Table 17: The table of parameters for the three dimensional Reduced NC-RUM model for the social anxiety disorder data. Some parameters in model could not be estimated due to structure of Reduced NC-RUM model. diagnostic classification models. We established four theorems on identifiability under various model settings including binary response case, multi-categorical response case and multi-level attribute case. In particular, we provide easy-to-check sufficient conditions in Theorem 4 that are applicable to a general class of latent class models. Further in Theorem 5, we show the existence of a consistent estimator which can asymptotically identify partial information structure of items. We construct an appropriate estimator by using latent Dirichlet allocation method, which does not require pre- specification of the number of latent classes. The simulation results show the proposed method works well under a variety of settings. There are some recent works on identifiability for certain latent class models. Xu, Shang, Zhang (2016) [37, 36, 38] give several identifiability results for Q -matrix and parameters under the restricted latent class setting with binary item responses. Their main result provides a slightly weaker sufficient condition version of our corollary 1. This is because that our results are applicable to general latent class models with multi-categorical responses and diagnostic classification models with multi-level attributes. Because the existing methods require specification of a particular Q -matrix based model, it is of interest to develop model checking methods for any departure from the model specifications. 30 Item $^{φ}$· $^{r}$· , 1 $^{r}$· , 2 $^{r}$· , 3 1 φ$_{1}$ = 0 . 95 r$_{1}$$_{,}$$_{1}$ = 0 . 26 - - 2 φ$_{2}$ = 0 . 95 r$_{2}$$_{,}$$_{1}$ = 0 . 10 r$_{2}$$_{,}$$_{2}$ = 0 . 79 - 3 φ$_{3}$ = 0 . 9 r$_{3}$$_{,}$$_{1}$ = 0 . 03 r$_{3}$$_{,}$$_{2}$ = 0 . 67 - 4 φ$_{4}$ = 0 . 95 r$_{4}$$_{,}$$_{1}$ = 0 . 16 r$_{4}$$_{,}$$_{2}$ = 0 . 80 r$_{4}$$_{,}$$_{3}$ = 0 . 80 5 φ$_{5}$ = 0 . 7 - r$_{5}$$_{,}$$_{2}$ = 0 . 21 - 6 φ$_{6}$ = 0 . 3 - r$_{6}$$_{,}$$_{2}$ = 0 . 20 - 7 φ$_{7}$ = 0 . 75 - r$_{7}$$_{,}$$_{2}$ = 0 . 40 r$_{7}$$_{,}$$_{3}$ = 0 . 80 8 φ$_{8}$ = 0 . 8 - r$_{8}$$_{,}$$_{2}$ = 0 . 37 r$_{8}$$_{,}$$_{3}$ = 0 . 75 9 φ$_{9}$ = 0 . 4 - - r$_{9}$$_{,}$$_{3}$ = 0 . 08 10 φ$_{10}$ = 0 . 9 - - r$_{10}$$_{,}$$_{3}$ = 0 . 22 11 φ$_{11}$ = 0 . 9 - - r$_{11}$$_{,}$$_{3}$ = 0 . 17 12 φ$_{12}$ = 0 . 6 - r$_{12}$$_{,}$$_{2}$ = 0 . 55 r$_{13}$$_{,}$$_{3}$ = 0 . 55 13 φ$_{13}$ = 0 . 45 - r$_{13}$$_{,}$$_{2}$ = 0 . 20 r$_{13}$$_{,}$$_{3}$ = 0 . 20