It is felt the position of the Scheffé polynomials as the primary, or sometimes sole recourse for practitioners of mixture experiments leads to a lack of enquiry regarding the type of blending behaviour that is used to describe the response and that this could be detrimental to achieving experimental objectives. Consequently, a new class of models and new experimental designs are proposed allowing a more thorough exploration of the experimental region with respect to different blending behaviours, especially those not associated with established models for mixtures, in particular the Scheffé polynomials. The proposed General Blending Models for Mixtures (GBMM) are a powerful tool allowing a broad range of blending behaviour to be described. These include those of the Scheffé polynomials (and its reparameterisations) and Becker's models. The potential benefits to be gained from their application include greater model parsimony and increased interpretability. Through this class of models it is possible for a practitioner to reject the assumptions inherent in choosing to model with the Scheffé polynomials and instead adopt a more open approach, flexible to many different types of behaviour. These models are presented alongside a fitting procedure, implementing a stepwise regression approach to the estimation of partially linear models with multiple nonlinear terms. The new class of models has been used to develop designs which allow the response surface to be explored fully with respect to the range of blending behaviours the GBMM may describe. These designs may additionally be targeted at exploring deviation from the behaviour described by the established models. As such, these designs may be thought to possess an enhanced optimality with respect to these models. They both possess good properties with respect to optimality criterion, but are also designed to be robust against model uncertainty.

In this research, Bayesian two-stage D-D optimal designs for mixture experiments with or without process variables under model uncertainty are developed. A Bayesian 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 optimality procedure that collaborates with the improved model from first stage data. Our results show that the Bayesian two-stage D-D optimal design is more efficient than both the Bayesian one-stage D-optimal design and the non-Bayesian one-stage D-optimal design in most cases. We also use simulations to investigate the ratio between the sample sizes for two stages and to observe least sample size for the first stage. On the other hand, we discuss D-optimal second or higher order designs, and show that Ds-optimal designs are a reasonable alternative to D-optimal designs. / Ph. D.

Many materials we use in daily life are comprised of a mixture; plastics, gasoline, food, medicine, etc. Mixture experiments, where factors are proportions of components and the response depends only on the relative proportions of the components, are an integral part of product development and improvement. However, when the number of components is large and there are complex constraints, experimentation can be a daunting task. We study screening methods in a mixture setting using the framework of the Cox mixture model [1]. We exploit the easy interpretation of the parameters in the Cox mixture model and develop methods for screening in a mixture setting. We present specific methods for adding a component, removing a component and a general method for screening a subset of components in mixtures with complex constraints. The variances of our parameter estimates are comparable with the typically used Scheff ́e model variances and our methods provide a reduced run size for screening experiments with mixtures containing a large number of components. We then further extend the new screening methods by using Evolutionary Operation (EVOP) developed by Box and Draper [2]. EVOP methods use small movement in a subset of process parameters and replication to reveal effects out of the process noise. Mixture experiments inherently have small movements (since the proportions can only range from zero to unity) and the effects have large variances. We update the EVOP methods by using sequential testing of effects opposed to the confidence interval method originally proposed by Box and Draper. We show that the sequential testing approach as compared with a fixed sample size reduced the required sample size as much as 50 percent with all other testing parameters held constant. We present two methods for adding a component and a general screening method using a graphical sequential t-test and provide R-code to reproduce the limits for the test.

A mixture experiment is an experiment in which the q-ingredients {x_i,i=1,2,...,q} are nonnegative and ubject to the simplex restriction £Ux_i=1 on the (q-1)-dimensional probability simplex S^{q-1}. It is usually assumed that the observations are uncorrelated, although in many applications the observations are correlated. We study the difference between the ordinary least square estimator and the Gauss Markov estimator under correlated observations. It is shown that for certain models and a special covariance structure for the mixture experiments, the unknown parameter vector for the ordinary least square estimators and the Gauss Markov estimators are the same. Moreover, we also show that the corresponding optimal designs may be obtained from previous D- and A-optimal designs for uncorrelated observations. The models studied here includ Scheff'e models, log contrast models, models containing homogeneous functions, and models containing inverse terms.

