In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453–464.] the concepts of a g(n,e)-design and a g(n,e)-matrix are introduced to study designs of n factor two-level experiments which can unbiasedly estimate the mean, the n main effects and e specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g-matrix. This paper deals with the construction of D-optimal g(n,1)-matrices. A standard form for a g(n,1)-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)-matrix in the class of all g(n,1)-matrices are obtained and an approach to construct D-optimal g(n,1)-matrices is given for 2⩽n⩽8. For two specific subclasses, namely a certain class of g(n,1)-matrices within the class of g(n,1)-matrices of index one and the class C(H) of g(8t+2,1)-matrices constructed from a normalized Hadamard matrix H of order 8t+4(t⩾1) two techniques for the construction of the restricted D-optimal matrices are given.

The concept of a strongly threefold orthogonal (STO) matrix is studied in A.S. Hedayat and H. Pesotan [J. Statist. Plann. Inference 15 (1986) 11–17; Linear Algebra Appl. 136 (1990) 1–23]. Let C(R,n) be the class of all STO matrices with R rows and index n. We define Lmax(C(R,n))=L if and only if there is a STO matrix in C(R,n) with L columns and every matrix in C(R,n) has at most L columns. A matrix in C(R,n) is called column optimal if it has Lmax(C(R,n)) columns. The column optimality problem for C(R,n) consists in determining Lmax(C(R,n)) and constructing a column optimal matrix in C(R,n). In this paper we study the column optimality problem for the sequence of classes C(8t+4,8),t⩾3. It is shown that (1) Lmax (C(28,8)) = 6, (2) Lmax(C(16t+12,8))=9 when t⩾46 and (3) 8⩽Lmax(C(16t+4,8))⩽9 when t⩾27. Corresponding column optimal STO matrices in the classes C(28, 8) and C(16t+12,8), t⩾46 are also constructed.

An orthogonal polynomial model is used to model the response influenced by n two level factors. Such a model is represented by an undirected graph g with n vertices and e edges. The vertices identify the n main effects and the e edges identify the two-factor interactions of interest which together with the mean are the parameters of interest. A g-design is a saturated design which can provide an unbiased estimator for these parameters and its design matrix is called a g-matrix. The latter two concepts were introduced by Hedayat and Pesotan (Statistica Sinica 2 (1992), 453–464). In this paper methods of constructing g-matrices are studied since such constructions are equivalent to the construction of g-designs. Some bounds on the absolute value of a determinant of a g-matrix are given and D-optimality results on certain classes of g-matrices are presented.

In Table 4 of Draper and Lin (1990) the authors present eleven 2k − pR designs for which doubt is expressed as to whether they are saturated. This note removes doubt on nine of these designs indicating that they are indeed saturated. Some observations are made in terms of the resolution of a design for the remaining two designs among the 11. The conclusions are derived by properly interpreting an updated table of binary linear codes of Verhoeff (1987).

From a Practical viewpoint the first decision to be made in the construction of a design of a two-level factorial experiment is the choice of the parameters of interest. It is convenient to represent such a choice by considering an undirected graph g with n vertices and e edges. The vertices and edges of g are used respectively to identify the main effects of n two-level factors and the e two-factor interactions Of interest. The parameters identified by g together with the general mean are taken to be the parameters of interest. A design d of the 2n factorial will be called a g-design if and only if d is saturated and is capable of providing an unbiased estimator of the parameters of interest relative to the orthogonal polynomial model. In this paper (i) a g-design is constructed for each graph g and certain features of g-designs are noted, (ii) some D-optimality results for g-designs within the class of all g-designs are obtained.

In this paper the authors obtained a determinant optimal embedding of an s(s+1) saturated orthogonal main effect plan in s2 runs into an s(s+1) x 2 expediment in s2 + 1 runs when s = 2m for a positive integer m. It is shown that an optimal embedding does not exist for this situation if s is an odd prime or a power of an odd prime.

Strongly threefold orthogonal (STO) matrices are introduced as a generalization of exact triply balanced matrices studied by Hedayat and Pesotan (1986). The latter concept grew out of the related notion of nearly triply balanced matrices, which occur in estimating and studying mean square errors of nonlinear statistics in survey samplings. In this paper we present some properties of STO matrices and use them to give some constructions of these arrays. In particular, column optimal STO matrices within certain specified classes of STO matrices are constructed. Further, it is shown that a column optimal STO matrix with 20 rows and index 8 must have 6 columns. Finally, a statistical application of these arrays is given when they are interpreted as fractions of a factorial experiment with each factor at two levels relative to the orthogonal polynomial model.

Under the setting of the columnwise orthogonal polynomial model in the context of the general factorial it is shown that (i) the determinant of the information matrix of a design relative to an admissible vector of effects is invariant under a permutation of levels; (ii) the unbiased estimation of a linear function of an admissible vector of effects can be obtained under equal probability randomization. These results extend the work on invariance and randomization carried out under the more restrictive assumption of the orthonormal polynomial model by Srivastava, Raktoe and Pesotan (1976) and Pesotan and Raktoe (1981). Moreover, the problem of the construction of D-optimal main effect designs in the symmetrical factorial is reduced to a study of a special class of (0,1)-matrices using the Helmat matrix model. Using this class of (0,1)-matrices and the determinant invariance result, some classes of D-optimal main effect designs of the s2 and s3 factorial respectively are presented.

