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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. 

