indicates whether or not f$_{j}$ depends on α $^{k}$. For instance, if q$_{j}$ = (1 , 1 , 0), then 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 . 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 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. 6 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 ) . p$_{j,}$ α = (1 − s$_{j}$ ) ξ j $_{DINA}$( α ,Q $^{)}$g $^{1}$− ξ j $_{DINA}$( α ,Q ) j . (3) ξ j $_{DINA}$( α , Q ) = K ∏ k =1 ( α$_{k}$ ) q$_{jk}$ = I ( α$_{k}$ ≥ q$_{jk}$ for all k ) (2) f$_{j}$ ( y | θ, α ) = f$_{j}$ ( y | θ, α $^{1}$, α$^{2}$) . 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