D-optimal design for a quadratic log contrast model for experiments with mixtures

A practical method is suggested for solving complicated D-optimal design problems analytically. Using this method the author has solved the problem for a quadratic log contrast model for experiments with mixtures introduced by J. Aitchison and J. Bacon-Shone. It is found that for a symmetric subspace of the finite dimensional simplex, the vertices and the centroid of this subspace are the only possible support points for a D-optimal design. The weights that must be assigned to these support points contain irrational numbers and are constrained by a system of three simultaneous linear equations, except for the special cases of 1- and 2-dimensional simplexes where the situation is much simpler. Numerical values for the solution are given up to the 19-dimensional simplex     

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.     

Optimal designs for experiments with mixtures: a survey

This is a survey article on known results about analytic solutions and numerical solutions of optimal designs for various regression models for experiments with mixtures. The regression models include polynomial models, models containing homogeneous functions, models containing inverse terms and ratios, log contrast models, models with quantitative variables, and mod els containing the amount of mixture, Optimality criteria considered include D-, A-, E-,φ_p- and I_λ-Optimalities. Uniform design and uniform optimal design for mixture components, and efficiencies of the {q,2} simplex-controid design are briefly discussed.     

Efficient maximin distance designs for experiments in mixtures

In this paper, different dissimilarity measures are investigated to construct maximin designs for compositional data. Specifically, the effect of different dissimilarity measures on the maximin design criterion for two case studies is presented. Design evaluation criteria are proposed to distinguish between the maximin designs generated. An optimization algorithm is also presented. Divergence is found to be the best dissimilarity measure to use in combination with the maximin design criterion for creating space-filling designs for mixture variables.     

A- and D-optimal designs for a log contrast model for experiments with mixtures

A- and D-optimal designs are investigated for a log contrast model suggested by Aitchison & Bacon-Shone for experiments with mixtures. It is proved that when the number of mixture components q is an even integer, A- and D-optimal designs are identical; and when q is an odd integer, A- and D-optimal designs are different, but they share some common support points and are very close to each other in efficiency. Optimal designs with a minimum number of support points are also constructed for 3, 4, 5 and 6 mixture components.     

Parametric estimation of mixtures of two uniform distributions

In this paper, we consider a mixture of two uniform distributions and derive L-moment estimators of its parameters. Three possible ways of mixing two uniforms, namely with neither overlap nor gap, with overlap, and with gap, are studied. The performance of these L-moment estimators in terms of bias and efficiency is compared to that obtained by means of the conventional method of moments (MM), modified maximum likelihood (MML) method and the usual maximum likelihood (ML) method. These intensive simulations reveal that MML estimators are the best in most of the cases, and the L-moment estimators are less subject to bias in estimation for some mixtures and more efficient in most of the cases than the conventional MM estimators. The L-moment estimators are, in some cases, more efficient than the ML and MML estimators.     

Optimal design for discrete choice experiments

In this paper we derive locally optimal designs for discrete choice experiments. As in Kanninen (2002) we consider a multinomial logistic model, which contains various qualitative attributes as well as a quantitative one, which may range over a sufficiently large interval. The derived optimal designs improve upon those given in the literature, but have the feature that every choice set contains alternatives, which coincide in all but the quantitative attributes. The multinomial logistic model will then lead to a response behavior, which is apparently unrealistic.     

Response model analysis for robust design experiments

Taguchi's robust design technique, also known as parameter design, focuses on making product and process designs insensitive (i.e., robust) to hard to control variations. In some applications, however, his approach of modeling expected loss and the resulting “product array” experimental format leads to unnecessarily expensive and less informative experiments. The response model approach to robust design proposed by Welch, Ku, Yang, and Sacks (1990), Box and Jones (1990), Lucas (1989), and Shoemaker, Tsui and Wu (1991) offers more flexibility and economy in experiment planning and more informative modeling. This paper develops a formal basis for the graphical data-analytic approach presented in Shoemaker et al. In particular, we decompose overall response variation into components representing the variability contributed by each noise factor, and show when this decomposition allows us to use individual control-by-noise interaction plots to minimize response variation. We then generalize the control-by-noise interaction plots to extend their usefulness, and develop a formal analysis strategy using these plots to minimize response variation.     

