content_image/297924.jpg content_image/297922.jpg
<img src='content_image/297924.jpg'>

We apply the K-means method to the item response probabilities of each item. The number of clusters are selected by the silhouette method. The partial information is then obtained via this cluster analysis. The results are summarized in Table 15. From Table 15, we can see the 13 items may be divided into three groups according to their functioning. Items 1-4 can differentiate between classes 1,2,3,5 and class 4. Items 5-8 differentiate classes 1,5 and classes 2,3,4. Items 9-13 differentiate classes 3,5 and classes 1,2,4. Furthermore, we can see that items 2,3,7,8,12,13 differentiate multiple groups, indicating that these items may be more informative. 

We further construct a parametrization of the item response function based on the estimated partial information. It turns out to be the Reduced NC-RUM model with number of attributes K = 3 whose item response function is defined as 

<img src='content_image/297922.jpg'>

where φ$_{j}$ is the correct response probability for subjects who possess all required attributes and 

