content_image/295315.jpg content_image/295314.jpg
procedure, we select the largest M such that 

<img src='content_image/295315.jpg'>

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. 

## 5 Simulation study 

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. 

## 5.1 NIDA model simulation 

Here, we consider a three-attribute NIDA model. The response probabilities are given by 

<img src='content_image/295314.jpg'>

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. 

