Characterization of Prediction Variance Properties of Rotatable Central Composite Designs for 3 to 10 Factors

Abstract

Rotatability is a very important property of the central composite designs (CCD) in predicting responses with stable prediction variance throughout the design region. A central composite design is made rotatable by the choice of α, the axial distance from the centre of the design region. In this work, we evaluate the prediction variance properties of the CCD with rotatable α by replicating the cube and star portions of the CCD. Three design optimality criteria, the D-efficiency, G-efficiency and V-criterion are utilized in evaluating the performances of the designs. The fraction of design space (FDS) plots for the scaled and unscaled prediction variances are employed in studying the performance characteristics of the prediction variance of the designs throughout the design region. The results show that, for k = 3 to 10 factors and with three centre points, the cube-replicated CCDs are D-efficient. Replicating the cube or star portions of the CCD improves the prediction capability of the designs. However, none of the design options, cube-replicated and star-replicated, is consistently superior to the others with respect to G-efficiency, V-criterion and FDS plots for any of the k factors considered. Analytical formulae for obtaining the G-efficiency and V-criterion when portions of the CCD are replicated are also given.