Statistical Modelling 8 (2008), 4166
Nonparametric Bayesian modelling for item response
Kristin A Duncan
Department of Mathematics and Statistics,
San Diego State University,
5500 Campanile Drive,
San Diego, CA 92182
USA
eMail:
duncan@sciences.sdsu.edu
Steven N MacEachern
Department of Statistics,
The Ohio State University
USA
Abstract:
Item response theory is widely used in standardized testing to model
the relationship between test takers’ unobserved ability levels and
their responses to items on the test. Item characteristic curves give
the probability of a correct response to an item as a function of
ability and are most often modelled with logistic curves. In this paper
we demonstrate how to model the item characteristic curve with
nonparametric Bayesian methods through the use of Dirichlet process
priors and present a complementary model in which the ability
distribution is modelled nonparametrically while the item characteristic
curves are logistic. We compare the nonparametric models with the
two-parameter logistic Bayesian model on data from an exam in an
introductory statistics course. We find that the nonparametric curve
model produces significantly different item characteristic curves for
a few of the items and that the corresponding ability estimates also
change substantially for some individuals.
Keywords:
Dirichlet process; item response theory; latent trait distribution;
monotone item response curve
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