abstract: Generalized Linear Models (GLMs) are widely used for modeling responses with non-normal error distributions. When the values of the covariates in such models are controllable, finding an optimal (or at least efficient) design could greatly facilitate the work of collecting and analyzing data. In fact, many theoretical results are obtained on a case-by-case basis, while in other situations, researchers also rely heavily on computational tools for design selection. Three topics are investigated in this dissertation with each one focusing on one type of GLMs. Topic I considers GLMs with factorial effects and one continuous covariate. Factors can have interactions among each other and there is no restriction on the possible values of the continuous covariate. The locally D-optimal design structures for such models are identified and results for obtaining smaller optimal designs using orthogonal arrays (OAs) are presented. Topic II considers GLMs with multiple covariates under the assumptions that all but one covariate are bounded within specified intervals and interaction effects among those bounded covariates may also exist. An explicit formula for D-optimal designs is derived and OA-based smaller D-optimal designs for models with one or two two-factor interactions are also constructed. Topic III considers multiple-covariate logistic models. All covariates are nonnegative and there is no interaction among them. Two types of D-optimal design structures are identified and their global D-optimality is proved using the celebrated equivalence theorem. / Dissertation/Thesis / Doctoral Dissertation Statistics 2018

In this thesis we revisit some classical results about the MacWilliams equivalence theorems for codes over fields and rings. These theorems deal with the question whether, for a given weight function, weight-preserving isomorphisms between codes can be described explicitly. We will show that a condition, which was already known to be sufficient for the MacWilliams equivalence theorem, is also necessary. Furthermore we will study a local-global property that naturally generalizes the MacWilliams equivalence theorems. Making use of F-partitions, we will prove that for various subgroups of the group of invertible matrices the local-global extension principle is valid.

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.

A mixture experiment is an experiment in which the q-ingredients {xi,i=1,...,q} are nonnegative and subject to the simplex restriction sum_{i=1}^q x_i=1 on the (q-1)-dimensional probability simplex S^{q-1}. In this work, we investigate the robust D-optimal designs for mixture experiments with consideration on uncertainties in the Scheffe's linear, quadratic and cubic model without 3-way effects. The D-optimal designs for each of the Scheffe's models are used to find the robust D-optimal designs. With uncertianties on the Scheffe's linear and quadratic models, the optimal convex combination of the two model's D-optimal designs can be proved to be a robust D-optimal design. For the case of the Scheffe's linear and cubic model without 3-way effects, we have some numerical results about the robust D-optimal designs, as well as that for Scheffe's linear, quadratic and cubic model without 3-way effects. Ultimately, we discuss the efficiency of a maxmin type criterion D_r under given the robust D-optimal designs for the Scheffe's linear and quadratic models.

Consider a linear regression model with a two-dimensional control vector (x_1, x_2) and an m-dimensional response vector y = (y_1, . . . , y_m). The components of y are correlated with a known covariance matrix. Based on the assumed regression model, there are two problems of interest. The first one is to estimate unknown control vector x_c corresponding to an observed y, where xc will be estimated by the classical estimator. The second one is to obtain a suitable estimation of the control vector x_T corresponding to a given target T = (T_1, . . . , T_m) on the expected responses. Consideration in this work includes the deviation of the expected response E(y_i) from its corresponding target value T_i for each component and defines the optimal control vector x, say x_T , to be the one which minimizes the weighted sum of squares of standardized deviations within the range of x. The objective of this study is to find c-optimal designs for estimating x_c and x_T , which minimize the mean squared error of the estimator of xc and x_T respectively. The comparison of the difference between the optimal calibration design and the optimal design for estimating x_T is provided. The efficiencies of the optimal calibration design relative to the uniform design are also presented, and so are the efficiencies of the optimal design for given target vector relative to the uniform design.

abstract: Bivariate responses that comprise mixtures of binary and continuous variables are common in medical, engineering, and other scientific fields. There exist many works concerning the analysis of such mixed data. However, the research on optimal designs for this type of experiments is still scarce. The joint mixed responses model that is considered here involves a mixture of ordinary linear models for the continuous response and a generalized linear model for the binary response. Using the complete class approach, tighter upper bounds on the number of support points required for finding locally optimal designs are derived for the mixed responses models studied in this work. In the first part of this dissertation, a theoretical result was developed to facilitate the search of locally symmetric optimal designs for mixed responses models with one continuous covariate. Then, the study was extended to mixed responses models that include group effects. Two types of mixed responses models with group effects were investigated. The first type includes models having no common parameters across subject group, and the second type of models allows some common parameters (e.g., a common slope) across groups. In addition to complete class results, an efficient algorithm (PSO-FM) was proposed to search for the A- and D-optimal designs. Finally, the first-order mixed responses model is extended to a type of a quadratic mixed responses model with a quadratic polynomial predictor placed in its linear model. / Dissertation/Thesis / Doctoral Dissertation Statistics 2020

This paper is concerned with the problem of constructing A-optimal design for polynomial regression with analytic weight function on the interval [m-a,m+a]. It is shown that the structure of the optimal design depends on a and weight function only, as a close to 0. Moreover, if the weight function is an analytic function a, then a scaled version of optimal support points and weights is analytic functions of a at $a=0$. We make use of a Taylor expansion which coefficients can be determined recursively, for calculating the A-optimal designs.

This paper is devoted to studying the problem of constructing Ds-optimal design for d-th degree polynomial regression with analytic weight function on the interval [m-a,m+a],m,a in R. It is demonstrated that the structure of the optimal design depends on d, a and weight function only, as a close to 0. Moreover, the Taylor polynomials of the scaled versions of the optimal support points and weights can be computed via a recursive formula.

In this dissertation we first consider a parallel linear model with correlated dual responses on a symmetric compact design region and construct locally optimal designs for estimating the location-shift parameter. These locally optimal designs are variant under linear transformation of the design space and depend on the correlation between the dual responses in an interesting and sensitive way. Subsequently, minimax and maximin efficient designs for estimating the location-shift parameter are derived. A comparison of the behavior of efficiencies between the minimax and maximin efficient designs relative to locally optimal designs is also provided. Both minimax or maximin efficient designs have advantage in terms of estimating efficiencies in different situations. Thirdly, we consider a linear regression model with a one-dimensional control variable x and an m-dimensional response variable y=(y_1,...,y_m). The components of y are correlated with a known covariance matrix. The calibration problem discussed here is based on the assumed regression model. It is of interest to obtain a suitable estimation of the corresponding x for a given target T=(T_1,...,T_m) on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration includes the deviation of the expected response E(y_i) from its corresponding target value T_i for each component and the optimal value of calibration point x, say x_0, is considered to be the one which minimizes the weighted sum of squares of such deviations within the range of x. The objective of this study is to find a locally optimal design for estimating x_0, which minimizes the mean square error of the difference between x_0 and its estimator. It shows the optimality criterion is approximately equivalent to a c-criterion under certain conditions and explicit solutions with dual responses under linear and quadratic polynomial regressions are obtained.

Consider the minimally-supported D-optimal designs for dth degree polynomial regression with bounded and positive weight function on a compact interval. We show that the optimal design converges weakly to the arcsin distribution as d goes to infinity. Comparisons of the optimal design with the arcsin distribution and D-optimal arcsin support design by D-efficiencies are also given. We also show that if the design interval is [−1, 1], then the minimally-supported D-optimal design converges to the D-optimal arcsin support design with the specific weight function 1/¡Ô(£\-x^2), £\>1, as £\¡÷1+.