psychometrika—vol. 83, no. 1, 89–108
BAYESIAN ESTIMATION OF THE DINA Q MATRIX
UNIVERSITY OF NEVADA, RENO
Steven Andrew Culpepper, Yuguo Chen and
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Cognitive diagnosis models are partially ordered latent class models and are used to classify students
into skill mastery proﬁles. The deterministic inputs, noisy “and” gate model (DINA) is a popular psycho-
metric model for cognitive diagnosis. Application of the DINA model requires content expert knowledge
of a Q matrix, which maps the attributes or skills needed to master a collection of items. Misspeciﬁcation
of Q has been shown to yield biased diagnostic classiﬁcations. We propose a Bayesian framework for
estimating the DINA Q matrix. The developed algorithm builds upon prior research (Chen, Liu, Xu, &
Ying, in J Am Stat Assoc 110(510):850–866, 2015) and ensures the estimated Q matrix is identiﬁed. Monte
Carlo evidence is presented to support the accuracy of parameter recovery. The developed methodology is
applied to Tatsuoka’s fraction-subtraction dataset.
Key words: cognitive diagnosis models, deterministic inputs, noisy “and” gate (DINA) model, Q matrix,
Bayesian statistics, fraction-subtraction data.
Cognitive diagnosis models (CDMs) were developed to provide educators and researchers
with instructionally relevant assessments (Huff & Goodman, 2007; Leighton & Gierl, 2007).
CDMs use cognitive theory within a psychometric framework to, “...explain achievement test
performance by providing insight into whether it is students’ understanding (or lack of it) or
something else that is the primary cause of their performance” (Norris, Macnab, & Phillips, 2007,
p. 61). That is, CDMs are partially ordered latent class models that assist researchers in classifying
students as either masters or non-masters on a collection of skills deemed important for success
on educational tasks. A beneﬁt is that the CDM framework provides educators more detailed
diagnostic information regarding student skills/attributes than is available with more broadly
deﬁned continuous traits in item response models.
The application of CDMs is dependent upon the availability of cognitive theory that spec-
iﬁes the skills and/or attributes necessary for success on a collection of tasks. In particular, let
j = 1,...,J and k = 1,...,K index items and skills, respectively. Content expert knowledge
is codiﬁed in a J × K Q matrix with individual elements denoted by q
= 1 if skill k is needed
to respond correctly to item j as a master and zero otherwise. Given a speciﬁed Q and item
responses, the CDM item and latent class parameters can be accurately estimated. In fact, CDMs
have received broader application among educational and psychological researchers in large part
due to the availability of computer code and software (de la Torre, 2009; Chiu & Köhn, 2016;
Electronic supplementary material The online version of this article (doi:10.1007/s11336-017-9579-4) contains
supplementary material, which is available to authorized users.
Correspondence should be made to Steven Andrew Culpepper, Department of Statistics, University of Illinois at
Urbana-Champaign, 725 South Wright Street, Champaign, IL 61820, USA. Email: firstname.lastname@example.org
© 2017 The Psychometric Society