Abstract: In scientific research major studies are often designed after a pilot study has been carried out. This paper uses Bayesian methodology to incorporate the results of the first study (or studies) into the design of the follow-up study. It is assumed that the error variances of the pilot and of the follow-up studies are unknown but that the experimenters are able to give intervals for the possible values of the two variances. This assumption, together with a noninformative prior distribution of the treatment means of the pilot study, leads to a class of prior distributions for the new experiment.
Since the properties of an experimental design can only be judged i n reference to a particular estimator, and since the choice of an estimator is in itself a very important problem, we combine the two tasks and simultaneously search for the estimator and the design which achieve the maximin efficiency over the class of posterior distributions.
The method is illustrated on data from ``6-Month Drinking Water Stu dies of Sodium Fluoride'', a study, which showed that rats exposed to very high levels of sodium fluoride had diminished growth rates.
Key words and phrases: Analysis of variance, Bayesian experimental design, efficiency robust design and estimation.