content_image/310760.jpg content_image/310759.jpg content_image/310758.jpg
indicates whether or not f$_{j}$ depends on α $^{k}$. For instance, if q$_{j}$ = (1 , 1 , 0), then 

<img src='content_image/310760.jpg'>

The Q -matrix qualitatively describes the relationship between items and attributes. The specific forms of f$_{j}$ as functions of α are determined by the parametric model subject to the constraints implied by the Q -matrix. The specification also consists of additional item parameters. An im- portant feature of the item response function is that f$_{j}$ ( y | θ, α ) as a function of α usually does not necessarily depend on all α $^{k}$, k = 1 , . . . , K . 

## 2.2 Examples of diagnostic classification models 

In what follows, we provide a few examples of parametric diagnostic classification models. 

Example 1 (DINA model, [14]) For item j and attribute vector α , we define the ideal response 

<img src='content_image/310759.jpg'>

that is, whether α has all the attributes required by item j . For each item, there are two additional parameters s$_{j}$ and g$_{j}$ , known as the slipping and guessing parameters. The response probability p$_{j,}$ α takes the form 

<img src='content_image/310758.jpg'>

If ξ j $_{DINA}$( α , Q ) = 1 (the subject is capable of solving a problem), then the positive response proba- bility is 1 − s$_{j}$ ; otherwise, the probability is g$_{j}$ . The item parameter vector is ( s$_{j}$, g$_{j}$ : j = 1 , · · · , J ) . 

The DINA model assumes a conjunctive (non-compensatory) relationship among attributes. It is necessary to possess all the attributes indicated by the Q -matrix to be capable of providing a positive response. In addition, having additional unnecessary attributes does not compensate for the lack of necessary attributes. The DINA model is popular in the educational testing applications and is often used for modeling exam problem solving processes. 

