Partial A-optimal balanced fractional \(2^ m\) factorial designs with 6\(\leq m\leq 8\) (Q914305)

We shall consider a \(2^ m\)-BFF design derived from a \(BA(m,6;z_ 0^{(m)},z_ 1^{(m)},...,z_ m^{(m)})\) such that the general mean and the main effects (or the main effects only) are estimable under the situation in which all four-factor and higher order interactions are assumed to be negligible. Such a design will be called a \(2^ m\)-BFF design having resolution \(R^*(\{0,1\}| P)\) (or \(R^*(\{1\}| P))\) where \(P=\{0,1,2,3\}.For\) a given pair (N,m), there are so many \(2^ m\)-BFF designs having resolution \(R^*(\{0,1\}| P)\) (or \(R^*(\{1\}| P))\). We may note that these designs may be superior to resolution IV designs in the sense that the confounding of the three- factor interactions and the main effects can be always avoided even though the latter exist. A design considered here is explicitly described by some specified simple array (S-array) for the cases of \(m=6,7,8\) and \(N<\sum^{3}_{i=0}\left( \begin{matrix} m\\ i\end{matrix} \right)(=v_ 3\), say) where \(v_ 3\) is the total number of factorial effects up to the three-factor interactions. In Section 4, for the cases of \(m=6,7\) and 8, partial A-optimal \(2^ m\)-BFF designs having resolution \(R^*(\{0,1\}| P)\) and \(R^*(\{1\}| P))\) will be presented for each value of \(N(<v_ 3)\). The covariance matrix of the estimates and the value of its trace are also given for such designs.