Matching Items (5)
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- All Subjects: Complete class
- All Subjects: Optimal designs (Statistics)
- Creators: Kao, Ming-Hung
Description
Statistical model selection using the Akaike Information Criterion (AIC) and similar criteria is a useful tool for comparing multiple and non-nested models without the specification of a null model, which has made it increasingly popular in the natural and social sciences. De- spite their common usage, model selection methods are not driven by a notion of statistical confidence, so their results entail an unknown de- gree of uncertainty. This paper introduces a general framework which extends notions of Type-I and Type-II error to model selection. A theo- retical method for controlling Type-I error using Difference of Goodness of Fit (DGOF) distributions is given, along with a bootstrap approach that approximates the procedure. Results are presented for simulated experiments using normal distributions, random walk models, nested linear regression, and nonnested regression including nonlinear mod- els. Tests are performed using an R package developed by the author which will be made publicly available on journal publication of research results.
ContributorsCullan, Michael J (Author) / Sterner, Beckett (Thesis advisor) / Fricks, John (Committee member) / Kao, Ming-Hung (Committee member) / Arizona State University (Publisher)
Created2018
Description
Functional or dynamic responses are prevalent in experiments in the fields of engineering, medicine, and the sciences, but proposals for optimal designs are still sparse for this type of response. Experiments with dynamic responses result in multiple responses taken over a spectrum variable, so the design matrix for a dynamic response have more complicated structures. In the literature, the optimal design problem for some functional responses has been solved using genetic algorithm (GA) and approximate design methods. The goal of this dissertation is to develop fast computer algorithms for calculating exact D-optimal designs.
First, we demonstrated how the traditional exchange methods could be improved to generate a computationally efficient algorithm for finding G-optimal designs. The proposed two-stage algorithm, which is called the cCEA, uses a clustering-based approach to restrict the set of possible candidates for PEA, and then improves the G-efficiency using CEA.
The second major contribution of this dissertation is the development of fast algorithms for constructing D-optimal designs that determine the optimal sequence of stimuli in fMRI studies. The update formula for the determinant of the information matrix was improved by exploiting the sparseness of the information matrix, leading to faster computation times. The proposed algorithm outperforms genetic algorithm with respect to computational efficiency and D-efficiency.
The third contribution is a study of optimal experimental designs for more general functional response models. First, the B-spline system is proposed to be used as the non-parametric smoother of response function and an algorithm is developed to determine D-optimal sampling points of a spectrum variable. Second, we proposed a two-step algorithm for finding the optimal design for both sampling points and experimental settings. In the first step, the matrix of experimental settings is held fixed while the algorithm optimizes the determinant of the information matrix for a mixed effects model to find the optimal sampling times. In the second step, the optimal sampling times obtained from the first step is held fixed while the algorithm iterates on the information matrix to find the optimal experimental settings. The designs constructed by this approach yield superior performance over other designs found in literature.
First, we demonstrated how the traditional exchange methods could be improved to generate a computationally efficient algorithm for finding G-optimal designs. The proposed two-stage algorithm, which is called the cCEA, uses a clustering-based approach to restrict the set of possible candidates for PEA, and then improves the G-efficiency using CEA.
The second major contribution of this dissertation is the development of fast algorithms for constructing D-optimal designs that determine the optimal sequence of stimuli in fMRI studies. The update formula for the determinant of the information matrix was improved by exploiting the sparseness of the information matrix, leading to faster computation times. The proposed algorithm outperforms genetic algorithm with respect to computational efficiency and D-efficiency.
The third contribution is a study of optimal experimental designs for more general functional response models. First, the B-spline system is proposed to be used as the non-parametric smoother of response function and an algorithm is developed to determine D-optimal sampling points of a spectrum variable. Second, we proposed a two-step algorithm for finding the optimal design for both sampling points and experimental settings. In the first step, the matrix of experimental settings is held fixed while the algorithm optimizes the determinant of the information matrix for a mixed effects model to find the optimal sampling times. In the second step, the optimal sampling times obtained from the first step is held fixed while the algorithm iterates on the information matrix to find the optimal experimental settings. The designs constructed by this approach yield superior performance over other designs found in literature.
ContributorsSaleh, Moein (Author) / Pan, Rong (Thesis advisor) / Montgomery, Douglas C. (Committee member) / Runger, George C. (Committee member) / Kao, Ming-Hung (Committee member) / Arizona State University (Publisher)
Created2015
Description
Accelerated life testing (ALT) is the process of subjecting a product to stress conditions (temperatures, voltage, pressure etc.) in excess of its normal operating levels to accelerate failures. Product failure typically results from multiple stresses acting on it simultaneously. Multi-stress factor ALTs are challenging as they increase the number of experiments due to the stress factor-level combinations resulting from the increased number of factors. Chapter 2 provides an approach for designing ALT plans with multiple stresses utilizing Latin hypercube designs that reduces the simulation cost without loss of statistical efficiency. A comparison to full grid and large-sample approximation methods illustrates the approach computational cost gain and flexibility in determining optimal stress settings with less assumptions and more intuitive unit allocations.
