On the characteristic polynomial of the information matrix of balanced fractional sm factorial designs of resolution Vp,q

An explicit expression for the characteristic polynomial of the information matrix for a balanced fractional s^m factorial design of resolution V_{p, q} (in particular, when p = q = s − 1, of resolution V) is obtained by utilizing the decomposition of a multidimensional relationship algebra into its four two-sided ideals. Furthermore, by use of the algebraic structure of the underlying multidimensional relationship, the trace and the determinant of the covariance matrix of the estimates of effects to be interest are derived.     

On invariance and randomization under factor permutation in fractional factorial designs

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.     

Designs for two-level factorial experiments with linear models containing main effects and selected two-factor interactions

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.     

Characteristic polynomials of the information matrices of balanced fractional 3m factorial designs of resolution v

An explicit expression for the characteristic polynomial of the information matrix M_T of a balanced fractional 3^m factorial (3^m-BFF) design T of resolution V is obtained by utilizing the algebraic structure of the underlying multidimentional relationship. Also by using of the multidimensional relationship algebra, the trace and the determinant of the covariance matrix of the estimates of effects are derived.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

Optimal partially replicated parallel-flats designs for 2×3 mixed factorial experiments

Augmenting additional repeated runs to an unreplicated factorial design provides an economical scheme of obtaining an unbiased estimate for the error variance based on pure replicates. The augmented partially replicated design usually performs satisfactorily in identifying truly active effects regardless of whether the effect sparsity principle holds. Liao and Chai (2009) proposed a set of sufficient conditions for a partially replicated two-level design to be D-optimal over the class of parallel-flats designs. In this article, we generalize their result to the 2^n₁×3^n₂ mixed factorial under D-, A- and E-optimality, and include the 2ⁿ and 3ⁿ symmetric factorials as special cases. In addition, some examples are given to illustrate the main results.     

On some optimal fractional 2m factorial designs of resolution V

This paper presents the trace of the covariance matrix of the estimates of effects based on a fractional 2^m factorial (2^m-FF) design T of resolution V for the following two cases: One is the case where T is constructed by adding some restricted assemblies to an orthogonal array. The other is one where T is constructed by removing some restricted assemblies from an orthogonal array of index unity. In the class of 2^m-FF designs of resolution V considered here, optimal designs with respect to the trace criterion, i.e. A-optimal, are presented for m = 4, 5, and 6 and for a range of practical values of N (the total number of assemblies). Some of them are better than the corresponding A-optimal designs in the class of balanced fractional 2^m factorial designs of resolution V obtained by Srivastava and Chopra (1971b) in such a sense that the trace of the covariance matrix of the estimates is small.     

Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

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)$g(n,e)$-design and a g(n,e)$g(n,e)$-matrix are introduced to study designs of n$n$ factor two-level experiments which can unbiasedly estimate the mean, the n$n$ main effects and e$e$ 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)$g(n,1)$-matrices. A standard form for a g(n,1)$g(n,1)$-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)$g(n,1)$-matrix in the class of all g(n,1)$g(n,1)$-matrices are obtained and an approach to construct D-optimal g(n,1)$g(n,1)$-matrices is given for 2?n?8$2?n?8$. For two specific subclasses, namely a certain class of g(n,1)$g(n,1)$-matrices within the class of g(n,1)$g(n,1)$-matrices of index one and the class C(H)$C(H)$ of g(8t+2,1)$g(8t+2,1)$-matrices constructed from a normalized Hadamard matrix H   of order 8t+4(t?1)$8t+4(t?1)$ two techniques for the construction of the restricted D-optimal matrices are given.     

Indicator function and complex coding for mixed fractional factorial designs

In a general fractional factorial design, the n levels of a factor are coded by the nth roots of the unity. This device allows a full generalization to mixed-level designs of the theory of the polynomial indicator function which has already been introduced for two-level designs in a joint paper with Fontana. The properties of orthogonal arrays and regular fractions are discussed.     

Optimal fractional factorial designs for estimating interactions of one factor with all others: 3m Series

In some experimental situations, only one factor is expected to interact with other factors. Designs which permit estimation of all main effects and the interactions of one factor ‘With All Others’, are termed WAO designs. This paper discusses the existence and construction of s^m WAO designs. A series of WAO designs are presented for the 3^m factorial, for m = 6, 7, ... , 14. The p non-negligible effects are estimable in 9f^? runs, where f^? is the smallest integer such that 9f^? ≥p. These designs are determinant optimal within the class of parallel flats fractions and, except for the case f^? = 9, are new. They are ideally suited for sequential experiments.     

Asymptotic relative efficiency of designs for factorial paired comparison experiments

A general approach for comparing designs of paired comparison experiments on the basis of the asymptotic relative efficiencies, in the Bahadur sense, of their respective likelihood ratio tests is discussed and extended to factorials. Explicit results for comparing five designs of 2^q factorial paired comparison experiments are obtained. These results indicate that some of the designs which require comparison of fewer distinct pairs of treatments than does the completely balanced design are, generally, more efficient for detecting main effects and/or certain interactions. The developments of this paper generalize the work of Littell and Boyett (1977) for comparing two designs of R x C factorial paired comparison experiments.     

