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     

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.     

A new model and class of designs for mixture experiments with process variables

Experiments that involve the blending of several components are known as mixture experiments. In some mixture experiments, the response depends not only on the proportion of the mixture components, but also on the processing conditions, A new combined model is proposed which is based on Taylor series approximation and is intended to be a compromise between standard mixture models and standard response surface models. Cost and/or time constraints often limit the size of industrial experiments. With this in mind, we present a new class of designs that will accommodate the fitting of the new combined model.     

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.     

Optimization of a product performance using mixture experiments including process variables

This article presents a case study of a chemical compound used in the delay mechanism to start a rocket engine. The compound consists in a three-component mixture. Besides the components proportions, two process variables are considered. The aim of the study is to investigate the mix components proportions and the levels of process variables that set the expected delay time as close as possible to the target value and, at the same time, minimize the width of prediction interval for the response. A linear regression model with normal responses was fitted. Through the model developed, the optimal components proportions and the levels of the process variables were determined. For the model selection, the use of the backward method with an information criterion proved to be efficient in the case under study.     

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.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

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.     

Modified D-optimal design for logistic model

Asymptotic theory of using the Fisher information matrix may provide poor approximation to the exact variance matrix of maximum likelihood estimation in nonlinear models. This may be due to not obtaining an efficient D-optimal design. In this article, we propose a modified D-optimality criterion, using a more accurate information matrix, based on the Bhattacharyya matrix. The proposed information matrix and its properties are given for two parameters simple logistic model. It is shown that the resulted modified locally D-optimal design is more efficient than the previous one; particularly, for small sample size experiments.     

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.     

Particle filters for mixture models with an unknown number of components

We consider the analysis of data under mixture models where the number of components in the mixture is unknown. We concentrate on mixture Dirichlet process models, and in particular we consider such models under conjugate priors. This conjugacy enables us to integrate out many of the parameters in the model, and to discretize the posterior distribution. Particle filters are particularly well suited to such discrete problems, and we propose the use of the particle filter of Fearnhead and Clifford for this problem. The performance of this particle filter, when analyzing both simulated and real data from a Gaussian mixture model, is uniformly better than the particle filter algorithm of Chen and Liu. In many situations it outperforms a Gibbs Sampler. We also show how models without the required amount of conjugacy can be efficiently analyzed by the same particle filter algorithm.     

Bayesian two-stage optimal design for mixture models

In this paper, a Bayesian two-stage D–D optimal design for mixture experimental models under model uncertainty is developed. A Bayesian D-optimality criterion is used in the first stage to minimize the determinant of the posterior variances of the parameters. The second stage design is then generated according to an optimalityprocedure that collaborates with the improved model from the first stage data. The results show that a Bayesian two-stage D–D-optimal design for mixture experiments under model uncertainty is more efficient than both the Bayesian one-stage D-optimal design and the non-Bayesian one-stage D-optimal design in most situations. Furthermore, simulations are used to obtain a reasonable ratio of the sample sizes between the two stages.     

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.     

A useful approximation for planning block designs with potential missing data

Many empirical studies are planned with the prior knowledge that some of the data may be missed. This knowledge is seldom explicitly incorporated into the experiment design process for lack of a candid methodology. This paper proposes an index related to the expected determinant of the information matrix as a criterion for planning block designs. Due to the intractable nature of the expected determinantal criterion an analytic expression is presented only for a simple 2x2 layout. A first order Taylor series approximation function is suggested for larger layouts. Ranges over which this approximation is adequate are shown via Monte Carlo simulations. The robustness of information in the block design relative to the completely randomized design with missing data is discussed.     

Multiple linear regression model with stochastic design variables

In a simple multiple linear regression model, the design variables have traditionally been assumed to be non-stochastic. In numerous real-life situations, however, they are stochastic and non-normal. Estimators of parameters applicable to such situations are developed. It is shown that these estimators are efficient and robust. A real-life example is given.     

Small orthogonal main effect plans with four factors

In this paper we study orthogonal main effect plans with four factors, A table of such designs, where each factor has at most 10 levels, and there are at most 40 runs, is generated. We determine the spectrum of the degrees of freedom of pure error for these designs.     

A practical sampling approach for a Bayesian mixture model with unknown number of components

Recently, mixture distribution becomes more and more popular in many scientific fields. Statistical computation and analysis of mixture models, however, are extremely complex due to the large number of parameters involved. Both EM algorithms for likelihood inference and MCMC procedures for Bayesian analysis have various difficulties in dealing with mixtures with unknown number of components. In this paper, we propose a direct sampling approach to the computation of Bayesian finite mixture models with varying number of components. This approach requires only the knowledge of the density function up to a multiplicative constant. It is easy to implement, numerically efficient and very practical in real applications. A simulation study shows that it performs quite satisfactorily on relatively high dimensional distributions. A well-known genetic data set is used to demonstrate the simplicity of this method and its power for the computation of high dimensional Bayesian mixture models.     

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.