Implicit in the design criteria of current ALT designs is the assumption that the form of the acceleration model is correct. This is unrealistic assumption in many real-world problems. Chapter 3 provides an approach for ALT optimum design for model discrimination. We utilize the Hellinger distance measure between predictive distributions. The optimal ALT plan at three stress levels was determined and its performance was compared to good compromise plan, best traditional plan and well-known 4:2:1 compromise test plans. In the case of linear versus quadratic ALT models, the proposed method increased the test plan's ability to distinguish among competing models and provided better guidance as to which model is appropriate for the experiment.
Chapter 4 extends the approach of Chapter 3 to ALT sequential model discrimination. An initial experiment is conducted to provide maximum possible information with respect to model discrimination. The follow-on experiment is planned by leveraging the most current information to allow for Bayesian model comparison through posterior model probability ratios. Results showed that performance of plan is adversely impacted by the amount of censoring in the data, in the case of linear vs. quadratic model form at three levels of constant stress, sequential testing can improve model recovery rate by approximately 8% when data is complete, but no apparent advantage in adopting sequential testing was found in the case of right-censored data when censoring is in excess of a certain amount.
Implicit in the design criteria of current ALT designs is the assumption that the form of the acceleration model is correct. This is unrealistic assumption in many real-world problems. Chapter 3 provides an approach for ALT optimum design for model discrimination. We utilize the Hellinger distance measure between predictive distributions. The optimal ALT plan at three stress levels was determined and its performance was compared to good compromise plan, best traditional plan and well-known 4:2:1 compromise test plans. In the case of linear versus quadratic ALT models, the proposed method increased the test plan's ability to distinguish among competing models and provided better guidance as to which model is appropriate for the experiment.
Chapter 4 extends the approach of Chapter 3 to ALT sequential model discrimination. An initial experiment is conducted to provide maximum possible information with respect to model discrimination. The follow-on experiment is planned by leveraging the most current information to allow for Bayesian model comparison through posterior model probability ratios. Results showed that performance of plan is adversely impacted by the amount of censoring in the data, in the case of linear vs. quadratic model form at three levels of constant stress, sequential testing can improve model recovery rate by approximately 8% when data is complete, but no apparent advantage in adopting sequential testing was found in the case of right-censored data when censoring is in excess of a certain amount.
ContributorsNasir, Ehab (Author) / Pan, Rong (Thesis advisor) / Runger, George C. (Committee member) / Gel, Esma (Committee member) / Kao, Ming-Hung (Committee member) / Montgomery, Douglas C. (Committee member) / Arizona State University (Publisher)
Created2014
Description
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.
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.
ContributorsKhogeer, Hazar Abdulrahman (Author) / Kao, Ming-Hung (Thesis advisor) / Stufken, John (Committee member) / Reiser, Mark R. (Committee member) / Zheng, Yi (Committee member) / Cheng, Dan (Committee member) / Arizona State University (Publisher)
Created2020
Description
This study concerns optimal designs for experiments where responses consist of both binary and continuous variables. Many experiments in engineering, medical studies, and other fields have such mixed responses. Although in recent decades several statistical methods have been developed for jointly modeling both types of response variables, an effective way to design such experiments remains unclear. To address this void, some useful results are developed to guide the selection of optimal experimental designs in such studies. The results are mainly built upon a powerful tool called the complete class approach and a nonlinear optimization algorithm. The complete class approach was originally developed for a univariate response, but it is extended to the case of bivariate responses of mixed variable types. Consequently, the number of candidate designs are significantly reduced. An optimization algorithm is then applied to efficiently search the small class of candidate designs for the D- and A-optimal designs. Furthermore, the optimality of the obtained designs is verified by the general equivalence theorem. In the first part of the study, the focus is on a simple, first-order model. The study is expanded to a model with a quadratic polynomial predictor. The obtained designs can help to render a precise statistical inference in practice or serve as a benchmark for evaluating the quality of other designs.
ContributorsKim, Soohyun (Author) / Kao, Ming-Hung (Thesis advisor) / Dueck, Amylou (Committee member) / Pan, Rong (Committee member) / Reiser, Mark R. (Committee member) / Stufken, John (Committee member) / Arizona State University (Publisher)
Created2017