Saha and Mohanty (1970) presented a main effect fold-over design consisting of 14 treatment combinations of the 24×33 factorial, which had the nice property of being even balanced. Calling this design DSM, this paper establishes the following specific results: (i) DSM is not d-optimal in the subclass Δe of all 14 point even balanced main effect fold-over designs of the 24×33 factorial; (ii) DSM is not d-optimal in the subclass Δ∗⊂Δe of all 14 point even and odd balanced main effect fold-over designs of the 24×33 factorial; (iii) DSM is even optimal in Δ∗ and Δe. In addition to these results two 14 point designs in Δ∗ are presented which are d-optimal and via a counter example it is shown that these designs are not odd optimal. Finally, several general matrix algebra results are given which should be useful in resolving d-optimality problems of fold-over designs of the kn11×kn22 factorial.

This paper deals with the existence and nonexistence of BIB designs with repeated blocks. The approach is an algebraic one. The concept of a support matrix is introduced and some of its basic properties are noted. Some basic examples of support matrices are given when the block size is 3. The connection between full column rank proper support matrices and irreducible designs is explored and some examples of such matrices are given.

A class of designs with property C(t) are introduced for the first time, and their applications in group testing of samples are studied.

R × L triply balanced matrices arise in estimating the mean square errors of nonlinear statistics in survey samplings. It is shown that: (1) Any R × L exact triply balanced matrix and an orthogonal array OA(R, L, 2, 3; λ) are one and the same object up to a possible notational change of the two symbols of the array. (2) R is a multiple of 8 and L ≤ 12R. (3) The problem of the construction of R × L exact triply balanced matrices, 3 ≤ L ≤ 12R, is completely resolved modulo the existence of Hadamard matrices of order 12R. (4) There is no sequence of R × L matrices which are nearly triply balanced in the sense of Rao and Wu (1985) if R < 2L.

This paper establishes the spectrum invariance of the information matrix under an arbitrary subgroup Г of the group Δ of factor permutations. In addition, it provides the randomized unbiased estimation of a linear parametric function under the composition Ω∘Г, where Ω is the group of level permutations. These two results are achieved for the most practical partitioning of the whole parametric vector using the concepts of Г-closed and admissibility of a parametric subvector. Applications are given with an explicit illustration using the minimal resolution III design setting for the 23 factorial.

Let D be a saturated fractional factorial design of the general K1 x K2 ...x Kt factorial such that it consists of m distinct treatment combinations and it is capable of providing an unbiased estimator of a subvector of m factorial parameters under the assumption that the remaining k-m,t (k = H it ) factorial parameters are negligible. Such a design will not provide an unbiased estimator of the varianceσ2 Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatmentcombinations with the aim to estimate 2 Suppose that D is an optimal design with respect to some optimality criterion (e.g. d-optimality, a-optimality or e-optimality) and it is desirable to augment D with c treatment combinations with the aim to estimate σ2 unbiasedly. The problem then is how to select the c treatment combinations such that the augmented design D retains its optimality property. This problem, in all its generality is extremely complex. The objective of this paper is to provide some insight in the problem by providing a partial answer in the case of the 2tfactorial, using the d-optimality criterion.

Dependence of the global orthogonality of a regular fraction of the s factorial s as introduced by Raktoe £t al_. f1980], on the basic matrix of contrasts used is brought out and studied in this paper. Some basic matrices for s = 3, 4, 5, 8, and a series of other higher values are presented for which any Aubapace-type regular design is globally orthogonal. For s « 4, under suitable basic matrices of contrasts, all regular fractions of the 4 factorial are shown to be globally orthogonal. A similar result for s = 2 is obtained by Raktoe ett ail. [1980J. Finally subspace-type fractions that are globally orthogonal under any basic matrix of contrasts are also identified

The partitioning of the complete parametric vector from the experimenter's viewpoint for the general mixed factorial gives rise to the usual four exhaustive cases, (i), (ii), (iii), and (iv). Cases (i) and (iv) may be viewed as special cases of cases (ii) and (iii) respectively in connection with the problem of spectrum invariance under the group of level permutations and the problem of unbiased estimation under a uniform randomized design. Srivastava, Raktoe and Pesotan (1976) resolved these problems for case (ii) and this paper does the same for case (iii) so that the study is now complete.

AbstractThe concepts of defining contrast (DC), generalized defining relationship (GDR) and aliasing structure (AS) are now well established in the terminology of regression analysis and factorial design theory. There is no complete agreement in the literature about the meaning of regular and irregular fractional factorial designs. This paper provides a workable definition of a regular fraction from a symmetrial prime‐powered factorial. It characterizes the uniqueness of the GDR for fractions from the most general factorial. Results are also présentés on the uniqueness of the GDR for regular designs, on orthogonality aspects of regular and irregular designs, and on group‐theoretic generation of the complete aliasing structure. Examples are provided to illustrate the developments.