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.     

Conditions for fourth order rotatability in K dimensions

In this paper, conditions for a set of points to form a rotatable design of order four have been worked out. These condi- tions form the basis for obtaining a set of rotatable designs of order four.     

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.     

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.     

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.     

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.     

Orthogonal blocking of response surface split-plot designs

When all experimental runs cannot be performed under homogeneous conditions, blocking can be used to increase the power for testing the treatment effects. Orthogonal blocking provides the same estimator of the polynomial effects as the one that would be obtained by ignoring the blocks. In many real-life design scenarios, there is at least one factor that is hard to change, leading to a split-plot structure. This paper shows that for a balanced ordinary least square–generalized least square equivalent split-plot design, orthogonal blocking can be achieved. Orthogonally blocked split-plot central composite designs are constructed and a catalog is provided.     

Efficient designs for constrained mixture experiments

Many practical experiments on mixtures (where the components sum to one) include additional lower or upper bounds on components, or on linear combinations of them. Usually theory cannot be used to obtain a good design, and algorithmic methods are necessary. Some of the available methods are discussed. Their performance is evaluated on some examples, and the form of the optimal design is investigated.     

A fourth order rotatable design in four dimensions

Patel and Arap Koske (1985) gave the necessary and sufficient conditions for a set of points to form a rotatable design of order four, A set of 256 points for four factors which satisfies these conditions is given in this paper.     

An investigation of first-order and second-order designs for extrapolation outside a hypersphere

Methods for selecting a first-order or second-order rotatable response surface design when both variance and bias error exist are applied to a situation in which it is desired to extrapolate the fitted model in all directions outside of a sphere within which all the experiments are to be made. The extrapolation region is a spherical shell.     

Graphical methods for evaluating ridge regression estimator in mixture experiments

When the component proportions in mixture experiments are restricted by lower and upper bounds, multicollinearity appears all too frequently. Thus, we can suggest the use of ridge regression as a mean for stabilizing the coefficient estimates in the fitted model. We propose graphical methods for evaluating the effect of ridge regression estimator with respect to the predicted response value and the prediction variance.     

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.     

A-optimal designs for mixture central polynomial model with qualitative factors

Mixture central polynomial models with qualitative factors are widely applied in many fields of research. In this paper, a method of finding A-optimal design for two degree mixture central polynomial model with qualitative factors will be proposed. The variance function will be given for getting the support points of the design. The A-optimality is confirmed by the equivalence theorem. In addition, this method also works effectively with higher degree models.     

Extension of Central Composite Design for Second-Order Response Surface Model Building

The central composite design (CCD) is perhaps the most popular class of second-order response surface designs. Even though the CCDs are popular for response surface designs, this class of design has some limitations such as it does not sometimes possess good statistical properties, and it does not fit complicated models well. In this article, we propose extended central composite designs (ECCDs) to overcome these limitations. We compare ECCDs with CCDs in terms of average prediction variance, and find that ECCDs are better than CCDs.     

On some DS-optimal designs in spherical regions

For polynomial regression over spherical regions the d-th order D_s-optimal designs for the λ-th order models are derived for 1 ≤ λ ≤ d ≤ 4. Efficiencies of these designs with respect to the λ-th order D-optimal designs are obtained. The effects of estimating addtional parameters due to an m-th order model (d ≥ m >>λ) on the efficiencies are investigated.     

Alternative modeling techniques for the quantal response data in mixture experiments

Mixture experiments are commonly encountered in many fields including chemical, pharmaceutical and consumer product industries. Due to their wide applications, mixture experiments, a special study of response surface methodology, have been given greater attention in both model building and determination of designs compared with other experimental studies. In this paper, some new approaches are suggested on model building and selection for the analysis of the data in mixture experiments by using a special generalized linear models, logistic regression model, proposed by Chen et al. [7]. Generally, the special mixture models, which do not have a constant term, are highly affected by collinearity in modeling the mixture experiments. For this reason, in order to alleviate the undesired effects of collinearity in the analysis of mixture experiments with logistic regression, a new mixture model is defined with an alternative ratio variable. The deviance analysis table is given for standard mixture polynomial models defined by transformations and special mixture models used as linear predictors. The effects of components on the response in the restricted experimental region are given by using an alternative representation of Cox's direction approach. In addition, odds ratio and the confidence intervals of odds ratio are identified according to the chosen reference and control groups. To compare the suggested models, some model selection criteria, graphical odds ratio and the confidence intervals of the odds ratio are used. The advantage of the suggested approaches is illustrated on tumor incidence data set.     

A graphical evaluation of logistic ridge estimator in mixture experiments

In comparison to other experimental studies, multicollinearity appears frequently in mixture experiments, a special study area of response surface methodology, due to the constraints on the components composing the mixture. In the analysis of mixture experiments by using a special generalized linear model, logistic regression model, multicollinearity causes precision problems in the maximum-likelihood logistic regression estimate. Therefore, effects due to multicollinearity can be reduced to a certain extent by using alternative approaches. One of these approaches is to use biased estimators for the estimation of the coefficients. In this paper, we suggest the use of logistic ridge regression (RR) estimator in the cases where there is multicollinearity during the analysis of mixture experiments using logistic regression. Also, for the selection of the biasing parameter, we use fraction of design space plots for evaluating the effect of the logistic RR estimator with respect to the scaled mean squared error of prediction. The suggested graphical approaches are illustrated on the tumor incidence data set.     

Conditions of Slope-Rotatability for Third-Order Polynomial Regression Models

In experimental design for response surface analysis, it is sometimes of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of the response surface is important. In particular, Hader and Park (1978 Hader , R. J. , Park , S. H. ( 1978 ). Slope-rotatable central composite designs . Technometrics 20 : 413 – 417 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggested the concept of slope-rotatability and studied slope rotatable central composite designs. Until now, many response surface designs including central composite designs have been suggested for fitting second order response surface models. However, we often need to fit third-order polynomial regression models. In this article, we suggest extended central composite designs (ECCDs) to fit third-order models and find the necessary and sufficient conditions for slope-rotatability over all directions in the third-order polynomial models.