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.     

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.     

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.     

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.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

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.     

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.     

Bayesian D-optimal designs for exponential regression models

We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.     

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.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

A REVIEW OF BOUNDS FOR THE EFFICIENCY FACTOR OF BLOCK DESIGNS

This paper draws together bounds for the efficiency factor of block designs, starting with the papers of Conniffe & Stone (1974) and Williams & Patterson (1977). By extending the methods of Jarrett (1983), firstly to cover supercomplete block designs and then to cover resolvable designs, a set of bounds is obtained which provides the best current bounds for any block design with equal replication and equal block size, including resolvable designs and two-replicate resolvable designs as special cases. The bounds given for non-resolvable designs apply strictly only to designs which are either regular-graph (John & Mitchell, 1977) or whose duals are regular-graph. It is conjectured (John & Williams, 1982) that they are in fact global bounds. Similar qualifications apply to the bounds for resolvable designs.     

Block-circulant matrices for constructing optimal Latin hypercube designs

Computer simulations are usually needed to study a complex physical process. In this paper, we propose new procedures for constructing orthogonal or low-correlation block-circulant Latin hypercube designs. The basic concept of these methods is to use vectors with a constant periodic autocorrelation function to obtain suitable block-circulant Latin hypercube designs. A general procedure for constructing orthogonal Latin hypercube designs with favorable properties and allowing run sizes being different from a power of 2 (or a power of 2 plus 1), is presented here for the first time. In addition, an expansion of the method is given for constructing Latin hypercube designs with low correlation. This expansion is useful when orthogonal Latin hypercube designs do not exist. The properties of the generated designs are further investigated. Some examples of the new designs, as generated by the proposed procedures, are tabulated. In addition, a brief comparison with the designs that appear in the literature is given.     

Designs for cropping systems research

The present article establishes equivalence between extended group divisible (EGD) designs and designs for crop sequence experiments. This equivalence has encouraged the agricultural experimenters to use EGD designs for their experimentation. Some real life applications of EGD designs have been given. It has also been shown that several existing association schemes are special cases of EGD association scheme. Some methods of construction of EGD designs are also given. A catalogue of EGD designs obtainable through methods of construction along with efficiency factors of various factorial effects is also presented. In some crop sequence experiments that are conducted to develop suitable integrated nutrient supply system of a crop sequence, the treatments do not comprise of a complete factorial structure. The experimenter is interested in estimating the residual and direct effect of the treatments along with their cumulative effects. For such experimental settings block designs with two sets of treatments applied in succession are the appropriate designs. The correspondence established between row–column designs and block designs for two stage experiments by Parsad et al. [2003. Structurally incomplete row–column designs. Comm. Statist. Theory Methods 32(1), 239–261] has been exploited in obtaining designs for such experimental situations. Some open problems related to designing of crop sequence experiments are also given.     

Orthogonal designs for computer experiments

Computer experiments are becoming increasingly popular surrogates for physical experiments in recent years. In this paper, we propose a general procedure for constructing orthogonal designs with many levels and distinct runs. These designs are suitable for computer experiments. The basic idea of the method is to fold-over suitable orthogonal matrices. The properties of the derived designs are studied and a brief comparison with the designs in the literature is given.     

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.     

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.     

Screening designs for model discrimination

We introduce new criteria for model discrimination and use these and existing criteria to evaluate standard orthogonal designs. We show that the capability of orthogonal designs for model discrimination is surprisingly varied. In fact, for specified sample sizes, number of factors, and model spaces, many orthogonal designs are not model discriminating by the definition given in this paper, while others in the same class of orthogonal designs are. We also use these criteria to construct optimal two-level model-discriminating designs for screening experiments. The efficacy of these designs is studied, both in terms of estimation efficiency and discrimination success. Simulation studies indicate that the constructed designs result in substantively higher likelihoods of identifying the correct model.     

D-optimal designs for beta regression models with single predictor

The problem of construction of D-optimal designs for beta regression models involving one predictor is considered for the mean-precision parameterization suggested by Ferrari and Cribari-Neto [Beta regression for modelling rates and proportions. J Appl Stat. 2004;31:799–815]. Here we use the logit link function for the mean sub-model. These designs are presented and discussed for unrestricted as well as restricted design regions by considering the precision parameter as (1) a known constant and (2) an unknown constant. Efficiency comparison of obtained designs with commonly used equi-weighted, equi-spaced designs is made to recommend designs for practical use. Real-life applications are given to show the usefulness of these designs.     

Some adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

Minimax A-, c-, and I-optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A-, c-, and I-optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.