Some d-optimal weighing designs for n≡ (mod 4)

A number of D-optimal weighing designs are constructed with the help of block matrices. The D-optimal designs (n,k,s)=(19,13,10), (19,14,7), (19,14,8), (19,15,7), (19,15,8), (19,17,6), (19,18,6), (23,16,8), (23,17,8), (23,18,8), (4n?1,2n+3,(3n+4)/2), (4n?1,2n+4,n+3), (4n?1,2n+4,n+2) where n≡0 mod 4 and a skew H_n exists, (31,24,8), (31,25,8) and many others are constructed. A computer routine leading to locally D-optimal designs is presented.     

Construction and analysis of edge designs from skew-symmetric supplementary difference sets

The purpose of screening experiments is to identify the dominant variables from a set of many potentially active variables which may affect some characteristic y. Edge designs were recently introduced in the literature and are constructed by using conferences matrices and were proved to be robust. We introduce a new class of edge designs which are constructed from skew-symmetric supplementary difference sets. These designs are particularly useful since they can be applied for experiments with an even number of factors and they may exist for orders where conference matrices do not exist. Using this methodology, examples of new edge designs for 6, 14, 22, 26, 38, 42, 46, 58, and 62 factors are constructed. Of special interest are the new edge designs for studying 22 and 58 factors because edge designs with these parameters have not been constructed in the literature since conference matrices of the corresponding order do not exist. The suggested new edge designs achieve the same model-robustness as the traditional edge designs. We also suggest the use of a mirror edge method as a test for the linearity of the true underlying model. We give the details of the methodology and provide some illustrating examples for this new approach. We also show that the new designs have good D-efficiencies when applied to first order models.     

On the optimality of chemical balance weighing designs

In this paper we consider the problem of optimally weighing n objects with N weighings on a chemical balance. Several previously known results are generalized. In particular, the designs shown by Ehlich (1964a) and Payne (1974) to be D-optimal in various classes of weighing designs where N≡2 (mod4) are shown to be optimal with respect to any optimality criterion of Type I as defined in Cheng (1980). Several results on the E-optimality of weighing designs are also given.     

The A-optimal non-saturated weighing designs for n = 15 observations

The problem of constructing A-optimal weighing and first order fractional factorial designs for n ≡ 3 mod 4 observations is considered. The non-existence of the weighing design matrices for n = 15 observations and k = 13, 14 factors, for which the corresponding information matrices have inverses with minimum trace, is proved. These designs are the first non-saturated cases (k < n) in which the unattainability of Sathe and Shenoy's (1989) lower bound on A-optimality is shown. Using an algorithm proposed in Farmakis (1991) we construct 15 × k (+1, −1)-matrices for k = 13, 14 and we prove their A-optimality using the improved (higher) lower bounds on A-optimality established by Chadjiconstantinidis and Kounias (1994). Also the A-optimal designs for n = 15, k ⩽ 12 are given.     

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.     

On invariance and randomization in factorial designs with applications to D-optimal main effect designs of the symmetrical factorial

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.     

Optimal and efficient designs for 2-parameter nonlinear models

By Carathéodory's theorem, for a k-parameter nonlinear model, the minimum number of support points for any D-optimal design is between k and k(k+1)/2. Characterizing classes of models for which a D-optimal design sits on exactly k support points is of great theoretical interest. By utilizing the equivalence theorem, we identify classes of 2-parameter nonlinear models for which a D-optimal design is precisely supported on 2 points. We also introduce the theory of maximum principle from differential equations into the design area and obtain some results on characterizing the minimally supported nonlinear designs. Examples are given to demonstrate our results. Designs with minimum number of support points may not always be suitable in practice. To alleviate this problem, we utilize some geometric and analytical methods to obtain some efficient designs which provide more opportunity for the model checking and prevent biases due to mis-specified initial parameters.     

A note on robustness of D-optimal block designs for two-colour microarray experiments

Two-colour microarray experiments form an important tool in gene expression analysis. Due to the high risk of missing observations in microarray experiments, it is fundamental to concentrate not only on optimal designs but also on designs which are robust against missing observations. As an extension of Latif et al. (2009), we define the optimal breakdown number for a collection of designs to describe the robustness, and we calculate the breakdown number for various D-optimal block designs. We show that, for certain values of the numbers of treatments and arrays, the designs which are D-optimal have the highest breakdown number. Our calculations use methods from graph theory.     