Experimentation order with good properties for 2 k factorial designs

Randomizing the order of experimentation in a factorial design does not always achieve the desired effect of neutralizing the influence of unknown factors. In fact, with some very reasonable assumptions, an important proportion of random orders achieve the same degree of protection as that obtained by experimenting in the design matrix standard order. In addition, randomization can induce a large number of changes in factor levels and thus make experimentation expensive and difficult. De Leon et al. [Experimentation order in factorial designs with 8 or 16 runs, J. Appl. Stat. 32 (2005), pp. 297–313] proposed experimentation orders for designs with eight or 16 runs that combine an excellent level of protection against the influence of unknown factors, with the minimum number of changes in factor levels. This article presents a new methodology to obtain experimentation orders with the desired properties for designs with any number of runs.     

Optimal designs for asymmetrical factorial paired comparison experiments

Three forms of a general null hypothesis Ho on the factorial parameters of a general asymmetrical factorial paired comparison experiment are considered. A class of partially balanced designscorresponding to each form of H₀ is constructed and the A,D and ioptimal design, minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix of related maximumlikelihoodestimators, in that class is determined. Moreover, the optimal design in each class maximizes the noncentrality parameter λ² of the asymptotic noncentral chi-square distribution of the likelihood ratiostatistic -2 log λ for testing H_o under defined local alternatives. These results apply directly to symmetrical factorial paired comparison experiments as special casesExamples are given forillustrating applications of the developed results     

Isomorphism check in fractional factorial designs via letter interaction pattern matrix

Two fractional factorial designs are considered isomorphic if one can be obtained from the other by relabeling the factors, reordering the runs, and/or switching the levels of factors. To identify the isomorphism of two designs is known as an NP hard problem. In this paper, we propose a three-dimensional matrix named the letter interaction pattern matrix (LIPM) to characterize the information contained in the defining contrast subgroup of a regular two-level design. We first show that an LIPM could uniquely determine a design under isomorphism and then propose a set of principles to rearrange an LIPM to a standard form. In this way, we can significantly reduce the computational complexity in isomorphism check, which could only take O(2^p)+O(3k³)+O(2^k) operations to check two 2^k−p designs in the worst case. We also find a sufficient condition for two designs being isomorphic to each other, which is very simple and easy to use. In the end, we list some designs with the maximum numbers of clear or strongly clear two-factor interactions which were not found before.     

On some DS-optimal designs in spherical regions

For polynomial regression over spherical regions the d-th order D_s-optimal designs for the λ-th order models are derived for 1 ≤ λ ≤ d ≤ 4. Efficiencies of these designs with respect to the λ-th order D-optimal designs are obtained. The effects of estimating addtional parameters due to an m-th order model (d ≥ m >>λ) on the efficiencies are investigated.     

Weighted A-optimality for fractional 2m factorial designs of resolution V

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.     

Cyclic designs for a covariate model

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. We restrict our attention to classes of designs d for which the number of observations N to be taken is a multiple of V, i.e. N = V × R with R ≥2, and each treatment level is observed R times. Among these designs, called here equireplicated, there is a subclass characterized by the following: the allocation matrix of each treatment level (for short, allocation matrix) is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. We call these designs cyclic. Besides having easy representation, the most efficient cyclic designs are often D-optimal in the class of equireplicated designs. A known upper bound for the determinant of the information matrix M(d) of a design, in the class of equireplicated ones, depends on the congruences of N and V modulo 4. For some combinations of parameter moduli, we give here methods of constructing families of D-optimal cyclic designs. Moreover, for some sets of parameters (N, V,K = V), where the upper bound on ∣M(d)∣ (for that specific combination of moduli) is not attainable, it is also possible to construct highly D-efficient cyclic designs. Finally, for N≤24 and V≤6, computer search was used to determine the most efficient design in the class of cyclic ones. They are presented, together with their respective efficiency in the class of equireplicated designs.     

Theory of optimal blocking of nonregular factorial designs

The authors introduce the notion of split generalized wordlength pattern (GWP), i.e., treatment GWP and block GWP, for a blocked nonregular factorial design. They generalize the minimum aberration criterion to suit this type of design. Connections between factorial design theory and coding theory allow them to obtain combinatorial identities that govern the relationship between the split GWP of a blocked factorial design and that of its blocked consulting design. These identities work for regular and nonregular designs. Furthermore, the authors establish general rules for identifying generalized minimum aberration (GMA) blocked designs through their blocked consulting designs. Finally they tabulate and compare some GMA blocked designs from Hall's orthogonal array OA(16,2¹5,2) of type III.