content_image/321968.jpg content_image/321969.jpg content_image/321970.jpg content_image/321971.jpg
α , that is, 

<img src='content_image/321968.jpg'>

This is known as the local independence assumption. The π α is the probability mass function of the latent class membership α . Suppose conditional distribution is parameterized through 

<img src='content_image/321969.jpg'>

where θ$_{j}$ is the item parameter specific to item j . Thus, the joint marginal distribution of Y can be expressed as 

<img src='content_image/321970.jpg'>

Diagnostic classification models in general admit the same distribution as in (1). However, it often imposes additional parametric or low-dimensional structures so that the item response function f$_{j}$ and the latent variable α have practical interpretations. Here, we adopt the following parameterization. The latent variable α is parameterized by a multidimensional binary or multi- category vector, i.e. α ∈ ( α $^{1}$, ..., α $^{K}$) where α k takes d$_{k}$ possible values. Each α k is known as an attribute or trait indicating the presence/absence or level of a latent characteristic. 

In addition to the multidimensional structure of the latent class space, the item response function f$_{j}$ also admits some parametric and usually low-dimensional structures that distinguish themselves from the general latent class models. To elaborate on this feature, we need to introduce the Q - matrix that is a J × K matrix taking binary values and indicating the item-attribute association. Each row of Q corresponds to an item and each column corresponds to an attribute. We write 

<img src='content_image/321971.jpg'>

where q$_{j}$ = ( q$_{j}$$_{1}$, ..., q$_{jK}$ ) is a K -dimensional row vector. As mentioned previously, q$_{jk}$ = 1 or 0, 

