D-minimax Second-order Designs Over Hypercubes for Extrapolation and Restricted Interpolation Regions

The D-minimax criterion for estimating slopes of a response surface involving k factors is considered for situations where the experimental region χ and the region of interest ? are co-centered cubes but not necessarily identical. Taking χ = [ ? 1, 1]^k and ? = [ ? R, R]^k, optimal designs under the criterion for the full second-order model are derived for various values of R and their relative performances investigated. The asymptotically optimal design as R → ∞ is also derived and investigated. In addition, the optimal designs within the class of product designs are obtained. In the asymptotic case it is found that the optimal product design is given by a solution of a cubic equation that reduces to a quadratic equation for k = 3?and?6. Relative performances of various designs obtained are examined. In particular, the optimal asymptotic product design and the traditional D-optimal design are compared and it is found that the former performs very well.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

On Some Two- and Three-dimensional D-minimax Designs for Estimating Slopes of a Third-order Response Surface

Design of experiments is considered for the situation where estimation of the slopes of a response surface is the main interest. Under the D-minimax criterion, the objective is to minimize the generalized variance of the estimated axial slopes at a point maximized over all points in the region of interest in the factor space. For the third-order model over spherical regions, the D-minimax designs are derived in two and three dimensions. The efficiencies of some two- and three-dimensional designs available in the literature are also investigated.     

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.     

Construction of uniform designs for mixture experiments with complex constraints

Abstract

Mixture experiments have attracted increasingly attention due to their great practical value in production and living, while uniform designs over irregular experimental regions have become a hot topic in the area of experimental designs in the past two decades. Noting that the experimental region of a mixture experiment with q components under some constraints is in fact a (q ? 1)-dimensional geometry, this article proposes a new method for searching nearly uniform designs for mixture experiments with any complex constraints. Two examples with some tables and figures are given to illustrate this method.     

Minimax designs for estimating slopes in a trigonometric regression model

Abstract

Designs for the first order trigonometric regression model over an interval on the real line are considered for the situation where estimation of the slope of the response surface at various points in the factor space is of primary interest. Minimization of the variance of the estimated slope at a point maximized over all points in the region of interest is taken as the design criterion. Optimal designs under the minimax criterion are derived for the situation where the design region and the region of interest are identical and a symmetric “partial cycle”. Some comparisons of the minimax designs with the traditional D- and A-optimal designs are provided. Efficiencies of some exact designs under the minimax criterion are also investigated.     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

The robustness of response surface designs with errors in factor levels

ABSRTACT

Since errors in factor levels affect the traditional statistical properties of response surface designs, an important question to consider is robustness of design to errors. However, when the actual design could be observed in the experimental settings, its optimality and prediction are of interest. Various numerical and graphical methods are useful tools for understanding the behavior of the designs. The D- and G-efficiencies and the fraction of design space plot are adapted to assess second-order response surface designs where the predictor variables are disturbed by a random error. Our study shows that the D-efficiencies of the competing designs are considerably low for big variance of the error, while the G-efficiencies are quite good. Fraction of design space plots display the distribution of the scaled prediction variance through the design space with and without errors in factor levels. The robustness of experimental designs against factor errors is explored through comparative study. The construction and use of the D- and G-efficiencies and the fraction of design space plots are demonstrated with several examples of different designs with errors.     

On Sampling Designs for Integral Estimation of a Random Process

Abstract

In this paper the problem of finding exactly optimal sampling designs for estimating the weighted integral of a stochastic process with a product covariance structure (R(s,t)=u(s)v(t), s<t) is discussed. The sampling designs for certain standard processes belonging to the product class are calculated. An asymptotic solution to the design problem also follows as a consequence.     

Optimal designs for statistical analysis with Zernike polynomials

The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and Φ _p -optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849–879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's Φ _p -criteria (p>?∞) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.     

Minimax Designs for the Stability of Slope Estimation on Second-order Response Surfaces

In this paper, designs for the stability of the slope estimation on a second-order response surface are considered. Minimization of the point dispersion measure, which is maximized over all points in the region of interest is taken as the optimality criterion, and the minimax properties in some class of designs are derived in spherical and cubic regions of interest. We study the efficiencies of the minimax designs relative to other optimal designs with various criteria.     

A catalogue of designs for partial profiles in paired comparison experiments with three groups of factors

A common strategy for avoiding information overload in multi-factor paired comparison experiments is to employ pairs of options which have different levels for only some of the factors in a study. For the practically important case where the factors fall into three groups such that all factors within a group have the same number of levels and where one is only interested in estimating the main effects, a comprehensive catalogue of D-optimal approximate designs is presented. These optimal designs use at most three different types of pairs and have a block diagonal information matrix.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.     

On d- and e- minimax optimal designs for estimating the axial slopes of a second-order response surface over hypercubic regions

Design of experiments for estimating the slopes of a response surface is considered. Design criteria analogous to the traditional ones but based upon the variance-covariance matrix of the estimated slopes along factor axes are proposed. Optimal designs under the proposed criteria are derived for second-order polynomial regression over hypercubic regions. Best de¬signs within some commonly used classes of designs are also obtained and their efficiencies are investigated.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

On the determination and construction of optimal designs for comparing a set of test treatments with a set of controls in the presence of a linear trend

In this paper we consider the problem of constructing optimal designs for experimental situations where c controls are to be compared to t test treatments and the treatments are to be applied to experimental units occurring in a linear array and where there may be an unknown linear trend. Methods are given for determining and constructing optimal designs for such situations.     

A-Optimal and MV-Optimal Incomplete Block Designs for Comparing Successive Treatments

Consider an incomplete block experiment in which observations are taken from t treatments using an incomplete block design with b blocks of size k < t. Suppose the interest is in estimating the differences of effects of successive treatments. This may occur, for example, if the treatments are different dosages or concentrations of a compound. This article presents A-optimal and MV-optimal incomplete block designs for estimating the the differences of successive treatment effects. Tables of optimal designs are given for k < t ≤ 5 with b ≤ 40.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

On constructing general minimum lower order confounding two-level block designs

In practice, to reduce systematic variation and increase precision of effect estimation, a practical design strategy is then to partition the experimental units into homogeneous groups, known as blocks. It is an important issue to study the optimal way on blocking the experimental units. Blocked general minimum lower order confounding (B¹-GMC) is a new criterion for selecting optimal block designs. The paper considers the construction of optimal two-level block designs with respect to the B¹-GMC criterion. By utilizing doubling theory and MaxC2 design, some optimal block designs with respect to the B¹-GMC criterion are obtained.