ABSTRACT
There are several ways to extend the ordered regression models considered thus far in order to account for more complex patterns and data structures. For example, researchers may have hypotheses that require the use of interaction terms, which is more complicated in categorical models such as ordered regression (Williams 2009; Mood 2010). The nonlinear nature of the probability model can cause problems due to potential differences in unobserved heterogeneity among groups or levels of a variable included in the interaction term. One may address this problem by explicitly modeling the residual variance and the categorical outcome simultaneously with a “heterogeneous choice” or “location-scale” model (Clogg and Shihadeh 1994; Williams 2009). Additionally, the models in this book assume a single level of analysis, but more complex data structures with multiple levels of analysis are very common in the social and behavioral sciences, which requires the use of a multilevel ordered regression model (Raudenbush and Bryk 2002; Gelman and Hill 2007). Finally, advances in the Bayesian approach to statistics offer several advantages over the more familiar “frequentist approach” (Gelman et al. 2014; Gill 2015), including the ability to test multiple hypotheses simultaneously. This is particularly useful for tests of the parallel regression assumption, which require simultaneous tests for coefficients in multiple cutpoint equations. This chapter focuses on these three extensions to the ordered regression models in this book. To simplify the presentation, we will focus on the logit link function for each model extension.
