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 $$P(Y_{ij}=1|{\alpha }_{lj}^{\ast })=\{\begin{array}{cc}{g}_j& \text{if}{\alpha }_{lj}^{\ast }\ne 1\\ 1-s_j& \text{otherwise}\end{array}$$

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

$$P(Y_{ij}=1)=\sum _{c=1}^C\underset{IRF}{\underset{}{P(Y_{ij}=1|{\alpha }_c)}}\underset{structural}{\underset{}{p({\alpha }_c)}}$$

Maris (1999) discussed a variety of approaches to parameterizing the joint attribute distribution, where the probability mass function of attribute profile ${\alpha }_c$ is denoted by ${\pi }_c=h\left(\lambda \right)$, where $\lambda$ 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