A model selection approach to analysis of variance and covariance

An alternative to analysis of variance is a model selection approach where every partition of the treatment means into clusters with equal value is treated as a separate model. The null hypothesis that all treatments are equal corresponds to the partition with all means in a single cluster. The alternative hypothesis correspond to the set of all other partitions of treatment means. A model selection approach can also be used for a treatment by covariate interaction, where the null hypothesis and each alternative correspond to a partition of treatments into clusters with equal covariate effects. We extend the partition-as-model approach to simultaneous inference for both treatment main effect and treatment interaction with a continuous covariate with separate partitions for the intercepts and treatment-specific slopes. The model space is the Cartesian product of the intercept partition and the slope partition, and we develop five joint priors for this model space. In four of these priors the intercept and slope partition are dependent. We advise on setting priors over models, and we use the model to analyze an orthodontic data set that compares the frictional resistance created by orthodontic fixtures.