A common finding in the educational and psychological measurement literature is the failure of simple IRT models to have good fit for certain items. In other words, an assumption regarding the functional form of the relationship between the latent construct and the item responses is violated. This observation sometimes occurs for psychopathology data, data with mixed populations, and for multiple choice educational tests where there is some effect of guessing. Some alternatives are notorioiusly difficult to estimate (3- and 4-parameter logistic models), and some nonparametric methods may not neatly fit the demands of operational testing programs in which policy and high-stakes decisions are made (e.g., patient-reported outcomes, academic entrance and licensure exams, large-scale international educational assessments). In such testing programs, it is often required that a candidate model is able to be used with multiple groups of participants (such as demographic groups, in order to investigate item or test bias) and planned missing data designs (in which participants do not complete all test items), can be estimated relatively quickly, and can be used in conjunction with a computer adaptive test. In addition, retaining items that would otherwise not fit is highly desirable as such items can cost thousands of dollars to develop. Thus, the overarching goal of this line of research is to develop more flexible modeling approaches that can more readily meet these operational demands.