A Bayesian perspective on factorial experiments using potential outcomes

Factorial designs have been widely used in many scientific and industrial settings, where it is important to distinguish "active'' or real factorial effects from "inactive" or noise factorial effects used to estimate residual or "error" terms. We propose a new approach to screen for active factorial effects from such experiments that utilizes the potential outcomes framework and is based on sequential posterior predictive model checks. One advantage of the proposed method lies in its ability to broaden the standard definition of active effects and to link their definition to the population of interest. Another important aspect of this approach is its conceptual connection to Fisherian randomization tests. As in the literature in design of experiments, the unreplicated case receives special attention and extensive simulation studies demonstrate the superiority of the proposed Bayesian approach over existing methods. The unreplicated case is also thoroughly explored. Extensions to three level and fractional factorial designs are discussed and illustrated using a classical seat belt example for the former and part of a stem-cell research collaborative project for the latter.