dina

DINA - Deterministic Inputs, Noisy “And” Gate

Model

Latent response vector from the students, given the Q-Matrix:

$${\eta }_{ij}=\prod _{k=1}^K\alpha {jk}^{q_{jk}}$$

However, the process is inherently stochastic, due to slip ($s_j=P(X_{ij}=0\mid {\eta }_{ij}=1)$) and guessing parameter ($g_j=P(X_{ij}=1\mid {\eta }_{ij}=0$) — students with “mastery” can slip and miss the answer, students with “non-mastery” can guess the answer. Taken these into consideration, the probability of examinee getting item $j$ right:

$$P_j({\alpha }_i)=P(X_{ij}=1\mid {\alpha }_i)=g_j^{1-{\eta }_{ij}}(1-s_j{)}^{{\eta }_{ij}}$$

Note that solving a question with skills not specified in Q-vector may look like “guessing”.

• DINA model is parsimonious and interpretable, but provides good model fit.
• DINA model deals with multidimensional binary latent skills, while traditional IRMs deal with unidimensional continuous latent traits.
• DINA model is a conditional distribution of $X_{ij}$ given a skills vector ${\alpha }_i$

NOTE

• $q$ - the skills needed
• $\alpha$ - the skills mastered
• $\eta$ - theoretically, disregarding the noises, are you supposed to answer it correctly?
• $g$ - guessed it?
• $s$ - slipped it?
• $P$ - the probability of answering it correctly
• $X$ - our prediction of answering correctness (think of it as an event)

Estimation