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Standard linear modeling approaches make potentially simplistic assumptionsregarding the structure of categorical effects that may obfuscate more complexrelationships governing data. For example, recent work focused on the two-wayunreplicated layout has shown that hidden groupings among the levels of onecategorical predictor frequently interact with the ungrouped factor. We extendthe notion of a "latent grouping factor" to linear models in general. Theproposed work allows researchers to determine whether an apparent grouping ofthe levels of a categorical predictor reveals a plausible hidden structuregiven the observed data. Specifically, we offer Bayesian model selection-basedapproaches to reveal latent group-based heteroscedasticity, regression effects,and/or interactions. Failure to account for such structures can producemisleading conclusions. Since the presence of latent group structures isfrequently unknown a priori to the researcher, we use fractional Bayes factormethods and mixture $g$-priors to overcome lack of prior information.

Detection of latent heteroscedasticity and group-based regression effects in linear models via Bayesian model selection