  Standard linear modeling approaches make potentially simplistic assumptions
regarding the structure of categorical effects that may obfuscate more complex
relationships governing data. For example, recent work focused on the two-way
unreplicated layout has shown that hidden groupings among the levels of one
categorical predictor frequently interact with the ungrouped factor. We extend
the notion of a "latent grouping factor" to linear models in general. The
proposed work allows researchers to determine whether an apparent grouping of
the levels of a categorical predictor reveals a plausible hidden structure
given the observed data. Specifically, we offer Bayesian model selection-based
approaches to reveal latent group-based heteroscedasticity, regression effects,
and/or interactions. Failure to account for such structures can produce
misleading conclusions. Since the presence of latent group structures is
frequently unknown a priori to the researcher, we use fractional Bayes factor
methods and mixture $g$-priors to overcome lack of prior information.
