A-optimal designs for optimum mixture in an additive quadratic mixture model

In the analysis of experiments with mixtures, quadratic models have been widely used. The optimum designs for the estimation of optimum mixing proportions in a quadratic mixture model have been studied by Pal and Mandal [Optimum designs for optimum mixtures. Statist Probab Lett. 2006;76:1369–1379] and Mandal et al. [Optimum mixture designs: a pseudo-Bayesian approach. J Ind Soc Agric Stat. 2008;62(2):174–182; Optimum mixture designs under constraints on mixing components. Statist Appl. 2008;6(1&2) (New Series): 189–205], using a pseudo-Bayesian approach. In this paper, a similar approach has been employed to obtain the A-optimal designs for the estimation of optimum proportions in an additive quadratic mixture model, proposed by Darroch and Waller [Additivity and interaction in three-component experiments with mixture. Biometrika. 1985;72:153–163], when the number of components is 3, 4 and 5. It has been shown that the vertices of the simplex are necessarily the support points of the optimum design, and the other support points include barycentres of depth at most 2.     

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.     

Outliers in incomplete multiresponse experiments

Identification of outlier vectors in block designs for incomplete multiresponse experiments has been considered. Design is composed of two sets of experimental units. Different numbers of response variables are observed from these two sets. Cook-statistic has been developed for identification of outliers. The developed statistic has been illustrated with a real-life data set. It has been shown that presence of outliers can distort the overall conclusion from an experiment.     

Type a incomplete multi-response designs

Srivastava considered the problem of designing and analysis of incomplete multi-response experiments, The main characteristics of the analysis given by her are investigated and a general class of incomplete multiresponse designs (Type A) is obtained on the basis of these investigations.     

Restricted region designs for multifactor mixture experiments

Designs for multifactor mixture experiments in which the component proportions are restricted by some specified lower and upper limits, are called restricted region designs (RRD) for multifactor mixture experiments. In this paper (i) a method of construction of designs for multifactor mixture experiments and (ii) two methods of constructing RRD for multifactor mixture experiments are described. Examples to illustrate the methods of construction have been given.     

Optimum mixture designs in a restricted region

In a mixture experiment, the response depends on the proportions of the mixing components. Canonical models of different degrees and also other models have been suggested to represent the mean response. Optimum designs for estimation of the parameters of the models have been investigated by different authors. In most cases, the optimum design includes the vertex points of the simplex as support points of the design, which are not mixture combinations in the true non-trivial sense. In this paper, optimum designs have been obtained when the experimental region is an ellipsoidal subspace of the entire factor space which does not cover the vertex points of the simplex.     

Designs for estimating the difference between two responses

In this paper the problem of designing experiments for estimating the difference between responses at two arbitrary points in the region of interest is considered. Under the average mean squared error criterion the design moment conditions have been worked out. The results have been applied to the class of central composite designs.     

Optimal designs to improve the power of multiresponse hypothesis tests

In this paper we develop a design criterion to improve the power of the multiresponse hypothesis tests, and give a sequential procedure for generating designs which satisfy the criterion.     

D-Optimum Designs for Optimum Mixture in a Quadratic Log Contrast Model

Optimal designs for estimating the optimum mixing proportions in a quadratic mixture model was first investigated by Pal and Mandal (2006). In this article, similar investigation is carried out when mean response in a mixture experiment is described by a quadratic log contrast model. It is found that in a symmetric subspace of the finite dimensional simplex, there exists a D-optimal design that puts weights at the centroid of the sub-space and the vertices of the experimental domain. The optimality is checked by numerical computation using Equivalence Theorem.     

Optimum Designs for Parameter Estimation in Linear Mixture Models with Synergistic Effects

In this article, we derive optimum designs for parameter estimation in a mixture experiment when the response function is linear in the mixing components with some synergistic effects. The D- and A-optimality criteria have been used for the purpose. The Equivalence Theorem has been used to check for the optimality of the proposed designs.     

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.     

Optimal orthogonal block designs for four-mixture components in two blocks based on F-squares for Becker's models and K-model

Orthogonal block designs in mixture experiments have been extensively studied by various authors. Aggarwal et al. [M.L. Aggarwal, P. Singh, V. Sarin, and B. Husain, Mixture designs in orthogonal blocks using F-squares, METRON – Int. J. Statist. LXVII(2) (2009), pp. 105–128] considered the case of components assuming the same volume fractions and obtained mixture designs in orthogonal blocks using F-squares. In this paper, we have used the class of designs presented by Aggarwal et al. and have obtained D-, A- and E-optimal orthogonal block designs for four components in two blocks for Becker's mixture models and K-model, respectively. Orthogonality conditions for the considered models are also given.     

