Date of Award


Document Type

Open Access Dissertation



First Advisor

Brian Habing


Item response theory (IRT) is widely applied to analyze educational and psychological assessments. Readily available IRT implementations allow for two common types of models: monotone models used for dominance scales (Guttman 1950; Rasch 1960/1980; Birnbaum 1968; Mokken 1971) and unfolding models used for proximity scales (Coombs, 1964; Andrich, 1996; Roberts, Donoghue and Laughlin, 2000).

When an exam contains items following both types of models, there is currently no method to distinguish the item types, estimate their characteristics, or estimate the examinee characteristics. Thus, there is no existing methodology to simultaneously analyze items like ``At a minimum, I am in favor of the economically disadvantaged receiving publicly funded private school vouchers" and ``I am in favor of publicly funded private school vouchers only for the economically disadvantaged" on the same survey. The former is a monotone increasing item (all those favoring some sort of voucher are likely to agree) whereas the latter is an unfolding item (one could disagree because they were against all vouchers, or because they were in favor of everyone receiving them). This situation forces analysts either to choose subscales of only one item type or risk incorrect and misleading results. The goal for this study is to find a reasonable means to solve this problem in the dichotomous (or binary) case when unidimensionality holds.

The first portion of this study is trying to identify these two item types in the mixed unfolding/monotone items exam (we call this a mixed exam). Two methods are adapted: the manifest monotonicity test (Sijtsma and Molenaar, 2002) and the p-value/biserial method based on classical item statistics. Simulation shows that the manifest monotonicity test does a good job of separating these two types when the items are located in the middle of the ability range. However, this method does not work for the extreme items (including higher or lower location values for both item types). The p-value/biserial method is limited to lower location values. But neither method functions very well for the extreme cases.

In the second portion of this study, a mixed model is proposed which combines both model types. The marginalized Bayes modal estimation algorithm is implemented to estimate the model parameters and examinees' abilities. The estimates for each parameter in the new model are also discussed. The new model and marginalized Bayes modal estimation can successfully identify the unfolding item with probability 1 and give posterior probabilities near 0.5 for the monotone items. Moreover, they give very good estimates for the item parameters.