Statistical properties of Rechtschaffner designs

Rechtschaffner designs are saturated designs of resolution V   in which main effects and two-factor interactions are estimable if three-factor and higher order interactions are negligible. Statistical properties of Rechtschaffner designs are studied in this paper. Best linear unbiased estimators of main effects and two-factor interactions are given explicitly and asymptotic properties of correlations between these estimators are studied as well. It is shown that designs recommended by Rechtschaffner [1967. Saturated fractions of 2ⁿ$2^n$ and 3ⁿ$3^n$ factorial designs, Technometrics 9, 569–576] are not only A-optimal but also D-optimal. Comparisons of Rechtschaffner designs with other A- and D-optimal designs of resolution V are also discussed.     

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.     

Some sufficient conditions for the type 1 optimality of block designs

In this paper, we investigate the problem of determining block designs which are optimal under type 1 optimality criteria within various classes of designs having υ treatments arranged in b blocks of size k. The solutions to two optimization problems are given which are related to a general result obtained by Cheng (1978) and which are useful in this investigation. As one application of the solutions obtained, the definition of a regular graph design given in Mitchell and John (1977) is extended to that of a semi-regular graph design and some sufficient conditions are derived for the existence of a semi-regular graph design which is optimal under a given type 1 criterion. A result is also given which shows how the sufficient conditions derived can be used to establish the optimality under a specific type 1 criterion of some particular types of semi- regular graph designs having both equal and unequal numbers of replicates. Finally,some sufficient conditions are obtained for the dual of an A- or D-optimal design to be A- or D-optimal within an appropriate class of dual designs.     

Some new factor screening designs using the search linear model

Factor screening designs for searching two and three effective factors using the search linear model are discussed. The construction of such factor screening designs involved finding a fraction with small number of treatments of a 2^m factorial experiment having the property P_2t (no 2t columns are linearly dependent) for t=2 and 3. A ‘Packing Problem’ is introduced in this connection. A complete solution of the problem in one case and partial solutions for the other cases are presented. Many practically useful new designs are listed.     

Supersaturated designs of projectivity P=3 or near P=3

Supersaturated designs offer a potentially useful way to investigate many factors in few experiments i.e. typical screening situations. Their design properties have mainly been evaluated based on their ability to identify and estimate main effects. Projective properties have received little attention. In this paper we show how to construct two-level supersaturated designs for 2(n−2) factors in n runs (n a multiple of four) of projectivity P=3 or near projectivity P=3 from orthogonal non-regular two-level designs. The designs obtained also have favourable properties such as low maximum absolute value of the inner product between a main effect column and a two-factor interaction column and relatively few types of different projections onto subsets consisting of three factor columns.     

Optimal designs for covariates’ models

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. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

Efficient three-level screening designs using weighing matrices

Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices.     

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.     

The intersection problem for K4 - e designs

A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph K_n and B is an edge-disjoint decomposition of K_n into copies of the simple graph G. Following design terminology, we call these copies “blocks”. Here K₄ - e denotes the complete graph K₄ with one edge removed. It is well-known that a K₄ - e design of order n exists if and only if n ≡ 0 or 1 (mod 5), n ⩾ 6. The intersection problem here asks for which k is it possible to find two K₄ - e designs (P,B₁) and (P,B₂) of order n, with |B₁ ∩ B₂| = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K₄ - e designs.     

D-optimal designs for Poisson regression models

Most of the current research on optimal experimental designs for generalized linear models focuses on logistic regression models. In this paper, D-optimal designs for Poisson regression models are discussed. For the one-variable first-order Poisson regression model, it has been found that the D-optimal design, in terms of effective dose levels, is independent of the model parameters. However, it is not the case for more complicated models. We investigate how the D-optimal designs depend on the model parameters for the one-variable second-order model and two-variable interaction model. The performance of some “standard” designs that appeal to practitioners is also studied.     

Foldover Conference Designs for Screening Experiments

A conference matrix is a square matrix C with zeros on the diagonal and ±1s off the diagonal, such that C ^T C = CC ^T  = (n ? 1)I, where I is the identity matrix. Conference matrices are an important class of combinatorial designs due to their many applications in several fields of science, including statistical-experimental designs, telecommunications, elliptic geometry, and more. In this article, conference matrices and their full foldover design are combined together to obtain an alternative method for screening active factors in complicated problems. This method provides a model-independent estimate of the set of active factors and also gives a linearity test for the underlying model.     

Optimal designs for second-degree Kronecker model mixture experiments

The paper investigates optimal designs in the second-degree Kronecker model for mixture experiments. Three groups of novel results are presented: (i) characterization of feasible weighted centroid designs for a maximum parameter system, (ii) derivations of D-, A-, and E-optimal weighted centroid designs, and (iii) numerically φ_p-optimal weighted centroid designs. Results on a quadratic subspace of invariant symmetric matrices containing the information matrices involved in the design problem serve as a main tool throughout the analysis.