Latent Group-Based Interaction Effects in Unreplicated Factorial Experiments

Statistical models which are used to analyze unreplicated factorial experiments typically do not include interaction effects between the experimental treatments. This is because the standard tests which determine whether or not interaction is present do not have an appropriate error term. Many authors have developed models and statistical tests based on restricted forms of interaction to address non-additivity for these unreplicated experiments. Our strategy is to instead assume that levels of one of the factors belong to some smaller number of latent groups or environments. For example, consider a randomized complete block design (RCBD) in an agricultural experiment. If some of the blocks come from a poor environment there may not be treatment effects for these blocks. Perhaps the rest of the blocks come from a second environment, where treatment effects are pronounced. Our model is able to capture this type of non-additivity since we allow these groups to interact with the other treatments, but we assume that treatment effects are additive within group. We focus primarily on the detection of this grouping structure in a one versus two group case. We also consider point and interval estimation of the simple effect of the treatments in each group. The randomized complete block design (RCBD) is used to describe our methods which include both an all configurations approach as well as a Bayes Factor approach to diagnosis of this latent group-based interaction.