[pt] Nesta dissertação é apresentada uma síntese das técnicas estatísticas necessárias ao planejamento e análise de experimentos com mistura e mistura-processo, e é apresentada uma metodologia original indicada para a seleção de modelos, especialmente para aqueles com forte colinearidade entre os níveis dos componentes da mistura, utilizando um critério baseado na Teoria da Informação. A metodologia é ilustrada com dois exemplos da literatura. Aplicando esta metodologia aos dois estudos de caso, tem-se como objetivo a seleção de modelos melhores dos que os apresentados inicialmente nos dois exemplos. Com os modelos definidos, foram determinadas as proporções ótimas dos componentes de mistura e os níveis ótimos das variáveis de processo para cada um dos exemplos. A metodologia desenvolvida para seleção de modelos, consistindo de duas etapas, provou ser eficiente nos dois casos estudados. / [en] This dissertation presents a summary of the statistical techniques necessary for planning and analysis of mixture and mixture-process experiments and presented an original methodology is indicated for the selection of models, especially for those with high collinearity between the components levels of the mixture, using a criterion based on information theory. The methodology is illustrated with two examples from the literature. Applying this methodology to two case studies has as its objective the selection of the best models than presented in the originally two examples. With the models defined, were determined optimal proportions of the mix components and the optimal levels of the process variables for each two examples. The methodology developed for model selection, which consists of two steps, proved to be efficient in both studied cases.

abstract: Mixture experiments are useful when the interest is in determining how changes in the proportion of an experimental component affects the response. This research focuses on the modeling and design of mixture experiments when the response is categorical namely, binary and ordinal. Data from mixture experiments is characterized by the perfect collinearity of the experimental components, resulting in model matrices that are singular and inestimable under likelihood estimation procedures. To alleviate problems with estimation, this research proposes the reparameterization of two nonlinear models for ordinal data -- the proportional-odds model with a logistic link and the stereotype model. A study involving subjective ordinal responses from a mixture experiment demonstrates that the stereotype model reveals useful information about the relationship between mixture components and the ordinality of the response, which the proportional-odds fails to detect. The second half of this research deals with the construction of exact D-optimal designs for binary and ordinal responses. For both types, the base models fall under the class of Generalized Linear Models (GLMs) with a logistic link. First, the properties of the exact D-optimal mixture designs for binary responses are investigated. It will be shown that standard mixture designs and designs proposed for normal-theory responses are poor surrogates for the true D-optimal designs. In contrast with the D-optimal designs for normal-theory responses which locate support points at the boundaries of the mixture region, exact D-optimal designs for GLMs tend to locate support points at regions of uncertainties. Alternate D-optimal designs for binary responses with high D-efficiencies are proposed by utilizing information about these regions. The Mixture Exchange Algorithm (MEA), a search heuristic tailored to the construction of efficient mixture designs with GLM-type responses, is proposed. MEA introduces a new and efficient updating formula that lessens the computational expense of calculating the D-criterion for multi-categorical response systems, such as ordinal response models. MEA computationally outperforms comparable search heuristics by several orders of magnitude. Further, its computational expense increases at a slower rate of growth with increasing problem size. Finally, local and robust D-optimal designs for ordinal-response mixture systems are constructed using MEA, investigated, and shown to have high D-efficiency performance. / Dissertation/Thesis / Doctoral Dissertation Industrial Engineering 2016

Mixture experiments are widely applied. The Scheffé quadratic polynomial is the most popular mixture model in industry due to its simplicity, but it fails to accurately describe the behaviour of response variables that deviate greatly from linear blending. Higherorder Scheffé polynomials do possess the ability to predict such behaviour but become increasingly more complex to use and the number of estimable parameters grow exponentially [15]. A parameter-parsimonious mixture model, developed from the linear blending rule with weighted power means and Wohl's Q-fractions, is introduced. Bootstrap is employed to analyse the model statistically. The model is proved to be flexible enough to model non-linear deviations from linear blending without losing the simplicity of the linear blending rule. / Dissertation (MSc)--University of Pretoria, 2011. / Chemical Engineering / unrestricted

