Item Q-matrix Parameters λ , η 1 1 0 0 λ$_{1}$ = 4 , η = − 2 2 1 0 0 λ$_{1}$ = 4 , η = 2 3 1 0 0 λ$_{1}$ = 4 , η = − 2 4 0 1 0 λ$_{2}$ = 4 , η = 2 5 0 1 0 λ$_{2}$ = 4 , η = − 2 6 0 1 0 λ$_{2}$ = 4 , η = 2 7 0 0 1 λ$_{3}$ = 4 , η = − 2 8 0 0 1 λ$_{3}$ = 4 , η = 2 9 0 0 1 λ$_{3}$ = 4 , η = − 2 10 1 1 0 λ$_{1}$ = 2 , λ$_{2}$ = 2 , λ$_{1}$$_{,}$$_{2}$ = 0 , η = − 2 11 1 1 0 λ$_{1}$ = 2 , λ$_{2}$ = 2 , λ$_{1}$$_{,}$$_{2}$ = 0 , η = 2 12 1 0 1 λ$_{1}$ = 2 , λ$_{3}$ = 2 , λ$_{1}$$_{,}$$_{3}$ = 0 , η = − 2 13 1 0 1 λ$_{1}$ = 2 , λ$_{3}$ = 2 , λ$_{1}$$_{,}$$_{3}$ = 0 , η = 2 14 0 1 1 λ$_{2}$ = 2 , λ$_{3}$ = 2 , λ$_{2}$$_{,}$$_{3}$ = 0 , η = − 2 15 0 1 1 λ$_{2}$ = 2 , λ$_{3}$ = 2 , λ$_{2}$$_{,}$$_{3}$ = 0 , η = 2 16 1 1 1 λ$_{1}$ = 1 , λ$_{2}$ = 1 , λ$_{3}$ = 1 , η = − 2 λ$_{1}$$_{,}$$_{2}$ = λ$_{1}$$_{,}$$_{3}$ = λ$_{2}$$_{,}$$_{3}$ = 0 , λ$_{1}$$_{,}$$_{2}$$_{,}$$_{3}$ = 1 Table 9: Q-matrix and parameter setting for LCDM model simulation. probability π$_{2}$ , (0 , 0 , 1) has probability π$_{3}$ , (1 , 1 , 0) has probability π$_{4}$ , (1 , 0 , 1) has probability π$_{5}$ , (0 , 1 , 1) has probability π$_{6}$ , (1 , 1 , 1) has probability π$_{7}$ , and (0 , 0 , 0) has probability π$_{8}$ . There are eight classes in total. π$_{1}$ = π$_{2}$ = π$_{3}$ = π$_{4}$ = π$_{5}$ = π$_{6}$ = π$_{7}$ = π$_{8}$ = 1 / 8. This model setup satisfies the identifiability condition. The estimated mixture probabilities (averaging over 100 independent replications) and their mean squared errors are given in Table 10. The estimated probabilities of the remaining classes other than these 8 classes are all less than 3e-3. Also, the estimated response probabilities for each item are listed in Table 11 We use the Silhouette method to cluster item response probabilities for each item j to estimate the partial information structure. The proportion of correctly estimated item partial information Table 10: The mean and mse of the estimated class probability of each class based on 100 simulation runs. C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 mean 0.123 0.121 0.122 0.119 0.121 0.121 0.125 0.125 mse 1.1e-4 1.1e-4 1.7e-4 1.1e-4 1.0e-4 1.3e-4 1.1e-4 1.0e-4 24