When linear models incorporate categorical predictors, the common strategy is to model the levels of these predictors using distinct means. However, practical analytic experience suggests that hidden groupings frequently exist within the levels of categorical predictors for a wide range of applications and study designs. We consider three possible effects that identifying such hidden clusters can have on the analysis. First, the model can be simplified when not all factor levels have a distinct mean; second, heteroscedasticity can be accounted for when clusters differ in their error variance; and third, clusters can statistically interact with other predictors in the model. In this talk, we describe a combinatoric search approach to reveal these latent structures via Bayesian model selection. We will propose these models in the context of a two-way layout, one-way ANOVA, and ANCOVA, discuss a simulation study to demonstrate the ability of our method to detect these latent structures and perform inference on parameters, and analyze data sets corresponding to research applications.