Design Efficiency and Optimal Values of Replicated Central Composite Designs with Full Factorial Portions

Efficiency and optimal properties of four varieties of Central Composite Design, namely, SCCD, RCCD, OCCD and FCCD and having r_f replicates of the full factorial portion, r_α replicates of the axial portion and r_c replicates of the center portion are studied in four to six design variables. Optimal combination,[r_f: r_α: r_c ] of design points associated with the three portions of each central composite design is presented. For SCCD, the optimal combinations resulting in A- and D- efficient designs generally put emphasis on replicating the center portion of the SCCD. However, replicating the center and axial portions allows for G-optimal and efficient designs. For RCCD, the optimal combinations resulting in A- and D- efficient designs generally put emphasis on replicating the factorial and center portions of the RCCD. However, replicating the center and axial portions allows for G-optimal and efficient designs. For OCCD, the optimal combinations resulting in A- optimal and efficient designs generally put emphasis on replicating the axial and center portion of the OCCD. The optimal combinations resulting in G- optimal and efficient designs generally put emphasis on replicating the factorial and axial portions of the OCCD. To achieve designs that are D-optimal and D-efficient, the optimal combination of design points generally put emphasis on replicating the center portion of the OCCD. For FCCD, the optimal combinations of design points resulting in A-efficient designs put emphasis on replicating the axial portion of the FCCD. The optimal combinations resulting in G- optimal and efficient designs as well as G-optimal and efficient designs generally put emphasis on replicating the factorial and axial portions of the FCCD. It is interesting to note that for FCCD in five design variables, any r^th complete replicate of the distinct design points of the combination [r_f: r_α: r_c ] resulted in a D-efficient design. Many super-efficient designs having efficiency values greater than 1.0 emerged under the D-criterion. Unfortunately, these designs did not perform very well under A- and G-criteria, having some efficiency values much below 0.5 or just about 0.6.