Balanced arrays of strength 2l and balanced fractional 2 m factorial designs

Abstract

SummaryA connection between a balanced fractional 2 ~ factorial design of resolution V and a balanced array of strength 4 with index set {Z0, Z~, /~2,/~3, Z~} has been established by Srivastava [3]. The purpose of this paper is to generalize his results by investigating the combinatorial property of a fraction T and the algebraic structure of the information matrix of the fractional design. Main results are: A necessary and sufficient condition for a fractional 2 ~ factorial design T of resolution 2/+1 to be balanced is that T is a balanced array of strength 2l with index set {/~0,/~1,/~,"',/~2t} provided the information matrix M is nonsingular.

Alias balanced and alias partially balanced fractional 2 m factorial designs of resolution 2l+1

“…In Shirakura [8], a 2~xl vector w=(e':e*')' in (1.1) is assumed to be normalized, i.e., w'w=l. As far as the problem of this paper is concerned, however, it may be assumed without loss of generality that such a vector w satisfies w'w=2~% Recall an (l+1) sets triangular type multidimensional partially balanced association algebra ~ defined in [14]. Then we have …”

“…In this paper, we introduce the concept of alias partially balanced (APB) designs as a generalization of AB designs. As will be seen from Yamamoto, Shirakura and Kuwada [14], this concept is similar to a generalization of orthogonal fractional designs to balanced fractional designs. We also discuss what designs are AB or APB designs.…”

“…The expected value of the observation y(j~,..., j~) for an assembly (j,,..., j~) can then be expressed as 57 (1.1) E(y(j~,..., j~)) :e'0-ke*'0*, where the elements of (e', e*') corresponding to 0~...~ k are given by d(j~,) 9 d(j~)...d (j~), and in particular the element to 0~ is given by 1 (see, e.g., Yamamoto, Shirakura and Kuwada [14]). Here d(j)=--I or 1 according as j=0 or 1.…”

On the Existence Conditions for Balanced Fractional $2^{m}$ Factorial Designs of Resolution $\mathrm{R}^{\ast}(\{1\}|\mathrm{\Omega}_{\ell})$ with $N<\nu_{\ell}(m)$

“…If the variance-covariance matrix of the estimators of the factorial effects to be of interest is invariant under any permutation on the factors, then a design is said to be balanced. Under certain conditions, a BA of strength 2ℓ turns out to be a balanced fractional 2 m factorial (2 m -BFF) design of resolution 2ℓ + 1 (see for ℓ = 2, Srivastava, 1970, and for general ℓ, Yamamoto et al, 1975), where 2ℓ ≤ m. The characteristic roots of the information matrix of a 2 m -BFF design of resolution V, i.e., ℓ = 2, were obtained by Srivastava and Chopra (1971). By using the triangular multidimensional partially balanced (TMDPB) association scheme and its algebra, their results were generalized by Yamamoto et al (1976) and Hyodo (1992) for a resolution 2ℓ + 1 design, where 2ℓ ≤ m and m < 2ℓ ≤ 2m, respectively.…”

Characterization of Balanced Fractional 2^<i>m</i> Factorial Designs of Resolution R^*({1}|3) and GA-optimal Designs

“…Under certain conditions, a B-array of strength 2 and two symbols turns out to be a balanced fractional 2 m factorial (2 m -BFF) design of resolution 2 + 1 (e.g., Srivastava (1970), and Yamamoto et al (1975)), where 2 ≤ m. The characteristic roots of the information matrix of a 2 m -BFF design of resolution V (i.e., = 2) were obtained by Srivastava and Chopra (1971). By applying the algebraic structure of the triangular multidimensional partially balanced (TMDPB) association scheme, their results were extended to 2 m -BFF designs of resolution 2 + 1 by Yamamoto et al (1976).…”