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.     

Extensions of D-optimal minimal designs for symmetric mixture models

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide the desirable response characteristics in finished products. D-optimal minimal designs have been considered for a variety of mixture models, including Scheffé's linear, quadratic, and cubic models. Usually, these D-optimal designs are minimally supported since they have just as many design points as the number of parameters. Thus, they lack the degrees of freedom to perform the lack-of-fit (LOF) tests. Also, the majority of the design points in D-optimal minimal designs are on the boundary: vertices, edges, or faces of the design simplex. In this article, extensions of the D-optimal minimal designs are developed for a general mixture model to allow additional interior points in the design space to enable prediction of the entire response surface. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. We compare the proposed designs with Cornell (1986 Cornell, J.A. (1986). A comparison between two ten-point designs for studying three-component mixture systems. J. Qual. Technol. 18(1):1–15.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) two 10-point designs for the LOF test by simulations.     

Optimum mixture designs in some constrained experimental regions

In a mixture experiment, the response depends on the mixing proportions of the components present in the mixture. Optimum designs are available for the estimation of parameters of the models proposed in such situations. However, these designs are found to include the vertex points of the simplex Ξ defining the experimental region, which are not mixtures in the true sense. Recently, Mandal et al. (2015 Mandal, N.K., Pal, M., Sinha, B.K., and Das, P. (2015). Optimum mixture designs in a restricted region. Stat. Pap. 56(1):105–119.[Crossref], [Web of Science ®] , [Google Scholar]) derived optimum designs when the experiment is confined to an ellipsoidal region within Ξ, which does not include the vertices of Ξ. In this paper, an attempt has been made to find optimum designs when the experimental region is a simplex or is cuboidal inside Ξ and does not contain the extreme points.     

Some new augmented fractional Box–Behnken designs

The augmented Box–Behnken designs are used in the situations in which Box–Behnken designs (BBDs) could not estimate the response surface model due to the presence of third-order terms in the response surface models. These designs are too large for experimental use. Usually experimenters prefer small response surface designs in order to save time, cost, and resources; therefore, using combinations of fractional BBD points, factorial design points, axial design points, and complementary design points, we augment these designs and develop new third-order response surface designs known as augmented fractional BBDs (AFBBDs). These AFBBDs have less design points and are more efficient than augmented BBDs.     

A Likelihood-Based Approximation to the Cumulative Distribution Function of the Noncentral t Distribution

Optimal designs for estimating the parameters and also the optimum factor combinations in multiresponse experiments have been considered by various authors. However, till date, in mixture experiments optimum designs have been studied only in the single response case. In this article, attempt has been made to investigate optimum designs for estimating optimum mixing proportions in a multiresponse mixture experiment.     

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.     

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 graphical method for evaluating mixture designs with respect to the slope

Good estimation of the slopes of the mixture response function may be important as well as estimation of mean mixture response. It is possible to evaluate and compare several mixture designs with respect to the slope. A graphical method is proposed that allows us to evaluate a given design's support for the fitted model in terms of slope variance. We can plot variances of slopes along Cox direction or axial direction according to existence of restriction of simplex region or not when comparing several different mixture designs.     

Optimal orthogonal block designs for a quadratic mixture model for three components

In experiments with mixtures that involve process variables, if the response function is expressed as the sum of a function of mixture components and a function of process variables, then the parameters in the mixture part and in the process part can be estimated independently using orthogonal block designs. This paper is concerned with such a block design for parameter estimation in the mixture part of a quadratic mixture model for three mixture components. The behaviour of the eigenvalues of the moment matrix of the design is investigated in detail, the design is optimized according to E- and Aoptimality criteria, and the results are compared together with a known result on Doptimality. It is found that this block design is robust with respect to these diff erent optimality criteria against the shifting of experimental points. As a result, we recommend experimental points of the form (a, b, c) in the simplex S2, where c=0, b=1-a, and a can be any value in the range 0.17+/-0.02.     

The Parshvanath yantram yields a constant-sum partially balanced incomplete block design

Yantrams have been used to generate mixture designs in the interior of a simplex. In this note, we show a connection between Parshvanath yantram and a particular partially balanced incomplete block design. This block design is rather special and somewhat unexpected due to the feature that sum of the treatment symbols in any block is constant.     

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.     

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.     

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

The design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T² control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T² control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.     

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.     

Mixture designs for product improvement studies

Many products are mixtures of several components (ingredient). Characteristics of the products such as the strength of steel, the efficacy of a chemical pesticide, or the viscosity of a liquid detergent, depend only on the relative proportions of the components in the mixture. Studying changes in a product' properties caused by varying the ingredient proportions is the objective of performing mixture experiments. The inherent restriction that the sum of the component proportions equal unity creates different design strategies than are usually employed with independent factors where factorial arrangements are quite common. Experimental designs for exploring the entire mixture simplex region as well as for exploring only a subregion of the simplex are presented. In those cases where four or more components are considered and a subregion is to be investigated, computer-aided designs are the rule rather than the exception. Design criterion based on the properties (variance and bias) of the prediction equation are mentioned briefly and some suggestions are made for future research in mixture experiments.     

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.     

Experimental designs for detecting synergy and antagonism between two drugs in a pre‐clinical study

The identification of synergistic interactions between combinations of drugs is an important area within drug discovery and development. Pre‐clinically, large numbers of screening studies to identify synergistic pairs of compounds can often be ran, necessitating efficient and robust experimental designs. We consider experimental designs for detecting interaction between two drugs in a pre‐clinical in vitro assay in the presence of uncertainty of the monotherapy response. The monotherapies are assumed to follow the Hill equation with common lower and upper asymptotes, and a common variance. The optimality criterion used is the variance of the interaction parameter. We focus on ray designs and investigate two algorithms for selecting the optimum set of dose combinations. The first is a forward algorithm in which design points are added sequentially. This is found to give useful solutions in simple cases but can lack robustness when knowledge about the monotherapy parameters is insufficient. The second algorithm is a more pragmatic approach where the design points are constrained to be distributed log‐normally along the rays and monotherapy doses. We find that the pragmatic algorithm is more stable than the forward algorithm, and even when the forward algorithm has converged, the pragmatic algorithm can still out‐perform it. Practically, we find that good designs for detecting an interaction have equal numbers of points on monotherapies and combination therapies, with those points typically placed in positions where a 50% response is expected. More uncertainty in monotherapy parameters leads to an optimal design with design points that are more spread out. Copyright © 2015 John Wiley & Sons, Ltd.     

Horwitz's rule, transforming both sides and the design of experiments for mechanistic models

Summary. The paper develops methods for the design of experiments for mechanistic models when the response must be transformed to achieve symmetry and constant variance. The power transformation that is used is partially justified by a rule in analytical chemistry. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples are given of optimum designs, not only for single-response models, but also for experiments in which multivariate responses are measured and for experiments in which the model is defined by a set of differential equations which cannot be solved analytically. The extension to designs for checking models is discussed.     

Minimax designs for approximately linear models with AR(1) errors

We obtain designs for linear regression models under two main departures from the classical assumptions: (1) the response is taken to be only approximately linear, and (2) the errors are not assumed to be independent, but to instead follow a first-order autoregressive process. These designs have the property that they minimize (a modification of) the maximum integrated mean squared error of the estimated response, with the maximum taken over a class of departures from strict linearity and over all autoregression parameters ρ,|ρ,| < 1, of fixed sign. Specific methods of implementation are discussed. We find that an asymptotically optimal procedure for AR(1) models consists of choosing points from that design measure which is optimal for uncorrelated errors, and then implementing them in an appropriate order.