# A semi-parametric approach to estimating item response functions

Abstract (Summary)

In Item Response Theory (IRT), normal ogive functions or logistic functions are
typically used to model the Item Characteristic Curve (ICC). Although the one parameter
(1PL), two parameter (2PL) or three parameter (3PL) logistic models have
been shown to be useful in a variety of situations, there are cases where these models
do not produce a good fit to the data. The Logistic function of a Monotonic
Polynomial (L-MP) is a model proposed in this dissertation aiming to improve the
model-data fit.
The L-MP model replaces the linear exponent of the 1PL or 2PL model with a
monotonic polynomial. It is a general model which includes the 1PL or 2PL model as
a special case. A surrogate-based two-stage approach is used to obtain the estimates
from the L-MP model.
The L-MP model is illustrated using both simulation studies and two real world
examples. Performance of the L-MP model in the simulation studies is evaluated by
examining the Root Integrated Mean Square Error (RIMSE) for the item curves and
the ability estimates, and also the rank correlations between the estimated and true
abilities. The L-MP model is compared with the 2PL model with Marginal Maximum
Likelihood (MML) estimates and Joint Maximum Likelihood (JML) estimates. It is
also compared with two nonparametric approaches, namely TESTGRAF which uses
a kernel smoothing method, and the Nonparametric Bayesian model. Results show
that: (1) The L-MP estimation method is able to recover the true values of person
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and of item parameters reasonably well. (2) If a standard logistic model holds, the L-
MP method can provide very close estimated ICCs to those of the MML method and
much better estimated ICCs than those of the JML method. For ability parameters,
?, the L-MP method can provide slightly better estimates than MML and much
better estimates than JML in terms of the RIMSE?. (3) When the true models are
not standard logistic functions, the L-MP model with a higher order polynomial is
preferable to the 2PL model. A comparison between TESTGRAF and L-MP shows
that generally L-MP and TESTGRAF produce very similar estimated ICCs for most
items. TESTGRAF has a slightly smaller RIMSE (in third decimal place) for the
estimated item curves, but L-MP model produces better estimates of abilities in
terms of RIMSE? and rank correlations. The comparison between L-MP and the
Nonparametric Bayesian model shows that these two methods produce very similar
results. The Nonparametric Bayesian model may yield better estimated ICCs than
the L-MP model, but differences are too small to interpret with any certainty. The
computational time for the Nonparametric Bayesian program is much longer than
for the L-MP program. In summary, our experiments indicate that results from the
L-MP model are comparable to the best of those from other approaches considered.
This demonstrates that the surrogate ability approach, adapted from TESTGRAF
and used in L-MP, yields results that are completely suitable for practical use.
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Bibliographical Information:

Advisor:

School:The Ohio State University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:item response theory psychometrics monotonic functions logistic distribution

ISBN:

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