Three methods of constructing pbib designs

Three construction methods of two- or three-associate partially balanced incomplete block (PBIB) designs are presented.     

A class of new pbib designs

Two series of PBIB designs, one with three associate classes and theother with four associate classes ars developed. Efficiency factors for two designs are computed.     

On algorithmic construction of maximin distance designs

Maximin distance designs are useful for conducting expensive computer experiments. In this article, we compare some global optimization algorithms for constructing such designs. We also introduce several related space-filling designs, including nested maximin distance designs, sliced maximin distance designs, and general maximin distance designs with better projection properties. These designs possess more flexible structures than their analogs in the literature. Examples of these designs constructed by the algorithms are presented.     

On construction of exact experimental designs from discrete ones

In the paper we try to evaluate approximately optimum exact designs of required goodness from a known approximately optimum discrete design. In dependence on this discrete design we define some subsets of the set of all considered exact designs.

Then we look for sufficiency conditions, on which elements of these subsets are approximately optimum (exact) designs of the required goodness. Moreover we offer a method to find such elements, if they exist at all. Finally we prove some statements concerning the existence of such elements for two classes of (optimum) criteria.     

On equivalence of fractional factorial designs based on singular value decomposition

The singular value decomposition of a real matrix always exists and is essentially unique. Based on the singular value decomposition of the design matrices of two general 2-level fractional factorial designs, new necessary and sufficient conditions for the determination of combinatorial equivalence or non-equivalence of the corresponding designs are derived. Equivalent fractional factorial designs have identical statistical properties for estimation of factorial contrasts and for model fitting. Non-equivalent designs, however, may have the same statistical properties under one particular model but different properties under a different model. Results extend to designs with factors at larger number of levels.     

On the construction of a class of efficient row-column designs

In this paper a method for the construction of a class of row-column designs with good statistical properties and high efficiency is presented. The class of designs produced is shown to exhibit balance, orthogonality and adjusted orthogonality. The efficiencies of these designs are investigated in detail, and they are shown to be very high, and possibly maximal in some cases.     

On the construction of optimal designs

Two strategies for specifying additional data to be included with the data of a non-orthogonal design are presented. The additional data increase the magnitude of the information matrix X′X and the orthogonality of the design matrix. Sequentially, the new points are augmented to the original design, such that each new point optimally increases the smallest eigenvalue of X′X. The new runs are created in a predefined spherical region and a rectangular region. Optimum number of additional observations is presented in order to orthogonalize the design matrix X and optimize some functions of the information matrix X′X. Comparisons of the results acquired with the proposed methods versus the most commonly used procedures for data augmentation are carried out. In addition, the advantages of the use of our techniques over the studied methods to solve the augmenting data problems are discussed.     

Algorithmic construction of nearly column-orthogonal designs

Computer models can describe complicated phenomena encountered in science and engineering fields. To use these models for scientific investigation, however, their generally long running time and mostly deterministic nature require a specially designed experiment. The column-orthogonal design (COD) is a popular choice for computer experiments. Because of the restriction on the orthogonality, however, only little CODs can be constructed. In this article, we propose two algorithms for constructing nearly CODs by rotating orthogonal arrays under two different criteria. Further, some obtained nearly CODs are nearly Latin hypercube designs. Some examples are provided to show the advantages of our algorithms. Some rotation matrices obtained via the algorithms are listed.     

On the analysis of experiments in affine resolvable designs

The paper gives explicit formulae for analysing an experiment carried out in an affine resolvable proper block design. They follow from a randomization model, decomposed into stratum submodels. Analyses within the four relevant strata, and then the combined analysis, are considered in details. The paper is essentially an extension of some results presented in recent books, by Caliński and Kageyama [2000. Block Designs: A Randomization Approach, Volume I: Analysis. Lecture Notes in Statistics, vol. 150. Springer, New York; 2003. Block Designs: A Randomization Approach, Volume II: Design. Lecture Notes in Statistics, vol. 170. Springer, New York].     

