of the required skills. The ‘‘and’’ gate component of the model refers to the con-
junctive process in determining Z
ij
in that a correct response to an item requires
the presence all the prescribed skills for the item. In the example above, students
who lack any one of the three required skills are not expected to answer the item
correctly. Thus, the DINA model exhibits the same property as standard non-
compensatory multidimensional IRMs. If the process is completely determinis-
tic (i.e., error-free or nonstochastic), the latent response vector is identical to the
manifest or observed response vector. However, because the underlying process
is inherently stochastic, the latent response vector represents only an ideal
response pattern. The noise introduced in the process is due to slip and guessing
parameters—that is, examinees who possess all the required skills for an item
can slip and miss the item, and examinees who lack at least one of the required
skills can guess and still answer the item correctly with typically nonzero prob-
abilities. In the DINA model, the slip and guessing parameters of item j are
defined as s
j
= PðX
ij
= 0|Z
ij
= 1) and g
j
= PðX
ij
= 1|Z
ij
= 0), respectively. There-
fore, the probability of examinee i with the skills vector α
i
answering item j cor-
rectly is given by
P
j
ðα
i
Þ = PðX
ij
= 1|α
i
Þ = g
1 − Z
ij
j
ð1 − s
j
Þ
Z
ij
: ð2Þ
From this equation, answering an item correctly requires an examinee who
has all the necessary skills to not slip and an examinee who lacks at least one of
the required skills to guess correctly. Note that if there is no guessing and no
slippage, the model probability of correct response to an item is either 0 or 1;
that is, the response is solely determined by the interaction of α and the Q-vector
for the item. However, as noted by de la Torre and Douglas (2004), guessing
in this context assumes a general interpretation; it is not confined to a correct
response arrived through a random response but rather includes the use of alter-
native strategies not articulated in the Q-matrix. For example, if an item can be
solved using a different set of skills, examinees who possess these skills but not
those prescribed in the Q-matrix may appear to be guessing but in fact are sys-
tematically solving the problem using a different strategy.
A graphical representation of the DINA model is given in Figure 1. As the
graph shows, the latent response Z
ij
is a function of the examinee’s skills fa
ik
g
and the requisites of the item fq
jk
g. Once Z
ij
has been determined, the probabil-
ity that examinee i will give a correct response to item j is g
j
if Z
ij
= 0 and 1 − s
j
if Z
ij
= 1:
The DINA model is a parsimonious and interpretable model that requires
only two parameters for each item (i.e., g
j
and s
j
) regardless of the number of
attributes being considered, and despite its simplicity, it has been shown to pro-
vide good model fit (e.g., de la Torre & Douglas, 2004, 2005). De la Torre and
Douglas (2004) and Junker and Sijtsma (2001) provide some applications of
the DINA model. Although labeled differently, other discussions of the DINA
DINA Model and Parameter Estimation
117
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