procedure, we select the largest M such that That is, we treat latent classes of very small proportion ( o ( 1 √$_{n}$ )) as practically non existing. In applying the K -means algorithm, we truncate the finite sample to ˆ p$_{j}$$_{1}$ , ˆ p$_{j}$$_{2}$ , ..., ˆ p$_{jM}$ . Then, the estimated partial information is given by α 1 j = α 2 if ˆ p$_{j}$ $^{α}$1 and ˆ p$_{j}$ $^{α}$2 are in the same cluster according to the K -means algorithm. In the following sections, we apply this method to both simulated and real data to assess its performance. In this section, simulation studies are conducted to assess the performance of the proposed method and identifiability results. We consider three models for the data generation: NIDA model, reduced NC-RUM model, and LCDM model. The results are presented assuming all the model parameters are unknown including the Q-matrix, attribute distribution, and the item response probabilities. Here, we consider a three-attribute NIDA model. The response probabilities are given by The Q -matrix and parameters s$_{k}$, g$_{k}$ are listed in Table 1. There are three attributes and eight classes in total. We consider sample size 2000. The latent class of attribute profile (1 , 0 , 0) has mixture probability π$_{1}$ , (0 , 1 , 0) has probability π$_{2}$ , (0 , 0 , 1) has probability π$_{3}$ , (1 , 1 , 0) has mixture probability π$_{4}$ , (1 , 0 , 1) has mixture probability π$_{5}$ , (0 , 1 , 1) has mixture probability π$_{6}$ , (1 , 1 , 1) has probability π$_{7}$ and (0 , 0 , 0) has mixture probability π$_{8}$ . And we set π$_{1}$ = π$_{2}$ = π$_{3}$ = π$_{7}$ = 0 . 15 , π$_{4}$ = π$_{5}$ = π$_{6}$ = π$_{8}$ = 0 . 1. This model setup satisfies the identifiability condition. p$_{j,}$ α = K ∏ k =1 [(1 − s$_{k}$ ) $^{α$_{k}$}$g $^{1}$− α$_{k}$ k ] $^{q$_{jk}$}$. ˆ π α ≪ n − 1 / 2 α = M + 1 , M + 2 , . . . . 5 Simulation study 5.1 NIDA model simulation 18