On pairs of Youden designs

It is well known that generalized Youden designs, or GYD's, enjoy a variety of optimality properties. Not being maximum trace designs, b copies of a non-regular GYD will not be optimum for b sufficiently large, opening the question of whether such a set will be so for any b. This paper explores the E-behavior of b = 2 non-regular GYDs. A general E-efficiency bound is derived and the E-optimality of a particular series is proven. That pairs of non-regular GYDs are not always E-optimal is shown by a counterexample.     

Comparison of designs for the three-fold nested random model

The quality of estimation of variance components depends on the design used as well as on the unknown values of the variance components. In this article, three designs are compared, namely, the balanced, staggered, and inverted nested designs for the three-fold nested random model. The comparison is based on the so-called quantile dispersion graphs using analysis of variance (ANOVA) and maximum likelihood (ML) estimates of the variance components. It is demonstrated that the staggered nested design gives more stable estimates of the variance component for the highest nesting factor than the balanced design. The reverse, however, is true in case of lower nested factors. A comparison between ANOVA and ML estimation of the variance components is also made using each of the aforementioned designs.     

On the problem of augmented fractional factorial designs

Let D be a saturated fractional factorial design of the general K₁ x K₂ ...x K_t 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 2^tfactorial, using the d-optimality criterion.     

On the E-optimality of complete block designs under a mixed interference model

In experiments in which the response to a treatment can be affected by other treatments, the interference model with neighbor effects is usually used. It is known that circular neighbor balanced designs (CNBDs) are universally optimal under such a model if the neighbor effects are fixed (Druilhet, 1999) or random (4 and 7). However, such designs cannot exist for every combination of design parameters. In the class of block designs with the same number of treatments as experimental units per block, a CNBD cannot exist if the number of blocks, b  , is equal to p(t−1)±1$p(t-1)\pm 1$, where p is a positive integer and t is the number of treatments. Filipiak et al. (2008) gave the structure of the left-neighboring matrix of E-optimal complete block designs with p  =1 under the model with fixed neighbor effects. The purpose of this paper is to generalize E-optimality results for designs with p∈N$p\in \mathbb{N}$ assuming random neighbor effects.     

On the construction of group divisible incomplete block designs

In this paper a method of constructing group-divisible incomplete block designs has been suggested. A series of balanced incomplete block designs has also been obtained.     

General orthogonal designs for parameter estimation of ANOVA models under weighted sum-to-zero constraints

ABSTRACT

Traditional studies on optimal designs for ANOVA parameter estimation are based on the framework of equal probabilities of appearance for each factor's levels. However, this premise does not hold in a variety of experimental problems, and it is of theoretical and practical interest to investigate optimal designs for parameters with unequal appearing odds. In this paper, we propose a general orthogonal design via matrix image, in which all columns’ matrix images are orthogonal with each other. Our main results show that such designs have A- and E-optimalities on the estimation of ANOVA parameters which have unequal appearing odds. In addition, we develop two simple methods to construct the proposed designs. The optimality of the design is also validated by a simulation study.     

On fourth order rotatable designs

Recently, Draper and Herzberg posed some open problems on fourth order rotatable designs. It has been attempted to settle these open problems.     

The analytic construction of D-optimal designs for the two-variable binary logistic regression model without interaction

Candidate locally D-optimal designs for the binary two-variable logistic model with no interaction, which comprise 3 and 4 support points lying in the first quadrant of the two-dimensional Euclidean space, were introduced by Haines et al. (D-optimal designs for logistic regression in two variables. In: Lopez-Fidalgo J, Rodrigez-Diaz JM, Torsney B, editors. MODA8 – advances in model-oriented designs and analysis. Heidelberg: Physica-Verlag; 2007. p. 91–98). The authors proved algebraically the global D-optimality of the 3-point design for the special case in which the intercept parameter is equal to?1.5434. However for other selected values of the intercept parameter, the global D-optimality of the proposed 3- and 4-point designs was only demonstrated numerically. In this paper, we provide analytical proofs of the D-optimality of these 3- and 4-point designs for all negative and zero intercept parameters of the binary two-variable logistic model with no interaction. The results are extended to the construction of D-optimal designs on a rectangular design space and illustrated by means of two examples of which one is a real example taken from the literature.