Maximum likelihood estimation of heterogeneous mixtures of Gaussian and uniform distributions

Existence and consistency of the Maximum Likelihood estimator of the parameters of heterogeneous mixtures of Gaussian and uniform distributions with known number of components are shown under constraints to prevent the likelihood from degeneration and to ensure identifiability. The EM-algorithm is discussed, and for the special case with a single uniform component a practical scheme to find a good local optimum is proposed. The method is compared theoretically and empirically to the estimation of a Gaussian mixture with “noise component” as introduced by Banfield and Raftery (1993) to find out whether it is a worthwhile alternative particularly in situations with outliers and points not belonging to the Gaussian components.     

Design and analysis of paired comparison experiments involving mixtures

In applications involving subjective judgements in the form of paired comparison experiments, the treatments to be compared may be a resulting mixture of several components in which the proportion, and not the actual amount of a given component is recorded. Springall's result relating classical response surface designs with designs for paired comparison experiments is used to construct rotatable and minimum bias designs for paired comparison experiments with mixtures.     

Optimal design of complex experiments with qualitative factors of influence

A complex experiment with qualirarive factors influencing the outcome of the experiment can be seen as a general ANOVA setup. A design of such an experiment will be the assignment at which of the possible levels of the factors the actual experiment should be performed. In this paper optimal designs of such experiments will be characterized with respect to three different optimality criteria including the so called uniform optimality of a design. The possible applications of the main optimization result providing these characterizations can be used to more general experiments. The particular results on these generalizations will be indicated at the end of this paper.     

An overview of design of experiments

An overview of the design of statistical experiments is presented, with special emphasis on response surface designs, block designs, neighbor designs. Applications are mentioned for industrial quality improvement, agricultural experiments, biometry. An outlook towards design optimality concludes the survey.     

DEXPERT: an expert system for the design of experiments

DEXPERT is an expert system, built using KEE, for the design and analysis of experiments. From a mathematical model, expected mean squares are computed, tests are determined, and the power of the tests computed. Comparisons between designs are aided by suggestions and verbal interpretations provided by DEXPERT. DEXPERT provides a layout sheet for the collection of the data and then analyzes and interprets the results using analytical and graphical methods.     

A computer program for the design of group testing experiments

This paper describes a computer program GTEST for designing group testing experiments for classifying each member of a population of items as “good” or “defective”. The outcome of a test on a group of items is either “negative” (if all items in the group are good) or “positive” (if at least one of the items is defective, but it is not known which). GTEST is based on a Bayesian approach. At each stage, it attempts to maximize (nearly) the expected reduction in the “entropy”, which is a quantitative measure of the amount of uncertainty about the state of the items. The user controls the procedure through specification of the prior probabilities of being defective, restrictions on the construction of the test group, and priorities that are assigned to the items. The nominal prior probabilities can be modified adaptively, to reduce the sensitivity of the procedure to the proportion of defectives in the population.     

An efficient algorithm for constructing optimal design of computer experiments

The long computational time required in constructing optimal designs for computer experiments has limited their uses in practice. In this paper, a new algorithm for constructing optimal experimental designs is developed. There are two major developments involved in this work. One is on developing an efficient global optimal search algorithm, named as enhanced stochastic evolutionary (ESE) algorithm. The other is on developing efficient methods for evaluating optimality criteria. The proposed algorithm is compared to existing techniques and found to be much more efficient in terms of the computation time, the number of exchanges needed for generating new designs, and the achieved optimality criteria. The algorithm is also very flexible to construct various classes of optimal designs to retain certain desired structural properties.     

Optimal design results for 2n factorial paired comparison experiments

Four general classes of partially balanced designs for 2ⁿ factorials, corresponding to four different forms of a general null hypothesis H on factorial effects, are presented. For the typical design in each class, the simplified form of the non-centrality parameter λ² of the asymptotic chi-square distribution of the likelihood ratio statistic for testing the corresponding form of H₀ is derived under defined local alternatives. Optimal designs d₁ maximizing λ² in the i-th class and minimizing the trace, determinant and largest eigenvalue of a defined covariance matrix, i =1,…,4, are determined.     

The design of experiments for correlated error structures: Layout and robustness

Observations with correlated error structures are sometimes unavoidable. Appropriate designs and analyses are reviewed for such situations. Serious problems can occur if conventional designs and analyses are used when correlated errors and layout are ignored or when the error structure is not known. Robust designs are discussed which guard against these problems.     

On the design of efficient experiments for choosing between two bernoulli populations

An experiment consists of taking samples one at a time from two Bernoulli populations. At the conclusion of the experiment one of the two populations is declared to have the higher probability of sucess.We propose two classes rules for sampling and for choosing a population which results which results good decisions made quickly.These rules are analysed and are compared suggested previously by means of Monte-Carlo simulations.