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[pt] Nesta tese é apresentada uma metodologia de seleção de modelos em experimentos mistura-processo e reunidas as técnicas estatísticas necessárias ao planejamento e análise de experimentos com mistura com ou sem variáveis de processo. Na pesquisa de seleção de modelos foi utilizado um experimento para determinar as proporções ótimas de um misto químico do mecanismo de retardo para ignição de um motor foguete. O misto químico consiste de uma mistura de três componentes. Além das proporções dos componentes da mistura, são consideradas duas variáveis de processo. O objetivo do estudo é investigar as proporções dos componentes da mistura e os níveis das variáveis de processo que colocam o valor esperado do tempo de retardo (resposta) o mais próximo possível do valor alvo e, ao mesmo tempo, minimizam o tamanho do intervalo de previsão de uma futura resposta. Foi ajustado um modelo de regressão linear com respostas normais. Com o modelo desenvolvido foram determinadas as proporções ótimas dos componentes da mistura e os níveis ótimos das variáveis de processo. Para a seleção do modelo foi utilizada uma metodologia de duas etapas, que provou ser eficiente no caso estudado. / [en] This thesis presents a methodology for model selection in mixture-process experiments and puts together the statistical techniques for the design and analysis of mixture experiments with or without process variables. An experiment of a three-component mixture of a delay mechanism to start a rocket engine was used in the research. Besides the mix components proportions, two process variables are considered. The aim of the study is to investigate the proportions of the mix components and the levels of the process variables that set the expected delay time (response) as close as possible to the target value and, at the same time, minimize the width of the prediction interval for the response. A linear regression model with normal responses was fitted. Through the developed model, the optimal proportions of the mix components and the levels of the process variables were determined. A two-stage methodology was used to select the model. This methodology for model selection proved to be efficient in the studied case.

[pt] Esta dissertação, além de apresentar uma abordagem de um caso prático real, fez reunir as técnicas estatísticas necessárias ao trato de experimentos envolvendo misturas. Foi visto que as metodologias adotadas em Projeto de Experimentos devem ser adaptadas para possibilitar o trato de problemas com misturas, já que há a necessidade de considerar a restrição básica desse tipo de experimento, o qual amarra a soma das proporções dos componentes, que deve ser sempre igual a 1, ou seja, 100%. O experimento do misto de retardo, objeto principal e motivador dessa dissertação, é um experimento com mistura, em que as proporções de todos os três componentes possuem restrições superiores e inferiores simultaneamente. Com essas restrições, o espaço fatorial restrito fica bem distorcido em relação ao simplex, havendo, portanto, a necessidade de geração de um design D-ótimo. Como houve a indicação de que a variância da resposta não é constante, no caso do misto de retardo, recorreu-se aos Modelos Lineares Generalizados, especificamente ao método da Quase- Verossimilhança. De posse do modelo adequado, pôde-se então determinar a proporção dos componentes do misto de retardo, tendo em vista o atendimento da especificação de projeto. / [en] This dissertation presents a real pactical case, and besides, it puts together the statistical techniques for the treatment of Mixture Experiments. It was presented, that the Design of Experiments techniques must be adapted in order to make possible the treatment of problems with mixtures, because the basic constraint in this type of experiment must be taken into account, that is, the sum of the proportions of all mixture components must be equal to 1 or 100%. The delay compound experiment, the main and motivating object in this dissertation, is a mixture experiment with simultaneous constraints in the proportions of all its three components. With these constraints, it is possible to observe a distortion in the restricted factorial design space in comparison to the simplex one. Therefore, it was necessary to generate a D-optimal design. When there was an indication that the response variance is not constant, in the case of the delay compound, the Generalized Linear Models, specifically the Quasi- Likelihood method was used to fit an adequate model. With the adequate model, it was possible to find the proportion of each component of the delay compound in order to attend the design specification.