Optimal designs for estimating linear and quadratic contrasts with three level factors, the case N ≡ 0 mod 3

Abstract

The purpose of this paper is to find and construct optimal designs for estimating the standardized linear and quadratic contrasts in fractional factorials with k factors, each at 3 levels, when the number of runs or assemblies is N. The case N=3m is examined, the notion of Balanced Arrays BA(N, k, 3, 2) or BA(N, k) for short, is introduced and the optimal BA(N, k) is specified. It is shown that for N=9m the orthogonal array OA(N, k, 3, 2) or OA(N, k) for short, is the φ-optimal design. If N=9m+3 and N=9m+6 the optimal designs are BA(N, k) which are specified for every value of N and k. In the case N=9m+3 and k=3 the optimal BA(N, k) are constructed by augmenting OA(N, k, 3, 2) by three rows which are specified. If the OA(N, k, 3, 2) does not exist, algorithms are developed to construct the optimal BA(N, k). For N=9m+6 and k=3 the optimal BA(N, k) are constructed by augmenting OA(N, k) by six rows, which are specified, otherwise algorithms are developed. Under optimal BA(N, k), the estimators of linear and quadratic contrasts are uncorrelated. The cases N=12,15,21,24,30,33 are examined in detail and optimal BA(N, k) are presented for different values of the number k of factors.