Mixture experiments with process variables: d-optimal orthogonal experimental designs

Blending experiments with mixture in the presence of process variables are considered. We present an experimental design for quadratic (or linear) blending. The design in two orthogonal blocks is D-optimized in the case where there are no restrictions on the blending in two orthogonal blocks is presented when there are arbitrary restrictions on the blending components. The pair of orthogonal blocks can be used with and arbitrary number of process variables. The number of design points needed when different orthogonal blocks are used is usually smaller than when a single block is repeated at the various process variables levels.     

Nearly optimal orthogonally blocked desings for a quadratic mixture model with q components

In experiments with mixtures involving process variables, orthogonal block designs may be used to allow estimation of the parameters of the mixture components independently of estimation of the parameters of the process variables. In the class of orthogonally blocked designs based on pairs of suitably chosen Latin squares, the optimal designs consist primarily of binary blends of the mixture components, regardless of how many ingredients are available for the mixture. This paper considers ways of modifying these optimal designs so that some or all of the runs used in the experiment include a minimum proportion of each mixture ingredient. The designs considered are nearly optimal in the sense that the experimental points are chosen to follow ridges of maxima in the optimality criteria. Specific designs are discussed for mixtures involving three and four components and distinctions are identified for different designs with the same optimality properties. The ideas presented for these specific designs are readily extended to mixtures with q>4 components.     

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.     

Experimental design for four mixture components with process variables

The experimental design to model the response of a mixture in four components in the presence of process variables is considered. Two different blocks of blends that are orthogonal for linear or quadratic blending are D-optimized. The two orthogonal blocks of blends are generalized and D-optimized in some cases (and possibly Doptimized in others) to deal with restrictions on the blending component proportions. The pair of orthogonal D-optimal blocks of blends can be used with an arbitrary number of process variables, and requires a reduced number of observations.     

Note on a mixture experiment with process variables

The experimental design to model the response of a mixture experiment in three blending components in the presence of process variables is considered. Czitrom (1988) gave an experimental design in two orthogonal blocks of blends that was "possibly" D-Optimal in the case of arbitrary restrictions on the blending component proportions. It will be shown that the design is indeed D-Optimal. The pair of orthogonal D-Optimal blocks of blends can be used with an arbitrary number of process variables and require a reduced number of observations     

Uniform Design for Experiments with Mixtures

The goal of uniform mixture design is to scatter the design points in the experimental region uniformly. The commonly used criteria, such as mean square distance, are based on the Euclidean distance. Based on the Lee distance, a new criterion is proposed in this article. And an algorithm, called NTLBG, is also proposed to refine the randomly generated design for the experimental design with mixtures. Some examples show that the design generated by the NTLBG algorithm has a lower criteria value.     

Mixture and mixture–process variable experiments for pharmaceutical applications

Many experiments in research and development in the pharmaceutical industry involve mixture components. These are experiments in which the experimental factors are the ingredients of a mixture and the response variable is a function of the relative proportion of each ingredient, not its absolute amount. Thus the mixture ingredients cannot be varied independently. A common variation of the mixture experiment occurs when there are also one or more process factors that can be varied independently of each other and of the mixture components, leading to a mixture–process variable experiment. We discuss the design and analysis of these types of experiments, using tablet formulation as an example. Our objective is to encourage greater utilization of these techniques in pharmaceutical research and development. Copyright © 2004 John Wiley & Sons Ltd.     

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.     

Symmetry in mixture experiments

Two types of symmetry can arise when the proportions of mixture components are constrained by upper and lower bounds. These two types of symmetry are shown to be useful for blocking first-order designs, as well as for finding the centroid of the experimental region. Orthogonal blocking of first-order mixture designs provides a method of including process variables in the mixture experiment, with the mixture terms orthogonal to the process factors. Symmetric regions are used to develop spherical and rotatable response surface designs for mixtures. The central composite design and designs based on the icosahedron and the dodecahedron are given for four-component mixtures. The uniform shell designs are three-level designs when applied to mixture experiments.     

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.     

Using conditional independence for parsimonious model-based Gaussian clustering

In the framework of model-based cluster analysis, finite mixtures of Gaussian components represent an important class of statistical models widely employed for dealing with quantitative variables. Within this class, we propose novel models in which constraints on the component-specific variance matrices allow us to define Gaussian parsimonious clustering models. Specifically, the proposed models are obtained by assuming that the variables can be partitioned into groups resulting to be conditionally independent within components, thus producing component-specific variance matrices with a block diagonal structure. This approach allows us to extend the methods for model-based cluster analysis and to make them more flexible and versatile. In this paper, Gaussian mixture models are studied under the above mentioned assumption. Identifiability conditions are proved and the model parameters are estimated through the maximum likelihood method by using the Expectation-Maximization algorithm. The Bayesian information criterion is proposed for selecting the partition of the variables into conditionally independent groups. The consistency of the use of this criterion is proved under regularity conditions. In order to examine and compare models with different partitions of the set of variables a hierarchical algorithm is suggested. A wide class of parsimonious Gaussian models is also presented by parameterizing the component-variance matrices according to their spectral decomposition. The effectiveness and usefulness of the proposed methodology are illustrated with two examples based on real datasets.     

Random search algorithm for optimal mixture experimental design

It is well known that it is difficult to obtain an accurate optimal design for a mixture experimental design with complex constraints. In this article, we construct a random search algorithm which can be used to find the optimal design for mixture model with complex constraints. First, we generate an initial set by the Monte-Carlo method, and then run the random search algorithm to get the optimal set of points. After that, we explain the effectiveness of this method by using two examples.     

