Longitudinal data involving multiple subjects is quite popular in medical and social science areas. I consider generalized linear mixed models (GLMMs) applied to such longitudinal data, and the optimal design searching problem under such models. In this case, based on optimal design theory, the optimality criteria depend on the estimated parameters, which leads to local optimality. Moreover, the information matrix under a GLMM doesn't have a closed-form expression. My dissertation includes three topics related to this design problem. The first part is searching for locally optimal designs under GLMMs with longitudinal data. I apply penalized quasi-likelihood (PQL) method to approximate the information matrix and compare several approximations to show the superiority of PQL over other approximations. Under different local parameters and design restrictions, locally D- and A- optimal designs are constructed based on the approximation. An interesting finding is that locally optimal designs sometimes apply different designs to different subjects. Finally, the robustness of these locally optimal designs is discussed. In the second part, an unknown observational covariate is added to the previous model. With an unknown observational variable in the experiment, expected optimality criteria are considered. Under different assumptions of the unknown variable and parameter settings, locally optimal designs are constructed and discussed. In the last part, Bayesian optimal designs are considered under logistic mixed models. Considering different priors of the local parameters, Bayesian optimal designs are generated. Bayesian design under such a model is usually expensive in time. The running time in this dissertation is optimized to an acceptable amount with accurate results. I also discuss the robustness of these Bayesian optimal designs, which is the motivation of applying such an approach.

In this dissertation two research questions in the field of applied experimental design were explored. First, methods for augmenting the three-level screening designs called Definitive Screening Designs (DSDs) were investigated. Second, schemes for strategic subdata selection for nonparametric predictive modeling with big data were developed.

Under sparsity, the structure of DSDs can allow for the screening and optimization of a system in one step, but in non-sparse situations estimation of second-order models requires augmentation of the DSD. In this work, augmentation strategies for DSDs were considered, given the assumption that the correct form of the model for the response of interest is quadratic. Series of augmented designs were constructed and explored, and power calculations, model-robustness criteria, model-discrimination criteria, and simulation study results were used to identify the number of augmented runs necessary for (1) effectively identifying active model effects, and (2) precisely predicting a response of interest. When the goal is identification of active effects, it is shown that supersaturated designs are sufficient; when the goal is prediction, it is shown that little is gained by augmenting beyond the design that is saturated for the full quadratic model. Surprisingly, augmentation strategies based on the I-optimality criterion do not lead to better predictions than strategies based on the D-optimality criterion.

Computational limitations can render standard statistical methods infeasible in the face of massive datasets, necessitating subsampling strategies. In the big data context, the primary objective is often prediction but the correct form of the model for the response of interest is likely unknown. Here, two new methods of subdata selection were proposed. The first is based on clustering, the second is based on space-filling designs, and both are free from model assumptions. The performance of the proposed methods was explored visually via low-dimensional simulated examples; via real data applications; and via large simulation studies. In all cases the proposed methods were compared to existing, widely used subdata selection methods. The conditions under which the proposed methods provide advantages over standard subdata selection strategies were identified.

Functional regression models are widely considered in practice. To precisely understand an underlying functional mechanism, a good sampling schedule for collecting informative functional data is necessary, especially when data collection is limited. However, scarce research has been conducted on the optimal sampling schedule design for the functional regression model so far. To address this design issue, efficient approaches are proposed for generating the best sampling plan in the functional regression setting. First, three optimal experimental designs are considered under a function-on-function linear model: the schedule that maximizes the relative efficiency for recovering the predictor function, the schedule that maximizes the relative efficiency for predicting the response function, and the schedule that maximizes the mixture of the relative efficiencies of both the predictor and response functions. The obtained sampling plan allows a precise recovery of the predictor function and a precise prediction of the response function. The proposed approach can also be reduced to identify the optimal sampling plan for the problem with a scalar-on-function linear regression model. In addition, the optimality criterion on predicting a scalar response using a functional predictor is derived when the quadratic relationship between these two variables is present, and proofs of important properties of the derived optimality criterion are also provided. To find such designs, an algorithm that is comparably fast, and can generate nearly optimal designs is proposed. As the optimality criterion includes quantities that must be estimated from prior knowledge (e.g., a pilot study), the effectiveness of the suggested optimal design highly depends on the quality of the estimates. However, in many situations, the estimates are unreliable; thus, a bootstrap aggregating (bagging) approach is employed for enhancing the quality of estimates and for finding sampling schedules stable to the misspecification of estimates. Through case studies, it is demonstrated that the proposed designs outperform other designs in terms of accurately predicting the response and recovering the predictor. It is also proposed that bagging-enhanced design generates a more robust sampling design under the misspecification of estimated quantities.

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.

