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.     

New classes of D-optimal edge designs

Edge designs are screening experimental designs that allow a model-independent estimate of the set of relevant variables, thus providing more robustness than traditional designs. In this paper we construct new classes of D-optimal edge designs. This construction uses weighing matrices of order n and weight k together with permutation matrices of order n to obtain D-optimal edge designs. One linear and one quadratic simulated screening scenarios are studied and compared using linear regression and edge designs analysis.     

Non-binary neighbor balance circular designs for v=2n and λ=2

Neighbor balance designs were first introduced by Rees (1967) in circular blocks for the use in serological research. Subsequently several researchers have defined the neighbor designs in different ways. In this paper, neighbor balance circular designs for (k≤v) block size are constructed for even number of treatments i.e. v=2n. No such series of designs is known in literature. Two theorems are developed for circular designs. Theorem 1 gives the non-binary circular blocks, whereas Theorem 2 generates binary circular blocks when n≤4 and non-binary blocks for n>4. In suggested designs no treatment is ever a neighbor of itself. Blocks are constructed in such a way that each treatment is a right and left neighbor of every other treatment for a fixed number of times say λ. Sizes of initial circular blocks are not same. One main guiding principle for such designs is to ensure economy in material use.     

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.     

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.     

2n−m designs with resolution III or IV containing clear two-factor interactions

For a fixed number of runs, not all 2^n−m designs with resolution III or IV have clear two-factor interactions. Therefore, it is highly desirable to know when resolution III or IV designs can have clear two-factor interactions. In this paper, we provide a unified geometrical study of this problem and give a complete classification of the existence of clear two-factor interactions in regular 2^n−m designs with resolution III or IV and reveal the structures of these designs.     

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.     

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.     

Efficient crossover designs in the presence of interactions between direct and carry-over treatment effects

Crossover designs, or repeated measurements designs, are used for experiments in which t treatments are applied to each of n experimental units successively over p time periods. Such experiments are widely used in areas such as clinical trials, experimental psychology and agricultural field trials. In addition to the direct effect on the response of the treatment in the period of application, there is also the possible presence of a residual, or carry-over, effect of a treatment from one or more previous periods. We use a model in which the residual effect from a treatment depends upon the treatment applied in the succeeding period; that is, a model which includes interactions between the treatment direct and residual effects. We assume that residual effects do not persist further than one succeeding period.A particular class of strongly balanced repeated measurements designs with n=t² units and which are uniform on the periods is examined. A lower bound for the A-efficiency of the designs for estimating the direct effects is derived and it is shown that such designs are highly efficient for any number of periods p=2,…,2t.     

Generalized Bhaskar Rao designs

Generalized Bhaskar Rao designs with non-zero elements from an abelian group G are constructed. In particular this paper shows that the necessary conditions are sufficient for the existence of generalized Bhaskar Rao designs with k=3 for the following groups: ?G? is odd, G=Z^r₂, and G=Z^r₂×H where 3? ?H? and r?1. It also constructs generalized Bhaskar Rao designs with υ=k, which is equivalent to υ rows of a generalized Hadamard matrix of order n where υ?n.     

Minimum aberration blocking schemes for 128-run designs

Several criteria have been proposed for ranking blocked fractional factorial designs. For large fractional factorial designs, the most appropriate minimum aberration criterion was one proposed by Cheng and Wu (2002). We justify this assertion and propose a novel construction method to overcome the computational challenge encountered in large fractional factorial designs. Tables of minimum aberration blocked designs are presented for N=128 runs and n=8–64 factors.     

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.     

Serial arrays and change-over designs

Consider change-over designs for the comparison of v treatments. Each application has a direct effect and up to n?1 residual and/or interaction effects. An array is defined to be a serial array if: (1) all treatments occur equally often in each row and (2) reading down columns, all possible permutations of n consecutive treatments occur equally often in the array. A method of constructing (λv+n?1) Xv^{n ?1}, λ integral, serial arrays is given. Use of these arrays as change-over designs is discussed with emphasis on the inclusion of interaction effects.     

Optimal U-type design for Bayesian nonparametric multiresponse prediction

This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.     

On block designs arising from rank 3 graphs

In this paper a criterion showing when the orbit of a subgraph of a given rank 3 graph forms a block design, is given. As an application several classes of block designs are derived from the triangular graph T(n) and the lattice graph L₂(n).     

Model-robust supersaturated and partially supersaturated designs

Supersaturated designs are an increasingly popular tool for screening factors in the presence of effect sparsity. The advantage of this class of designs over resolution III factorial designs or Plackett–Burman designs is that n, the number of runs, can be substantially smaller than the number of factors, m. A limitation associated with most supersaturated designs produced thus far is that the capability of these designs for estimating g active effects has not been discussed. In addition to exploring this capability, we develop a new class of model-robust supersaturated designs that, for a given n and m, maximizes the number g   of active effects that can be estimated simultaneously. The capabilities of model-robust supersaturated designs for model discrimination are assessed using a model-discrimination criterion, the subspace angle. Finally, we introduce the class of partially supersaturated designs, intended for use when we require a specific subset of _m1$m_1$ core factors to be estimable, and the sparsity of effects principle applies to the remaining (m-_m1$m-m_1$) factors.     

Weakly resolvable IV.3 search designs for the pn factorial experiment

A series of weakly resolvable search designs for the pⁿ factorial experiment is given for which the mean and all main effects are estimable in the presence of any number of two-factor interactions and for which any combination of three or fewer pairs of factors that interact may be detected. The designs have N = p(p–1)n+p runs except in one case where additional runs are required for detection and one case where $\text{(p?1)}\text{2}$ additional runs are needed to estimate all (p–1)² degrees of freedom for each pair of detected interactions. The detection procedure is simple enough that computations can be carried out with hand calculations.     

An MEP.2 plan in 3 factorial experiment and its probability of correct identification

The present work consider the problem of constructing search designs for searching at most two active hidden two-factor interactions in 3ⁿ factorial setup under the assumption that the three-factor and higher-order interactions are negligible. The designs presented here are also capable of estimating all the main effects, general mean and the effects of identified active two-factor interactions. The performance of the designs has been studied in the noisy case by computing probability of correct identification.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

Results for two-level fractional factorial designs of resolution IV or more

The main theorem of this paper shows that foldover designs are the only (regular or nonregular) two-level factorial designs of resolution IV (strength 3) or more for n   runs and n/3?m?n/2$n/3?m?n/2$ factors. This theorem is a generalization of a coding theory result of Davydov and Tombak [1990. Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry. Problems Inform. Transmission 25, 265–275] which, under translation, effectively states that foldover (or even) designs are the only regular two-level factorial designs of resolution IV or more for n   runs and 5n/16?m?n/2$5n/16?m?n/2$ factors. This paper also contains other theorems including an alternative proof of Davydov and Tombak's result.