Experimental designs for mixture systems with multicomponent constraints

In an earlier paper it was recommended that an experimental design for the study of a mixture system in which the components had lower and upper limits should consist of a subset of the vertices and centroids of the region defined by the limitson the components. This paper extends this methodology to the situation where linear combinations of two or more components (e.g., liquid content=x₃+x₄+≦0.35) are subject to lower and upper constraints. The CONSIM algorithm, developed by R. E. Wheeler, is recommended for computing the vertices of the resulting experimental region. Procedures for developing linear and quadratic mixture model designs are discussed. A five-component example which has two multiple-component constraints is included to illustrate the proposed methods of mixture experimentation.     

A variables repetitive group sampling plan under failure-censored reliability tests for Weibull distribution

We propose a variables repetitive group sampling plan under type-II or failure-censored life testing when the lifetime of a part follows a Weibull distribution with a known shape parameter. The acceptance criteria do not involve unknown scale parameter differently from the existing plans. To determine the design parameters of the proposed plan, the usual approach of using two points on the operating characteristic curve is adopted and an optimization problem is formulated so as to minimize the average number of failures observed. Tables for design parameters are constructed when the quality of parts is represented by the unreliability or the ratio of the mean lifetime to the specified life. It is found that the proposed sampling plan can reduce the sample size significantly than do the single sampling plan.     

Designing and analyzing a mixture experiment to optimize the mixing proportions of polyvinyl chloride composites

Polyvinyl chloride (PVC) products are typically complex composites, whose quality characteristics vary widely depending on the types and proportions of their components, as well as other processing factors. It is often required to optimize PVC production for specific applications at the highest cost efficiency. This study describes the design and analysis of a statistical experiment to investigate the effects of different parameters over the mechanical properties of PVC intended for use in electrical wire insulation. Four commonly used mixture components, namely, virgin PVC, recycled PVC, calcium carbonate, and a plasticizer, and two process variables, type of plasticizer and filler particle size, were examined. Statistical tools were utilized to analyze and optimize the mixture while simultaneously finding the proper process parameters. The mix was optimized to achieve required strength and ductility, as per ASTM D6096 while minimizing cost. The paper demonstrates how statistical models can help tailor complex polymeric composites in the presence of variations created by process variables.     

Bayesian D-optimal design for logistic regression model with exponential distribution for random intercept

ABSTRACT

Nowadays, generalized linear models have many applications. Some of these models which have more applications in the real world are the models with random effects; that is, some of the unknown parameters are considered random variables. In this article, this situation is considered in logistic regression models with a random intercept having exponential distribution. The aim is to obtain the Bayesian D-optimal design; thus, the method is to maximize the Bayesian D-optimal criterion. For the model was considered here, this criterion is a function of the quasi-information matrix that depends on the unknown parameters of the model. In the Bayesian D-optimal criterion, the expectation is acquired in respect of the prior distributions that are considered for the unknown parameters. Thus, it will only be a function of experimental settings (support points) and their weights. The prior distribution of the fixed parameters is considered uniform and normal. The Bayesian D-optimal design is finally calculated numerically by R3.1.1 software.     

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.     

On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points

Abstract.  The supremum difference between the cumulative sum diagram, and its greatest convex minorant (GCM), in case of non-parametric isotonic regression is considered. When the regression function is strictly increasing, and the design points are unequally spaced, but approximate a positive density in even a slow rate ( n ^−1/3), then the difference is shown to shrink in a very rapid (close to n ^−2/3) rate. The result is analogous to the corresponding result in case of a monotone density estimation established by Kiefer and Wolfowitz, but uses entirely different representation. The limit distribution of the GCM as a process on the unit interval is obtained when the design variables are i.i.d. with a positive density. Finally, a pointwise asymptotic normality result is proved for the smooth monotone estimator, obtained by the convolution of a kernel with the classical monotone estimator.     

Optimal experimental design criterion for discriminating semiparametric models

This paper studies the optimal experimental design problem to discriminate two regression models. Recently, López-Fidalgo et al. [2007. An optimal experimental design criterion for discriminating between non-normal models. J. Roy. Statist. Soc. B 69, 231–242] extended the conventional T-optimality criterion by Atkinson and Fedorov [1975a. The designs of experiments for discriminating between two rival models. Biometrika 62, 57–70; 1975b. Optimal design: experiments for discriminating between several models. Biometrika 62, 289–303] to deal with non-normal parametric regression models, and proposed a new optimal experimental design criterion based on the Kullback–Leibler information divergence. In this paper, we extend their parametric optimality criterion to a semiparametric setup, where we only need to specify some moment conditions for the null or alternative regression model. Our criteria, called the semiparametric Kullback–Leibler optimality criteria, can be implemented by applying a convex duality result of partially finite convex programming. The proposed method is illustrated by a simple numerical example.     

A method for obtaining randomized block designs in preclinical studies with multiple quantitative blocking variables

A method is proposed for block randomization of treatments to experimental units that can accommodate both multiple quantitative blocking variables and unbalanced designs. Hierarchical clustering in conjunction with leaf‐order optimization is used to block experimental units in multivariate space. The method is illustrated in the context of a diabetic mouse assay. A simulation study is presented to explore the utility of the proposed randomization method relative to that of a completely randomized approach, both in the presence and absence of covariate adjustment. An example R function is provided to illustrate the implementation of the method. Copyright © 2010 John Wiley & Sons, Ltd.