The dangers posed by saddle points, and other problems, when using central composite designs

Abstract

This paper discusses two problems, which can occur when using central composite designs (CCDs), that are not generally covered in the literature but can lead to wrong decisions-and therefore incorrect models-if they are ignored. Most industrialbased experimental designs are sequential. This usually involves running as few initial tests as possible, while getting enough information as is needed to provide a reasonable approximation to reality (the screening stage). The CCD design strategy generally requires the running of a full or fractional factorial design (the cube or hypercube) with one or more additional centre points. The cube is augmented, if deemed necessary, by additional experiments known as star-points. The major problems highlighted here concern the decision to run the star points or not. If the difference between the average response at the centre of the design and the average of the cube results is significant, there is probably a need for one or more quadratic terms in the predictive model. If not, then a simpler model that includes only main effects and interactions is usually considered sufficient. This test for 'curvature' in a main effect will often fail if the design space contains or surrounds a saddle-point. Such a point may disguise the need for a quadratic term. This paper describes the occurrence of a real saddle-point from an industrial project and how this was overcome. The second problem occurs because the cube and star point portions of a CCD are sometimes run as orthogonal blocks. Indeed, theory would suggest that this is the correct procedure. However in the industrial context, where minimizing the total number of tests is at a premium, this can lead to designs with star points a long way from the cube. In such a situation, were the curvature test to be found non-significant, we could end with a model that predicted well within the cube portion of the design space but that would be unreliable in the balance of the total area of investigation. The paper discusses just such a design, one that disguised the real need for a quadratic term.