Within the documentation and code implementation of ECDM models, we have opted to adopt certain notational conventions. The hope of consistent notation is to provide a base vocabulary to communicate about the ECDM models.
We adopt the following structure for variable naming.
To access values, we use the following indices setup:
The item matrix that contains dichotomous random variables corresponding to whether the answer was correct or incorrect is denoted by:
\[\mathbf{Y} = {\left( {{\mathbf{Y}_1}, \cdots ,{\mathbf{Y}_N}} \right)^T}_{N \times J}\]
Referencing a single observation from \(J\)-dimensional vector of binary responses for individual \(i\) is given by:
\[\mathbf{Y}_i = \left( {Y_{i1}, \cdots , Y_{iJ} }\right)_{J \times 1}^T\]
The \(\mathbf Q\) matrix provides the item to attribute mapping of the items on the assessment. We denote \(\bf Q\) as:
\[\mathbf{Q} =\left(q_{j1},\dots, q_{iK}\right)^T_{J \times K}\] where \(q_{jk}=1\) indicates that attribute \(k\) is required to answer item \(j\) and zero otherwise.
\[\mathbf{\alpha}_i=\left(\alpha_{i1},\dots,\alpha_{iK}\right)^T_{2^K \times 1}\]
where \(\alpha_{ik}=1\) states that subject \(i\) has attribute \(k\).
The presence of \(\boldsymbol a_c\in \left\{0,1\right\}^K\) indicates a value of \(2^K\).
\[\eta_{ij}=\mathcal {I} \left( \alpha_{ik}\geq q_{jk} \text{ for all $k$}\right)=\mathcal {I} \left(\mathbf{\alpha}_i'\mathbf{q}_j=\mathbf{q}_j'\mathbf{q}_j\right)\]