Cylindrical designs for response surface studies

Abstract

Central Composite Designs (CCDs) with cuboidal and spherical regions are among the most popular experimental designs for studying response surfaces. Cuboidal regions are typically used when the experimenter believes the levels of one or more of the factors are bounded while a spherical region is employed when there are no restrictions on the levels of any of the factors. We propose what we call a cylindrical design in which the levels of some factors are restricted while the other factors' levels need not be. Assuming the use of a second-order model, we give the general form for the model matrix X of such a design and give a closed form for the determinant of the X 0X matrix as well as its inverse. We use the results for the determinant and inverse of X 0X to compare designs using the alphabetic design optimality criteria. D-efficiencies, A-efficiencies, G-efficiencies, and IV-efficiencies for CCDs will be compared with those of the cylindrical design. Graphical assessment of the maximum spherical prediction variance will also be done. It will be shown that the cylindrical design is an excellent alternative when some but not all factors have restricted levels.