Entropy of unequal probability sampling designs

There exist many designs for unequal probability sampling. In this paper entropy, which is a measure of randomness, is used to compare eight designs. Both old and commonly used designs and more recent designs are included. Several different and general estimates of entropy are presented. In the quest of finding entropy, expressions for the probability function are derived for different designs. One of them is a recent general design called correlated Poisson sampling. Several designs are close to having maximum entropy, which means that the designs are robust. A few designs yield low entropy and should therefore in general be avoided.     

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 Exact K-optimal Designs Minimizing the Condition Number

A new design criterion based on the condition number of an information matrix is proposed to construct optimal designs for linear models, and the resulting designs are called K-optimal designs. The relationship between exact and asymptotic K-optimal designs is derived. Since it is usually hard to find exact optimal designs analytically, we apply a simulated annealing algorithm to compute K-optimal design points on continuous design spaces. Specific issues are addressed to make the algorithm effective. Through exact designs, we can examine some properties of the K-optimal designs such as symmetry and the number of support points. Examples and results are given for polynomial regression models and linear models for fractional factorial experiments. In addition, K-optimal designs are compared with A-optimal and D-optimal designs for polynomial regression models, showing that K-optimal designs are quite similar to A-optimal designs.     

Approximate theory of multiway block designs

Optimality properties of multiway block designs are deduced from the general results of J. Kiefer's approximate-design theory. In the model with additive effects these optimality properties solely depend on the two-dimensional marginals of the designs. Uniform designs, and designs whose two-dimensional marginals are products of the one-dimensional marginals, are shown to be optimal. Approximate Youden designs are introduced for the case when the support sets of the two-dimensional marginals are prescribed in advance. They are optimal in a relatively small class of competing designs only.     

Optimal block designs for triallel cross experiments

Optimal block designs for a certain type of triallel cross experiments are investigated. Nested balanced block designs are introduced and it is shown how these designs give rise to optimal designs for triallel crosses. Several .series of nested balanced block designs, leading to optimal designs for triallel crosses are reported.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

The analysis of mixed and random effects models for equireplicate variance-balanced block designs

Equirephcate variance-balanced block designs possess many desirable properties and include several important subclasses of designs, such as balanced incomplete block designs. However, while attention has been paid to the derivation of properties and the construction of these designs, methods of analyzing experiments using these designs have not been developed completely. Thus, we consider models appropriate to these designs, construct test statistics for basic hypotheses and derive the distributions for these statistics. Four additive models that include all of the possible combinations of fixed and random effects are examined. The analyses are generalizations of the analyses for balanced incomplete block designs, and include the intra-block, intra-treatment and inter-block analyses, Estimators of fixed treatment effects and their distributions also are given.     

Classification of efficient two-level fractional factorial designs of resolution IV or more

Two-level fractional factorial designs that are efficient in terms of aberration or other aliasing properties are classified into four types of designs of resolution IV or more: the half-fraction designs, the even designs, the five-column designs and the join designs. The designs are shown to have concise grid representations which provide simple interpretations of their aliasing structure. New efficient 128-run designs are presented and blocking of the designs is considered.     

Model-Robust Design of Conjoint Choice Experiments

Within the context of choice experimental designs, most authors have proposed designs for the multinomial logit model under the assumption that only the main effects matter. Very little attention has been paid to designs for attribute interaction models. In this article, three types of Bayesian D-optimal designs for the multinomial logit model are studied: main-effects designs, interaction-effects designs, and composite designs. Simulation studies are used to show that in situations where a researcher is not sure whether or not attribute interaction effects are present, it is best to take into account interactions in the design stage. In particular, it is shown that a composite design constructed by including an interaction-effects model and a main-effects model in the design criterion is most robust against misspecification of the underlying model when it comes to making precise predictions.     

Circular Binary Block Second- and Higher-Order Neighbor Designs

Some new neighbor designs are presented here. Second-order neighbor designs for different configurations are generated in circular binary blocks. Third-order and fourth-order neighbor designs for some cases are also constructed. In all cases, circular blocks are well separated and these designs are obtained through initial block/s. At the end of the study, some models for analysis of these designs are also 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.     