A series of single array 2m factorial search designs for even m

By means of a search design one is able to search for and estimate a small set of non‐zero elements from the set of higher order factorial interactions in addition to estimating the lower order factorial effects. One may be interested in estimating the general mean and main effects, in addition to searching for and estimating a non‐negligible effect in the set of 2‐ and 3‐factor interactions, assuming 4‐ and higher‐order interactions are all zero. Such a search design is called a ‘main effect plus one plan’ and is denoted by MEP.1. Construction of such a plan, for 2^m factorial experiments, has been considered and developed by several authors and leads to MEP.1 plans for an odd number m of factors. These designs are generally determined by two arrays, one specifying a main effect plan and the other specifying a follow‐up. In this paper we develop the construction of search designs for an even number of factors m, m≠6. The new series of MEP.1 plans is a set of single array designs with a well structured form. Such a structure allows for flexibility in arriving at an appropriate design with optimum properties for search and estimation.     

Design and analysis of mixture experiments with process variable

A mixture experiment is an experiment in which the response is assumed to depend on the relative proportions of the ingredients present in the mixture and not on the total amount of the mixture. In such experiment process, variables do not form any portion of the mixture but the levels changed could affect the blending properties of the ingredients. Sometimes, the mixture experiments are costly and the experiments are to be conducted in less number of runs. Here, a general method for construction of efficient mixture experiments in a minimum number of runs by the method for projection of efficient response surface design onto the constrained region is obtained. The efficient designs with a less number of runs have been constructed for 3rd, 4th, and 5th component of mixture experiments with one process variable.     

Developing mixture models,are we done

Forty-one years ago, Scheffé’s pioneering article “Experiments With Mixtures”, was the impetus for much of the research on mixture experiments performed during the next three decades and into the 1990’s. That article introduced the simplex lattice designs and the associated canonical polynomials, the latter often referred to as Scheffé’s models. Over the years, alternative designs and models have been developed mainly to deal with such modifications to the standard mixture problem as the placing of additional constraints on the mixture component proportions and the inclusion of nonmixture factors called process variables or the amount of the mixture. As we move into the 21st century and think about future research on models for mixture experiments, we are tempted to ask, “Are there additional challenges facing us or are we done” To answer this question truthfully we must look back to see where we’ve been and look to the future at what it is we want to accomplish. This paper is the beginning of this process.     

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 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.     

Experimentation order in factorial designs: new findings

Under some very reasonable hypotheses, it becomes evident that randomizing the run order of a factorial experiment does not always neutralize the effect of undesirable factors. Yet, these factors do have an influence on the response, depending on the order in which the experiments are conducted. On the other hand, changing the factor levels is many times costly; therefore it is not reasonable to leave to chance the number of changes necessary. For this reason, run orders that offer the minimum number of factor level changes and at the same time minimize the possible influence of undesirable factors on the experimentation have been sought. Sequences which are known to produce the desired properties in designs with 8 and 16 experiments can be found in the literature. In this paper, we provide the best possible sequences for designs with 32 experiments, as well as sequences that offer excellent properties for designs with 64 and 128 experiments. The method used to find them is based on a mixture of algorithmic searches and an augmentation of smaller designs.     

Optimal U-type design for Bayesian nonparametric multiresponse prediction

This paper presents an extension of the work of Yue and Chatterjee (2010) about U-type designs for Bayesian nonparametric response prediction. We consider nonparametric Bayesian regression model with p responses. We use U-type designs with n runs, m factors and q levels for the nonparametric multiresponse prediction based on the asymptotic Bayesian criterion. A lower bound for the proposed criterion is established, and some optimal and nearly optimal designs for the illustrative models are given.     

Numerical solution of likelihood equations for a first-order response surface model for mixture paired comparisons

Summary A simple procedure for numerical solution of the likelihood equations for estimating the regression parameters of a first-order response surface model for the treatment parameters of mixture paired comparison experiments is developed. It is demonstrated that, for defined rotatable designs, those regression parameters are simple functions of the main effect parameters of a corresponding factorial model with no interactions. The maximum likelihood estimators of those main effect parameters, and hence of their corresponding regression parameters, are obtained through using procedures of treatment contrasts, factorial and iterations. A numerical example is given to illustrate applications of the procedures developed in this paper.