On the norm of alias matrices in balanced fractional 2m factorial designs of resolution 2l + 1

The norm $\text{6A6 = {tr(A′A)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of the alias matrix A of a design can be used as a measure for selecting a design. In this paper, an explicit expression for 6A6 will be given for a balanced fractional 2^m factorial design of resolution 2l + 1 which obtained from a simple array with parameters (m; λ₀, λ₁,…, λ_m). This array is identical with a balanced array of strength m, m constraints and index set {λ₀, λ₁,…, λ_m}. In the class of the designs of resolution V (l = 2) obtained from S-arrays, ones which minimize 6A6 will be presented for any fixed N assemblies satisfying (i) m = 4, 11 ? N ? 16, (ii) m = 5, 16 ? N ? 32, and (iii) m = 6, 22 ? N ? 40.     

NECESSARY AND SUFFICIENT CONDITION FOR A BALANCED ARRAY OF STRENGTH 21 TO BE A BALANCED FRACTIONAL 2 FACTORIAL DESIGN OF RESOLUTION 2l

A necessary and sufficient condition for a balanced array of strength 2l to be a balanced fractional 2^m factorial design of resolution 2l is given. This design has the property that the main effects, two-factor interactions,.and (l-1)-factor interactions are estimable ignoring the (l + 1)-factor and higher order interactions, and that the covariance matrix of their estimates is invariant under any permutation of m factors. The above condition includes sufficient conditions given in earlier works of Shirakura (1976b, 1977).     

Search designs for 2m factorial experiments

Theorems 5, 6 and 10, and Tables 1–2 in Ghosh (1981) are corrected. These are concerned with search designs which permit the estimation of the general mean and main effects, and allow the search and estimation of one possibly unknown nonzero effect among the two- and three-factor interactions in 2^m factorial experiments. Some new results are presented.     

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

We consider a search design for the 2^m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2^m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2^m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).