Construction of circular partially balanced-Repeated measurement designs using cyclic shifts

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. These designs balance out the residual effects. The situations where balanced repeated measurements designs require a large number of the subjects, partially-balanced repeated measurements designs should be used. In this paper some infinite series are developed which provide circular partially-balanced repeated measurement designs for p (periods) even. Catalogues of circular partially-balanced repeated measurement designs are also presented for v (treatments) ≤ 100 with p = 5, 7 & 9.     

On the Construction of Nearest–Neighbour Balanced Row–Column Designs

Nearest–neighbour balance is considered a desirable property for an experiment to possess in situations where experimental units are influenced by their neighbours. This paper introduces a measure of the degree of nearest–neighbour balance of a design. The measure is used in an algorithm which generates nearest–neighbour balanced designs and is readily modified to obtain designs with various types of nearest–neighbour balance. Nearest–neighbour balanced designs are produced for a wide class of parameter settings, and in particular for those settings for which such designs cannot be found by existing direct combinatorial methods. In addition, designs with unequal row and column sizes, and designs with border plots are constructed using the approach presented here.     

The construction of nearly balanced and nearly strongly balanced uniform cross-over designs

In this paper we consider the class of uniform cross-over designs. Existing results on the universal optimality of uniform cross-over designs are reviewed and a general method of construction is described. The constructed designs fall into four families, which include the balanced and strongly balanced designs as special cases: the remaining designs we refer to as nearly strongly balanced, a term first introduced by Kunert (Ann. Statist. 11 (1983)), and nearly balanced. The nearly strongly balanced and nearly balanced designs form an important family of uniform cross-over designs which provide designs where balanced or strongly balanced designs do not exist. The construction method can be easily generalized for any number of periods and subjects, as long as they are both a multiple of the number of treatments. Some illustrative examples are included.     

A-optimal diallel crosses for test versus control comparisons

A-optimality of block designs for control versus test comparisons in diallel crosses is investigated. A sufficient condition for designs to be A-optimal is derived. Type S₀ designs are defined and A-optimal type S₀ designs are characterized. A lower bound to the A-efficiency of type S₀ designs is also given. Using the lower bound to A-efficiency, type S₀ designs are shown to yield efficient designs for test versus control comparisons.     

Balanced Factorial Designs for cDNA Microarray Experiments

Balanced factorial designs are introduced for cDNA microarray experiments. Single replicate designs obtained using the classical method of confounding are shown to be particularly useful for deriving suitable balanced designs for cDNA microarrays. Classical factorial designs obtained using methods other than the method of confounding are also shown to be useful. The paper provides a systematic method of deriving designs for microarray experiments as opposed to algorithmic and ad-hoc methods and generalizes several of the microarray designs given recently in the literature.     

Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

Some families of group divisible designs

Generalizing methods of constructions of Hadamard group divisible designs due to Bush (1979), some new families of semi-regular or regular group divisible designs are produced. Furthermore, new nonisomorphic solutions for some known group divisible designs are given, together with useful group divisible designs not listed in Clatworthy (1973).     

αn–Designs

This paper defines a broad class of resolvable incomplete block designs called α_n–designs, of which the original α–designs are a special case with n = 1. The statistical and mathematical properties of α–designs extend naturally to these n –dimensional designs. They are a flexible class of resolvable designs appropriate for use in factorial experiments, in constructing efficient t –latinized resolvable block designs, and for enhancing the existing class of α–designs for a single treatment factor.     

Computer experiment designs for accurate prediction

Computer experiments using deterministic simulators are sometimes used to replace or supplement physical system experiments. This paper compares designs for an initial computer simulator experiment based on empirical prediction accuracy; it recommends designs for producing accurate predictions. The basis for the majority of the designs compared is the integrated mean squared prediction error (IMSPE) that is computed assuming a Gaussian process model with a Gaussian correlation function. Designs that minimize the IMSPE with respect to a fixed set of correlation parameters as well as designs that minimize a weighted IMSPE over the correlation parameters are studied. These IMSPE-based designs are compared with three widely-used space-filling designs. The designs are used to predict test surfaces representing a range of stationary and non-stationary functions. For the test conditions examined in this paper, the designs constructed under IMSPE-based criteria are shown to outperform space-filling Latin hypercube designs and maximum projection designs when predicting smooth functions of stationary appearance, while space-filling and maximum projection designs are superior for test functions that exhibit strong non-stationarity.