This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be determined e¡Óciently by standard constrained concave programming algorithms.

Consider the D-optimal designs for a combined polynomial of degree d and trigonometric of order m regression on a partial circle [see Graybill (1976), p. 324]. It is shown that the structure of the optimal design depends only on the length of the design interval and that the support points are analytic functions of this parameter. Moreover, the Taylor expansion of the optimal support points can be determined efficiently by a recursive procedure.

Weighted polynomial regression of degree d with weight function Exp(£\ x) on an interval is considered. The D-optimal designs £i_d^* are completely characterized via three differential equations. Some invariant properties of £i_d^* under affine transformation are derived. The design £i_d^* as d goes to 1, is shown to converge weakly to the arcsin distribution. Comparisons of £i_d^* with the arcsin distribution are also made.

This paper discusses the approximate and the exact n-point D-optimal design problems for the common multivariate linear and quadratic polynomial regression on some convex design spaces. For the linear polynomial regression, the design space considered are q-simplex, q-ball and convex hull of a set of finite points. It is shown that the approximate and the exact n-point D-optimal designs are concentrated on the extreme points of the design space. The structure of the optimal designs on regular polygons or regular polyhedra is also discussed. For the quadratic polynomial regression, the design space considered is a q-ball. The configuration of the approximate and the exact n-point D-optimal designs for quadratic model in two variables on a disk are investigated.

Minimax D-optimal designs for regression models with heteroscedastic errors are studied and constructed. These designs are robust against possible misspecification of the error variance in the model. We propose a flexible assumption for the error variance and use a minimax approach to define robust designs. As usual it is hard to find robust designs analytically, since the associated design problem is not a convex optimization problem. However, the minimax D-optimal design problem has an objective function as a difference of two convex functions. An effective algorithm is developed to compute minimax D-optimal designs under the least squares estimator and generalized least squares estimator. The algorithm can be applied to construct minimax D-optimal designs for any linear or nonlinear regression model with heteroscedastic errors. In addition, several theoretical results are obtained for the minimax D-optimal designs. / Graduate

Understanding the choice behavior of customers is crucial for effective service management and marketing in the hospitality industry. The first purpose of this dissertation is to examine the differential effects that cognitive, affective, and sensory attributes have on consumer hotel choice. The second purpose is to examine the moderating effects of consumer choice context on the relationship between the cognitive, affective, and sensory attributes and hotel choice. To achieve these two purposes, this dissertation includes the design of a choice experiment to examine how cognitive, affective, and sensory attributes predict consumer hotel choice using multinomial logit (MNL) and random parameter (or mixed) logit (RPL) models. For choice experiments, the main objectives are to determine the choice attributes and attribute levels to be used for the choice modeling and to create an optimal choice design. I used a Bayesian D-optimal design for the choice experiment, which I assess from the DOE (design of experiment) procedure outlined in JMP 8.0. The primary analysis associated with discrete choice analysis is the log-likelihood ratio (LR) test and the estimation of the parameters (known as part-worth utilities), using LIMDEP 9.0. The results showed that the addition of affective and sensory attributes to the choice model better explained hotel choice compared to the model with only cognitive attributes. The second purpose is to examine the moderating effects of choice context on the relationship between cognitive, affective, and sensory attributes and hotel choice. Using a stated choice model, respondents were randomly divided into two different groups and asked to evaluate their preference for two differently manipulated choice sets. For this purpose, it is necessary to include interaction effects in the choice model. This study identified the differences among choice criteria based on two different contexts. Among eight interaction effects, four interaction effects with the contexts -- price, comfortable, room quality, and atmosphere -- were statistically significant on hotel choice. The findings provide hotel managers with important insights and implications in terms of target segmentation, product development, and marketing communication strategy. / Ph. D.

