A semi-parametric approach to estimating item response functions

by 1977- Liang, Longjuan

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 ii 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. iii