Analysis of Factorial Experiments.

Fractional factorial designs have long been a key tool for the industrial statistician. They have received renewed attention recently due to the movement toward quality improvement sparked by the success of the Japanese in penetrating markets formerly dominated by western countries. Fractional factorial designs are usually not replicated, so that it is not possible to estimate error variance in the usual way from repeat observations. Past methods of analysis have rested on an implicit hypothesis of effect sparsity, that most of the estimated effects measure only noise. Formalization of this hypothesis leads to a Bayesian analysis in which the posterior probability that an effect is active can be computed. A similar approach can be employed to obtain the posterior probability that a particular experimental factor is active. These probabilities are readily interpreted by graphical means, and provide a straightforward method for identifying active contrasts and active factors. In addition, the model is extended to the situation where there are possible outliers in the original observations. The posterior probability that an effect is active can be computed taking into account the possibility of bad values, and the posterior probability that an observation is bad can be computed taking into account that the identity of active effects is unknown. Author.