Efficiency of some experimental designs to estimate a response surface

Abstract

The election of the experimental design in response surface methodology, is concentrated on minimizing experimental costs, reducing experimental time and maximizing the efficiency of the estimated response. For their simplicity and easy interpretation in estimating a response surface, linear models of order lower than or equal three have been employed very frequently. However, often polynomials of fractional degree (pseudoquadratic models), give a better approximation on fertilizer experiments. Based on the individual or joint precision of estimators, several methods have been devised, to determine the effect of experimental designs on the precision of estimators of a response surface. In this paper the efficiency of several experimental designs to estimate a pseudoquadratic response in two or three factors has been evaluated, with the criteria of the integrated variance of the estimated response, of the determinant and the trace of the (X'X)^-1 matrix. With two factors, the following designs: repeated double square, double square, A-optimum with 13 treatments, the orthogonal San Cristobal with α=0.66, A-optimum with 16 treatments, orthogonal double square with α=2.20, Thompson's central composite with 12 and 16 treatments, orthogonal triple square, augmented San Cristobal, A-optimum with 9 treatments, Escobar's double square and Thompson's central composite with 13 treatments are all found more efficient than the 32 factorial. With three factors, the designs: A-optimum with 31 treatments, A-optimum with 23 treatments, double cube and A-optimum with 18 and 15 treatments are also found more efficient than the 33 factorial.