One of the premier technologies for studying human brain functions is the event-related functional magnetic resonance imaging (fMRI). The main design issue for such experiments is to find the optimal sequence for mental stimuli. This optimal design sequence allows for collecting informative data to make precise statistical inferences about the inner workings of the brain. Unfortunately, this is not an easy task, especially when the error correlation of the response is unknown at the design stage. In the literature, the maximin approach was proposed to tackle this problem. However, this is an expensive and time-consuming method, especially when the correlated noise follows high-order autoregressive models. The main focus of this dissertation is to develop an efficient approach to reduce the amount of the computational resources needed to obtain A-optimal designs for event-related fMRI experiments. One proposed idea is to combine the Kriging approximation method, which is widely used in spatial statistics and computer experiments with a knowledge-based genetic algorithm. Through case studies, a demonstration is made to show that the new search method achieves similar design efficiencies as those attained by the traditional method, but the new method gives a significant reduction in computing time. Another useful strategy is also proposed to find such designs by considering only the boundary points of the parameter space of the correlation parameters. The usefulness of this strategy is also demonstrated via case studies. The first part of this dissertation focuses on finding optimal event-related designs for fMRI with simple trials when each stimulus consists of only one component (e.g., a picture). The study is then extended to the case of compound trials when stimuli of multiple components (e.g., a cue followed by a picture) are considered.

Functional brain imaging experiments are widely conducted in many fields for study- ing the underlying brain activity in response to mental stimuli. For such experiments, it is crucial to select a good sequence of mental stimuli that allow researchers to collect informative data for making precise and valid statistical inferences at minimum cost. In contrast to most existing studies, the aim of this study is to obtain optimal designs for brain mapping technology with an ultra-high temporal resolution with respect to some common statistical optimality criteria. The first topic of this work is on finding optimal designs when the primary interest is in estimating the Hemodynamic Response Function (HRF), a function of time describing the effect of a mental stimulus to the brain. A major challenge here is that the design matrix of the statistical model is greatly enlarged. As a result, it is very difficult, if not infeasible, to compute and compare the statistical efficiencies of competing designs. For tackling this issue, an efficient approach is built on subsampling the design matrix and the use of an efficient computer algorithm is proposed. It is demonstrated through the analytical and simulation results that the proposed approach can outperform the existing methods in terms of computing time, and the quality of the obtained designs. The second topic of this work is to find optimal designs when another set of popularly used basis functions is considered for modeling the HRF, e.g., to detect brain activations. Although the statistical model for analyzing the data remains linear, the parametric functions of interest under this setting are often nonlinear. The quality of the de- sign will then depend on the true value of some unknown parameters. To address this issue, the maximin approach is considered to identify designs that maximize the relative efficiencies over the parameter space. As shown in the case studies, these maximin designs yield high performance for detecting brain activation compared to the traditional designs that are widely used in practice.

Historically, per capita water demand has tended to increase proportionately with population growth. However, the last two decades have exhibited a different trend; per capita water usage is declining despite a growing economy and population. Subsequently, city planners and water suppliers have been struggling to understand this new trend and whether it will continue over the coming years. This leads to inefficient water management practices as well as flawed water storage design, both of which have adverse impacts on the economy and environment. Water usage data, provided by the city of Santa Monica, was analyzed using a combination of hydro-climatic and demographic variables to dissect these trends and variation in usage. The data proved to be tremendously difficult to work with; several values were missing or erroneously reported, and additional variables had to be brought from external sources to help explain the variation. Upon completion of the data processing, several statistical techniques including regression and clustering models were built to identify potential correlations and understand the consumers’ behavior. The regression models highlighted temperature and precipitation as significant stimuli of water usage, while the cluster models emphasized high volume consumers and their respective demographic traits. However, the overall model accuracy and fit was very poor for the models due to the inadequate quality of data collection and management. The imprecise measurement process for recording water usage along with varying levels of granularity across the different variables prevented the models from revealing meaningful associations. Moving forward, smart meter technology needs to be considered as it accurately captures real-time water usage and transmits the information to data hubs which then implement predictive analytics to provide updated trends. This efficient system will allow cities across the nation to stay abreast of future water usage developments and conserve time, resources, and the environment.

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.

In the presence of correlation, generalized linear models cannot be employed to obtain regression parameter estimates. To appropriately address the extravariation due to correlation, methods to estimate and model the additional variation are investigated. A general form of the mean-variance relationship is proposed which incorporates the canonical parameter. The two variance parameters are estimated using generalized method of moments, negating the need for a distributional assumption. The mean-variance relation estimates are applied to clustered data and implemented in an adjusted generalized quasi-likelihood approach through an adjustment to the covariance matrix. In the presence of significant correlation in hierarchical structured data, the adjusted generalized quasi-likelihood model shows improved performance for random effect estimates. In addition, submodels to address deviation in skewness and kurtosis are provided to jointly model the mean, variance, skewness, and kurtosis. The additional models identify covariates influencing the third and fourth moments. A cutoff to trim the data is provided which improves parameter estimation and model fit. For each topic, findings are demonstrated through comprehensive simulation studies and numerical examples. Examples evaluated include data on children’s morbidity in the Philippines, adolescent health from the National Longitudinal Study of Adolescent to Adult Health, as well as proteomic assays for breast cancer screening.

The purpose of this study was to assess usage and satisfaction of a large university recreation fitness center. Data from 471 respondents was collected during Spring 2018. Although users were satisfied overall, we obtained useful information to guide center administration towards improved usage rates and experiences for users of the center.