A Bayesian Analysis of Very Small Unreplicated Experiments

It is not uncommon to deal with very small experiments in practice. For example, if the experiment is conducted on the production process, it is likely that only a very few experimental runs will be allowed. If testing involves the destruction of expensive experimental units, we might only have very small fractions as experimental plans. In this paper, we will consider the analysis of very small factorial experiments with only four or eight experimental runs. In addition, the methods presented here could be easily applied to larger experiments. A Daniel plot of the effects to judge significance may be useless for this type of situation. Instead, we will use different tools based on the Bayesian approach to judge significance. The first tool consists of the computation of the posterior probability that each effect is significant. The second tool is referred to in Bayesian analysis as the posterior distribution for each effect. Combining these tools with the Daniel plot gives us more elements to judge the significance of an effect. Because, in practice, the response may not necessarily be normally distributed, we will extend our approach to the generalized linear model setup. By simulation, we will show that not only in the case of discrete responses and very small experiments, the usual large sample approach for modeling generalized linear models may produce a very biased and variable estimators, but also that the Bayesian approach provides a very sensible results.