Introduction

Traditional, frequentist-based, statistical methods have provided researchers with a lens through which they could fruitfully examine the world. These well-known methods for both continuous (e.g., regression) and discrete outcome data (e.g., contingency table procedures) have proved their worth innumerable times. Yet, all methods have limits of applicability and these are no exceptions to that rule. The growth of analogous robust procedures have helped to provide the unwary analyst with protection against unexpected events – usually longer-tailed-than-Gaussian error distributions – but despite the proven robustness of these methods, careful thought must accompany any analysis. If one blindly utilizes statistical methods even those as well established as the Mantel–Haenszel (MH) test considered here, one does so with some risk. The computationally intensive procedures that we have used in fitting the Bayesian testlet models are continually becoming “cheaper and cheaper” and have made a Bayesian approach to many problems a practical lens through which phenomena can be viewed. Such methods provide further protection against the unknown. Remember Donald Rubin's comment, quoted in Chapter 7, that

In this chapter, we provide a vivid illustration of this in our study of differential item functioning (DIF).