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.     

Minimax Design for the Difference Between Estimated Responses for the Quadratic Model Over Hypercubic Regions

Abstract

A second-order model involving the intercept and only the pure quadratic terms is considered for regression over hypercubes. Minimization of the variance of the difference between estimated responses at two points, maximized over all pairs of points in the region of interest in factor space, is taken as the design criterion. Optimal design under the minimax criterion is derived and found to be the one which is also simultaneously A-, D-, and E-optimal for the parameters excluding the intercept. The minimax design is compared with other standard designs and is found to perform extremely well.     

Admissible two-stage designs for phase II cancer clinical trials that incorporate the expected sample size under the alternative hypothesis

two‐stage studies may be chosen optimally by minimising a single characteristic like the maximum sample size. However, given that an investigator will initially select a null treatment e?ect and the clinically relevant di?erence, it is better to choose a design that also considers the expected sample size for each of these values. The maximum sample size and the two expected sample sizes are here combined to produce an expected loss function to ?nd designs that are admissible. Given the prior odds of success and the importance of the total sample size, minimising the expected loss gives the optimal design for this situation. A novel triangular graph to represent the admissible designs helps guide the decision‐making process. The H ₀‐optimal, H ₁‐optimal, H ₀‐minimax and H ₁‐minimax designs are all particular cases of admissible designs. The commonly used H ₀‐optimal design is rarely good when allowing stopping for e?cacy. Additionally, the δ‐minimax design, which minimises the maximum expected sample size, is sometimes admissible under the loss function. However, the results can be varied and each situation will require the evaluation of all the admissible designs. Software to do this is provided. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust designs for experiments with blocks

ABSTRACT

For experiments running in field plots or over time, the observations are often correlated due to spatial or serial correlation, which leads to correlated errors in a linear model analyzing the treatment means. Without knowing the exact correlation matrix of the errors, it is not possible to compute the generalized least-squares estimator for the treatment means and use it to construct optimal designs for the experiments. In this paper, we propose to use neighborhoods to model the covariance matrix of the errors, and apply a modified generalized least-squares estimator to construct robust designs for experiments with blocks. A minimax design criterion is investigated, and a simulated annealing algorithm is developed to find robust designs. We have derived several theoretical results, and representative examples are presented.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal 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.     

Designs for weighted least squares regression, with estimated weights

We study designs, optimal up to and including terms that are O(n ^?1), for weighted least squares regression, when the weights are intended to be inversely proportional to the variances but are estimated with random error. We take a finite, but arbitrarily large, design space from which the support points are to be chosen, and obtain the optimal proportions of observations to be assigned to each point. Specific examples of D- and I-optimal design for polynomial responses are studied. In some cases the same designs that are optimal under homoscedasticity remain so for a range of variance functions; in others there tend to be more support points than are required in the homoscedastic case. We also exhibit minimax designs, that minimize the maximum, over finite classes of variance functions, value of the loss. These also tend to have more support points, often resulting from the breaking down of replicates into clusters.     

Non binary partially neighbor balanced designs for circular blocks

ABSTRACT

Neighbor designs are recommended for the cases where the performance of treatment is affected by the neighboring treatments as in biometrics and agriculture. In this paper we have constructed two new series of non binary partially neighbor balanced designs for v = 2n and v = 2n+1 number of treatments, respectively. The blocks in the design are non binary and circular but no treatment is ever a neighbor to itself. The designs proposed here are partially balanced in terms of nearest neighbors. No such series are known in the literature.     

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we obtain balanced resolution V plans for 2^m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal balanced resolution V designs obtained earlier in the work of Srivastava and Chopra.     

Asymptotic relative efficiency of designs for factorial paired comparison experiments

A general approach for comparing designs of paired comparison experiments on the basis of the asymptotic relative efficiencies, in the Bahadur sense, of their respective likelihood ratio tests is discussed and extended to factorials. Explicit results for comparing five designs of 2^q factorial paired comparison experiments are obtained. These results indicate that some of the designs which require comparison of fewer distinct pairs of treatments than does the completely balanced design are, generally, more efficient for detecting main effects and/or certain interactions. The developments of this paper generalize the work of Littell and Boyett (1977) for comparing two designs of R x C factorial paired comparison experiments.     

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.     

Projection designs for mixture experiments in orthogonal blocks

Mixture experiments are often carried out in the presence of process variables, such as days of the week or different machines in a manufacturing process, or different ovens in bread and cake making. In such experiments it is particularly useful to be able to arrange the design in orthogonal blocks, so that the model in tue mixture vanauies may ue iitteu inucpenuentiy or tne UIOCK enects mtrouuceu to take account of the changes in the process variables. It is possible in some situations that some of the ingredients in the mixture, such as additives or flavourings, are present in soian quantities, pernaps as iuw a.s 5% ur even !%, resulting in the design space being restricted to only part of the mixture simplex. Hau and Box (1990) discussed the construction of experimental designs for situations where constraints are placed on the design variables. They considered projecting standard response surface designs, including factorial designs and central composite designs, into the restricted design space, and showed that the desirable property of block orthogonality is preserved by the projections considered. Here we present a number of examples of projection designs and illustrate their use when some of the ingredients are restricted to small values, such that the design space is restricted to a sub-region within the usual simplex in the mixture variables.     

