CV, ECV, and Robust CV designs for replications under a class of linear models in factorial experiments

A class of linear models is considered for describing the data collected from an experiment. Any two models have some common as well as uncommon parameters. To discriminate between any two models, the uncommon parameters play a major role. A common variance (CV) design is proposed for collecting the data so that all the uncommon parameters are estimated with as similar variances as possible in all models. The variance equality for a CV design is attained exactly when there is one uncommon parameter for any two models within the class. A new concept “Robust CV designs for replications” having the possibility of replicated observations is introduced. The conditions are presented for a CV design having no replicated observations to be robust for general replicated observations. A CV design having no replicated observations is always robust for any equally replicated observations. In the class of linear models considered for factorial experiments, the common parameters for all models correspond to the general mean and main effects, and the other parameters correspond to two factor interactions. Two general CV designs are presented for three level factorial experiments. Examples of Efficient CV (ECV) designs as well as Robust CV designs for general replicated observations are also presented. A simple illustrative example of the complete 2×3 factorial design is demonstrated to be not a CV design and then the condition on replications of each run is obtained to turn it into a CV design.