abstract: Optimal design theory provides a general framework for the construction of experimental designs for categorical responses. For a binary response, where the possible result is one of two outcomes, the logistic regression model is widely used to relate a set of experimental factors with the probability of a positive (or negative) outcome. This research investigates and proposes alternative designs to alleviate the problem of separation in small-sample D-optimal designs for the logistic regression model. Separation causes the non-existence of maximum likelihood parameter estimates and presents a serious problem for model fitting purposes. First, it is shown that exact, multi-factor D-optimal designs for the logistic regression model can be susceptible to separation. Several logistic regression models are specified, and exact D-optimal designs of fixed sizes are constructed for each model. Sets of simulated response data are generated to estimate the probability of separation in each design. This study proves through simulation that small-sample D-optimal designs are prone to separation and that separation risk is dependent on the specified model. Additionally, it is demonstrated that exact designs of equal size constructed for the same models may have significantly different chances of encountering separation. The second portion of this research establishes an effective strategy for augmentation, where additional design runs are judiciously added to eliminate separation that has occurred in an initial design. A simulation study is used to demonstrate that augmenting runs in regions of maximum prediction variance (MPV), where the predicted probability of either response category is 50%, most reliably eliminates separation. However, it is also shown that MPV augmentation tends to yield augmented designs with lower D-efficiencies. The final portion of this research proposes a novel compound optimality criterion, DMP, that is used to construct locally optimal and robust compromise designs. A two-phase coordinate exchange algorithm is implemented to construct exact locally DMP-optimal designs. To address design dependence issues, a maximin strategy is proposed for designating a robust DMP-optimal design. A case study demonstrates that the maximin DMP-optimal design maintains comparable D-efficiencies to a corresponding Bayesian D-optimal design while offering significantly improved separation performance. / Dissertation/Thesis / Doctoral Dissertation Industrial Engineering 2019

Sigmoid curves have found broad applicability in biological sciences and biopharmaceutical research during the last decades. A well planned experiment design is essential to accurately estimate the parameters of the model. In contrast to a large literature and extensive results on optimal designs for linear models, research on the design for nonlinear, including sigmoid curve, models has not kept pace. Furthermore, most of the work in the optimal design literature for nonlinear models concerns the characterization of minimally supported designs. These minimal, optimal designs are frequently criticized for their inability to check goodness of fit, as there are no additional degrees of freedom for the testing. This design issue can be a serious problem, since checking the model adequacy is of particular importance when the model is selected without complete certainty. To assess for lack of fit, we must add at least one extra distinct design point to the experiment. The goal of this dissertation is to identify optimal or highly efficient designs capable of checking the fit for sigmoid curve models. In this dissertation, we consider some commonly used sigmoid curves, including logistic, probit and Gompertz models with two, three, or four parameters. We use D-optimality as our design criterion. We first consider adding one extra point to the design, and consider five alternative designs and discuss their suitability to test for lack of fit. Then we extend the results to include one more additional point to better understand the compromise among the need of detecting lack of fit, maintaining high efficiency and the practical convenience for the practitioners. We then focus on the two-parameter Gompertz model, which is widely used in fitting growth curves yet less studied in literature, and explore three-point designs for testing lack of fit under various error variance structures. One reason that nonlinear design problems are so challenging is that, with nonlinear models, information matrices and optimal designs depend on the unknown model parameters. We propose a strategy to bypass the obstacle of parameter dependence for the theoretical derivation. This dissertation also successfully characterizes many commonly studied sigmoid curves in a generalized way by imposing unified parameterization conditions, which can be generalized and applied in the studies of other sigmoid curves. We also discuss Gompertz model with different error structures in finding an extra point for testing lack of fit. / Statistics

The purpose of mixture experiments is to explore the optimum blends of mixture components, which will provide desirable response characteristics in finished products. D-Optimal minimal designs have been considered for a variety of mixture models, including Scheffe'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 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 dissertation, extensions of the D-Optimal minimal designs are developed to allow additional interior points in the design space to enable prediction of the entire response surface. First, the extensions of the D-Optimal minimal designs for two commonly used second-degree mixture models are considered. Second, the methodology for adding interior points to general mixture models is generalized. Also a new strategy for adding multiple interior points for symmetric mixture models is proposed. When compared with the standard mixture designs, the proposed extended D-Optimal minimal design provides higher power for the Lack of Fit tests with comparable D-efficiency. / Statistics

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. However, a difficulty in dealing with the Bayesian design is the lack of efficient computational methods. In this research, a hybrid computational method, which consists of the combination of a rough global optima search and a more precise local optima search, is proposed to efficiently search for the Bayesian D-optimal designs for multi-variable generalized linear models. Particularly, Poisson regression models and logistic regression models are investigated. Designs are examined for a range of prior distributions and the equivalence theorem is used to verify the design optimality. Design efficiency for various models are examined and compared with non-Bayesian designs. Bayesian D-optimal designs are found to be more efficient and robust than non-Bayesian D-optimal designs. Furthermore, the idea of the Bayesian sequential design is introduced and the Bayesian two-stage D-optimal design approach is developed for generalized linear models. With the incorporation of the first stage data information into the second stage, the two-stage design procedure can improve the design efficiency and produce more accurate and robust designs. The Bayesian two-stage D-optimal designs for Poisson and logistic regression models are evaluated based on simulation studies. The Bayesian two-stage optimal design approach is superior to the one-stage approach in terms of a design efficiency criterion. / Ph. D.