Adaptive testing models

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• $n$ is the number of questions
• $K$ the number of dimensions / knowledge components considered by the model
• $k$ is the rough number of nonzero elements per row (in the question parameters)

Latent Trait Models

Summative models.

Item Response Theory (IRT)

• Student $i$ has ability $\theta_i$.
• Question $i$ has difficulty $d_j$.

where $Phi : x \mapsto 1/(1 + e^{-x})$ is the logistic function.

Multidimensional IRT

• $K$ is the number of dimensions measured.
• Student $i$ has level $\theta_{ik}$ over dimensions $k = 1, \ldots, K$.
• Question $j$ has discrimination $d_{jk}$ over dimensions $k = 1, \ldots, K$ and easiness (bias) $\delta_j$.

SPARFA

Sparse Factor Analysis.

Same as MIRT, but:

• $d_{jk} \geq 0$ for every question $j$ and dimension $k$;
• $d_{jk} = 0$ for most of them ($D$ matrix is sparse).

Knowledge-based Models

Formative models.

DINA

• Knowledge components are $1, \ldots, K$.
• Student $i$ has knowledge $\in {0, 1}^K$.
• Question $j$ has requirements $\in {0, 1}^K$ and slip and guess parameters $s_j$ and $g_j$.

Attribute Hierarchy Model & Knowledge Space Theory

Knowledge components are linked in a graph.
$u \rightarrow v$ means $u$ has to be mastered before $v$.

GenMA

• Knowledge components $1, \ldots, K$.
• Student $i$

• $K$ is the number of knowledge components (KC) involved.
• $\theta_{ik}$ is student $i$’s level along KC $k$;
• $q_{jk}$ the $(j, k)$ element of the q-matrix that is 1 if KC $k$ is involved in the resolution of question $j$, 0 otherwise;
• $d_{jk}$ is the discrimination parameter of question $j$ over KC $k$;
• $\delta_j$ is the easiness (bias) of question $j$.

Benchmarks

See here for a comparison of those models.