Currently, standardized tests are widely used as a method to measure how well schools and students meet academic standards. As a result, measurement issues have become an increasingly popular topic of study. Unidimensional item response models are used to model latent abilities and specific item characteristics. This class of models makes assumptions that are not always met by "real" testing data. The aim of this work is to add to the current body of work addressing models that violate these common assumptions. The three studies look at issues concerning violations of unidimensionality and monotonicity. The first study compares two common methods used to investigate multidimensional item response theory models. Researchers often simulate multidimensional item response data using different structures that induce a correlation between dimensions. The first method simulates the ability distribution to have a correlated structure, while the second method simulates the ability distributions to have an uncorrelated structure and displaces the item discriminations from the ability axes to induce a correlation. The first goal of this study is to address why these two methods may not provide equivalent results and describe a method for correcting this discrepancy. The second study looks at describing how multidimensional an instrument is and ways of simplifying its description. While there are several methods for testing whether an exam is unidimensional and attempting to say how many dimensions it measures, there is no widely accepted measure for the "amount of dimensionality" that the exam exhibits. Additionally, if an exam has a complex structure, it is hard to interpret and understand how the subscales are related. The second goal of this work provides a set of summary statistics to help identify "how multidimensional" an exam is. Additionally, we are able to summarize a complex structure exam with an equivalent simple structure. This will help applied researchers estimate and interpret the relationships between the dimensions/constructs. The focus of the third study is developing an alternate item response theory model to account for non-monotonicity in the response curve due to a set of examinee's having partial knowledge. The most commonly used model that accounts for guessing is the three parameter logistic (3PL) model. This model assumes that there is a constant guessing effect, and does not account for some examinees eliminating incorrect choices. The final goal of this study is to derive an IRT model that accounts for the affects of partial knowledge in guessing that can be fit using only the correct/incorrect response data. The results of the first study add to the understanding of the common simulation settings used to study multidimensional item response theory models. Findings in the second study help practitioners summarize complex structure exams using an "equivalent" simple structure. Finally, the third study offers an alternative model to fit items that exhibit non-monotonicity that may be due to a group of examinees having partial knowledge. As a whole, this work helps researchers and practitioners understand the properties of multidimensional item response theory models and item response theory models that are not monotonic. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]