute profile (1 , 0 , 0) has probability π$_{1}$ , (0 , 1 , 0) has probability π$_{2}$ , (0 , 0 , 1) has probability π$_{3}$ , (1 , 1 , 0) has probability π$_{4}$ and (1 , 1 , 1) has probability π$_{5}$ . We set the probabilities for classes (1 , 0 , 1), (0 , 1 , 1), and (0 , 0 , 0) to zero. Hence, there are five classes in total. π$_{1}$ = π$_{2}$ = π$_{3}$ = π$_{4}$ = 1 / 6 , π$_{5}$ = 1 / 3. This model setup satisfies the identifiability condition. The estimated mixture probabilities (averaging over 100 independent replications) and their mean squared errors are listed in Table 7. The estimated response probabilities for each class and their mean squared errors are listed in Table 6. We see that the estimated probabilities of the remaining classes other than these 5 classes are very small (below 4e-3). This suggests that the model identifies exactly 5 classes. p$_{j,}$ α = φ$_{j}$ K ∏ k =1 ( r$_{jk}$ ) q$_{jk}$ $^{(1}$− α$_{k}$ $^{)}$. 21