A semidefinite programming study of the Elfving theorem

The theorem of Elfving is one of the most important and earliest results which have led to the theory of optimal design of experiments. This paper presents a fresh study of it from the viewpoint of modern semidefinite programming. There is one-to-one correspondence between solutions of the derived semidefinite programming problem (SDP) and c-optimal designs. We also derive a uniqueness theorem which ensures a unique optimal design without assuming the linear independence property over the largest set of supporting points. The SDP can also be cast as an ?₁‐convex program which has recently been extensively studied and often yields sparse solutions. Our numerical experiments on the trigonometric regression model confirm that the SDP does produce a sparse optimal design.     

Optimal design in average for inference in generalized linear models

This paper considers the problem of optimal design for inference in Generalized Linear Models, when prior information about the parameters is available. The general theory of optimum design usually requires knowledge of the parameter values. These are usually unknown and optimal design can, therefore, not be used in practice. However, one way to circumvent this problem is through so-called “optimal design in average”, or shortly, “ave optimal”. The ave optimal design is chosen to minimize the expected value of some criterion function over a prior distribution. We focus our interest on the aveD _A-optimality, including aveD- and avec-optimality and show the appropriate equivalence theorems for these optimality criterions, which give necessary conditions for an optimal design. Ave optimal designs are of interest when e.g. a factorial experiment with a binary or a Poisson response in to be conducted. The results are applied to factorial experiments, including a control group experiment and a 2×2 experiment.     

On Kronecker block designs for factorial experiments

In this paper the use of Kronecker designs for factorial experiments is considered. The two-factor Kronecker design is considered in some detail and the efficiency factors of the main effects and interaction in such a design are derived. It is shown that the efficiency factor of the interaction is at least as large as the product of the efficiency factors of the two main effects and when both the component designs are totally balanced then its efficiency factor will be higher than the efficiency factor of either of the two main effects. If the component designs are nearly balanced then its efficiency factor will be approximately at least as large as the efficiency factor of either of the two main effects. It is argued that these designs are particularly useful for factorial experiments.Extensions to the multi-factor design are given and it is proved that the two-factor Kronecker design will be connected if the component designs are connected.     

Constructing optimal projection designs

ABSTRACT

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.     

D‐optimal minimax fractional factorial designs

The D‐optimal minimax criterion is proposed to construct fractional factorial designs. The resulting designs are very efficient, and robust against misspecification of the effects in the linear model. The criterion was first proposed by Wilmut & Zhou (2011); their work is limited to two‐level factorial designs, however. In this paper we extend this criterion to designs with factors having any levels (including mixed levels) and explore several important properties of this criterion. Theoretical results are obtained for construction of fractional factorial designs in general. This minimax criterion is not only scale invariant, but also invariant under level permutations. Moreover, it can be applied to any run size. This is an advantage over some other existing criteria. The Canadian Journal of Statistics 41: 325–340; 2013 © 2013 Statistical Society of Canada     

Two-stage k-sample designs for the ordered alternative problem

In preclinical studies and clinical dose-ranging trials, the Jonckheere-Terpstra test is widely used in the assessment of dose-response relationships. Hewett and Spurrier (1979) presented a two-stage analog of the test in the context of large sample sizes. In this paper, we propose an exact test based on Simon's minimax and optimal design criteria originally used in one-arm phase II designs based on binary endpoints. The convergence rate of the joint distribution of the first and second stage test statistics to the limiting distribution is studied, and design parameters are provided for a variety of assumed alternatives. The behavior of the test is also examined in the presence of ties, and the proposed designs are illustrated through application in the planning of a hypercholesterolemia clinical trial. The minimax and optimal two-stage procedures are shown to be preferable as compared with the one-stage procedure because of the associated reduction in expected sample size for given error constraints.     

A simple method for constructing confounded designs for mixed factorial experiments

This paper presents the sinplesr procedure that uses wodular aryithmetic for constructing confounded designs for mixed factorial experiments. The present procedure and the classical one for confounding in symmetrical factorial experiments are both at the same mathema.tical level. The present procedure is written for

practitioners and is lllustrared with several examples.     

Optimal designs for asymmetrical factorial paired comparison experiments

Three forms of a general null hypothesis Ho on the factorial parameters of a general asymmetrical factorial paired comparison experiment are considered. A class of partially balanced designscorresponding to each form of H₀ is constructed and the A,D and ioptimal design, minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix of related maximumlikelihoodestimators, in that class is determined. Moreover, the optimal design in each class maximizes the noncentrality parameter λ² of the asymptotic noncentral chi-square distribution of the likelihood ratiostatistic -2 log λ for testing H_o under defined local alternatives. These results apply directly to symmetrical factorial paired comparison experiments as special casesExamples are given forillustrating applications of the developed results