In tmsalab/edm: Exploratory Diagnostic Models

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.

Variables

We adopt the following structure for variable naming.

• $N$ being the number of subjects.
• $J$ being the number of items.
• $K$ being the number of attributes.
• $s$ represents the slipping
• $g$ represents the guessing

To access values, we use the following indices setup:

• Subjects: $i = 1, \ldots, N$
• Items: $j = 1, \ldots, J$
• Attributes: $k = 1,\ldots, K$
• Time points: $t = 1,\ldots, T$

Response Matrix

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:

$$\mathbf{Y}i = \left( {Y{i1}, \cdots , Y_{iJ} }\right)_{J \times 1}^T$$

$\mathbf Q$ Matrix

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.

Attribute Matrix

$$\mathbf{\alpha}i=\left(\alpha{i1},\dots,\alpha_{iK}\right)^T_{2^K \times 1}$$

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 $2^K$.

$\eta$ Matrix

$$\eta_{ij}=\mathcal {I} \left( \alpha_{ik}\geq q_{jk} \text{ for all $k$}\right)=\mathcal {I} \left(\mathbf{\alpha}_i'\mathbf{q}_j=\mathbf{q}_j'\mathbf{q}_j\right)$$

Guessing and Slipping

• $g_j$ represents guessing or the probability of correctly answering item $j$ when at least one attribute is lacking, e.g. $g_j=P\left(Y_{ij}=1|\eta_{ij}=0\right)$
• $s_j$ represents slipping or the probability of an incorrect response for individuals with all of the required attributes, e.g. $s_j=P\left(Y_{ij}=0|\eta_{ij}=1\right)$.

Model Selection

DIC

$DIC = -2\left({\log p\left( {\mathbf{y}| \mathbf{\hat{\theta}} } \right) - 2\left( {\log p\left( {\mathbf{y}| \mathbf{\hat{\theta}} } \right) - \frac{1}{N}\sum\limits_{n = 1}^N {\log p\left( {\mathbf{y}|{\mathbf{\theta} _s}} \right)} } \right)} \right)$

tmsalab/edm documentation built on June 17, 2021, 6:46 a.m.