The A-optimal two-level fractional factorial resolution III saturated design with n = 21 observations
Abstract
In this paper we consider the problem of optimally estimation of the effects of N factors each at two levels making N observations in fractional factorial experiments, as well as the problem of optimally weighing N objects with N weighings on a chemical balance. The case N = 21(≠ 2s(s + 1) + 1) is examined under the A-optimality criterion. Making use the improved lower bounds on A-optimality for N ≡ 1 mod 4 established by Moyssiadis, Chadjiconstantinidis and Kounias [18], we develop a computational procedure to find all the N × N information matrices M with der(M) = α2 for some α ∈ Z and having inverses with trace smaller than a given one, which corresponds to the known D-optimal saturated first order design R* for N = 21, where M = (mij) is symmetric p.d., mij = 21, mij ≡ 1 mod 4, i ≠ j. There are found up to equivalence, besides M* = R* R*T, two more such matrices M1,/M2 and the verification that do not exist (+1, -1)-matrices Ri: 21 × 21 such that RiRiT = Mi, i = 1, 2, proves finally that the D-optimal first order saturated design for N = 21 observations is the A-optimal one too.
