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.     

Optimal design for nonlinear regression models

Two results for D _θ-optimal designs for nonlinear regression models are shown to follow directly from approximate design theory. The first result considered is one concerning the replication of exact designs with minimum support, first established by Atkinson and Hunter and by M.J. Box in 1968, while the second pertains to a heteroscedastic model introduced by Velilla and Llosa in 1992. An illustrative example is provided.     

Locally and maximin optimal designs for multi-factor nonlinear models

This paper considers the search for locally and maximin optimal designs for multi-factor nonlinear models from optimal designs for sub-models of a lower dimension. In particular, sufficient conditions are given so that maximin D-optimal designs for additive multi-factor nonlinear models can be built from maximin D-optimal designs for their sub-models with a single factor. Some examples of application are models involving exponential decay in several variables.     

Locally D-Optimal Design for a Logit Model in Discrete Choice Experiment

Conjoint analysis is concerned with understanding how people make choice between products or services (alternatives) or a combination of product and service (choice set), so that businesses can design new products or services that better meet customers needs. In this situation, logit model (Multinomial Logit Model) has been used to calculate the probability related to choosing an alternative in a choice set with the highest utility. Then I considered several choice sets instead of one. In this article, I have used the locally D-optimal design for the combination of the level of attributes (two attributes each with two levels) to create alternatives. The optimal combination of alternatives in choice sets which help us to have a suitable choice.     

Bayesian A- and D-optimal designs for gamma regression model with inverse link function

Bayesian optimal designs have received increasing attention in recent years, especially in biomedical and clinical trials. Bayesian design procedures can utilize the available prior information of the unknown parameters so that a better design can be achieved. With this in mind, this article considers the Bayesian A- and D-optimal designs of the two- and three-parameter Gamma regression model. In this regard, we first obtain the Fisher information matrix of the proposed model and then calculate the Bayesian A- and D-optimal designs assuming various prior distributions such as normal, half-normal, gamma, and uniform distribution for the unknown parameters. All of the numerical calculations are handled in R software. The results of this article are useful in medical and industrial researches.     

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.     

Model-Robust Design of Conjoint Choice Experiments

Within the context of choice experimental designs, most authors have proposed designs for the multinomial logit model under the assumption that only the main effects matter. Very little attention has been paid to designs for attribute interaction models. In this article, three types of Bayesian D-optimal designs for the multinomial logit model are studied: main-effects designs, interaction-effects designs, and composite designs. Simulation studies are used to show that in situations where a researcher is not sure whether or not attribute interaction effects are present, it is best to take into account interactions in the design stage. In particular, it is shown that a composite design constructed by including an interaction-effects model and a main-effects model in the design criterion is most robust against misspecification of the underlying model when it comes to making precise predictions.     

Sequential Designs for Binary Data with the Purpose to Maximize the Probability of Response

Two kinds of sequential designs are proposed for finding the point that maximizes the probability of response assuming a binary response variable and a quadratic logistic regression model. One is a parametric optimal design approach, and the other one is a nonparametric stochastic approximation approach. The suggested sequential designs are evaluated and compared in a simulation study. In summary, the parametric approach performed very well whereas its competitor failed in some cases.     

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.     

Modified Wynn's Sequential Algorithm for Constructing D-Optimal Designs: Adding Two Points at a Time

Partial least squares (PLS) is a class of methods for modeling relations between sets of observed variables by using the latent components where the predictors are highly collinear. SIMPLS is a commonly used PLS algorithm that calculates the latent components directly as linear combinations of the original variables. However, SIMPLS is known to be very sensible to outliers since it is based on the empirical cross-covariance matrix. RoPLS is a recently proposed iterative method for robust SIMPLS. In this article, the influence function for the RoPLS coefficient estimator is derived. It is demonstrated that under certain conditions, the RoPLS estimator has infinitesimal robustness.     

Optimal designs for multivariate logistic mixed models with longitudinal data

This paper considers the optimal design problem for multivariate mixed-effects logistic models with longitudinal data. A decomposition method of the binary outcome and the penalized quasi-likelihood are used to obtain the information matrix. The D-optimality criterion based on the approximate information matrix is minimized under different cost constraints. The results show that the autocorrelation coefficient plays a significant role in the design. To overcome the dependence of the D-optimal designs on the unknown fixed-effects parameters, the Bayesian D-optimality criterion is proposed. The relative efficiencies of designs reveal that both the cost ratio and autocorrelation coefficient play an important role in the optimal designs.     

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.     

Compound designs for dose‐finding in the presence of nondesignable covariates

Compound optimal designs are considered where one component of the design criterion is a traditional optimality criterion such as the D‐optimality criterion, and the other component accounts for higher efficacy with low toxicity. With reference to the dose‐finding problem, we suggest the technique to choose weights for the two components that makes the optimization problem simpler than the traditional penalized design. We allow general bivariate responses for efficacy and toxicity. We then extend the procedure in the presence of nondesignable covariates such as age, sex, or other health conditions. A new breast cancer treatment is considered to illustrate the procedures. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian D-optimal designs for exponential regression models

We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.     

Optimal two-point designs for the michaelis-menten model with heteroscedastic errors

We construct D-optimal designs for the Michaelis-Menten model when the variance of the response depends on the independent variable. However, this dependence is only partially known. A Bayesian approacn is used to find an optimal design by incorporating the prior lnformation about the variance structure. We demonstrate the method for a class of error variance structures and present efficiencies of these optimal designs under prior mis-specifications. In particular, we show that an erroneous assumption on the variance structure for the Michaelis-Menten model can have serious consequences.     

Optimal designs for non-competitive enzyme inhibition kinetic models

In this paper, we present a new method for determining optimal designs for enzyme inhibition kinetic models, which are used to model the influence of the concentration of a substrate and an inhibition on the velocity of a reaction. The approach uses a nonlinear transformation of the vector of predictors such that the model in the new coordinates is given by an incomplete response surface model. Although there exist no explicit solutions of the optimal design problem for incomplete response surface models so far, the corresponding design problem in the new coordinates is substantially more transparent, such that explicit or numerical solutions can be determined more easily. The designs for the original problem can finally be found by an inverse transformation of the optimal designs determined for the response surface model. We illustrate the method determining explicit solutions for the D-optimal design and for the optimal design problem for estimating the individual coefficients in a non-competitive enzyme inhibition kinetic model.     

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.     

Optimum Designs for Estimation of Parameters in a Quadratic Mixture-Amount Model

The authors propose a mixture-amount model, which is quadratic both in the proportions of mixing components and the amount of mixture. They attempt to find the A- and D-optimal designs for the estimation of the model parameters within a subclass of designs. The optimality of the derived designs in the entire class of competing designs has been investigated through equivalence theorem.     

Optimal experiments in the presence of a learning effect: a problem suggested by software production

In software engineering empirical comparisons of different ways of writing computer code are often made. This leads to the need for planned experimentation and has recently established a new area of application of DoE. This paper is motivated by an experiment on the production of multimedia services on the web, performed at the Telecom Research Centre in Turin, where two different ways of developing code, with or without a framework, were compared. As the experiment progresses, the programmers performance improves as he/she undergoes a learning process; this must be taken into account as it may affect the outcome of the trial. In this paper we discuss statistical models and D-optimal plans for such experiments and indicate some heuristics which allow a much speedier search for the optimum. Solutions differ according to whether we assume that the learning process depends or not on the treatments.