# DINA - Deterministic Inputs, Noisy "And" Gate ## Model Latent response vector from the students, given the [[q-matrix|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(\mathbf\alpha_i) = P(X_{ij} = 1 \mid \mathbf\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 $\mathbf\alpha_{i}$ > [!NOTE] > > - $q$ - the skills needed > - $\mathbf\alpha$ - 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