3.11 Joint Attribute Distribution

Recall that the item response function of a CDM defines the conditional probability of success given a specific attribute profile. For example, the DINA model can be written as $$\begin{equation} P(Y_{ij}=1|\mathbf{\alpha}_{lj}^*)= \begin{cases} g_j & \quad \text{if } \mathbf{\alpha}_{lj}^*\neq \mathbf{1}\\ 1-s_j & \quad \text{otherwise} \end{cases} \end{equation}$$

The IRF is only one part of a CDM. The other component is the joint attribute distribution or $p(\mathbf{\alpha}_c)$.

$$\begin{equation} P({{\rm{Y}}_{ij}}{\rm{ = 1) = }}\sum\limits_{c = 1}^C {\underbrace {P({{\rm{Y}}_{ij}}{\rm{ = 1|}}{\alpha _c}{\rm{)}}}_{{\rm{IRF}}}\underbrace {p({\alpha _c}{\rm{)}}}_{{\rm{structural}}}} \end{equation}$$

Maris (1999) discussed a variety of approaches to parameterizing the joint attribute distribution, where the probability mass function of attribute profile $\mathbf{\alpha}_c$ is denoted by $\pi_c=h(\mathbf{\lambda})$, where $\mathbf{\lambda}$ is a vector of unknown structural parameters.

References

Maris, E. (1999). Estimating multiple classification latent class models. Psychometrika, 64(2), 187–212. https://doi.org/10.1007/bf02294535