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- All Subjects: Design of Experiments
- Creators: Montgomery, Douglas C.
Description
Nonregular screening designs can be an economical alternative to traditional resolution IV 2^(k-p) fractional factorials. Recently 16-run nonregular designs, referred to as no-confounding designs, were introduced in the literature. These designs have the property that no pair of main effect (ME) and two-factor interaction (2FI) estimates are completely confounded. In this dissertation, orthogonal arrays were evaluated with many popular design-ranking criteria in order to identify optimal 20-run and 24-run no-confounding designs. Monte Carlo simulation was used to empirically assess the model fitting effectiveness of the recommended no-confounding designs. The results of the simulation demonstrated that these new designs, particularly the 24-run designs, are successful at detecting active effects over 95% of the time given sufficient model effect sparsity. The final chapter presents a screening design selection methodology, based on decision trees, to aid in the selection of a screening design from a list of published options. The methodology determines which of a candidate set of screening designs has the lowest expected experimental cost.
ContributorsStone, Brian (Author) / Montgomery, Douglas C. (Thesis advisor) / Silvestrini, Rachel T. (Committee member) / Fowler, John W (Committee member) / Borror, Connie M. (Committee member) / Arizona State University (Publisher)
Created2013
Description
Random Forests is a statistical learning method which has been proposed for propensity score estimation models that involve complex interactions, nonlinear relationships, or both of the covariates. In this dissertation I conducted a simulation study to examine the effects of three Random Forests model specifications in propensity score analysis. The results suggested that, depending on the nature of data, optimal specification of (1) decision rules to select the covariate and its split value in a Classification Tree, (2) the number of covariates randomly sampled for selection, and (3) methods of estimating Random Forests propensity scores could potentially produce an unbiased average treatment effect estimate after propensity scores weighting by the odds adjustment. Compared to the logistic regression estimation model using the true propensity score model, Random Forests had an additional advantage in producing unbiased estimated standard error and correct statistical inference of the average treatment effect. The relationship between the balance on the covariates' means and the bias of average treatment effect estimate was examined both within and between conditions of the simulation. Within conditions, across repeated samples there was no noticeable correlation between the covariates' mean differences and the magnitude of bias of average treatment effect estimate for the covariates that were imbalanced before adjustment. Between conditions, small mean differences of covariates after propensity score adjustment were not sensitive enough to identify the optimal Random Forests model specification for propensity score analysis.
ContributorsCham, Hei Ning (Author) / Tein, Jenn-Yun (Thesis advisor) / Enders, Stephen G (Thesis advisor) / Enders, Craig K. (Committee member) / Mackinnon, David P (Committee member) / Arizona State University (Publisher)
Created2013
Description
This dissertation explores different methodologies for combining two popular design paradigms in the field of computer experiments. Space-filling designs are commonly used in order to ensure that there is good coverage of the design space, but they may not result in good properties when it comes to model fitting. Optimal designs traditionally perform very well in terms of model fitting, particularly when a polynomial is intended, but can result in problematic replication in the case of insignificant factors. By bringing these two design types together, positive properties of each can be retained while mitigating potential weaknesses. Hybrid space-filling designs, generated as Latin hypercubes augmented with I-optimal points, are compared to designs of each contributing component. A second design type called a bridge design is also evaluated, which further integrates the disparate design types. Bridge designs are the result of a Latin hypercube undergoing coordinate exchange to reach constrained D-optimality, ensuring that there is zero replication of factors in any one-dimensional projection. Lastly, bridge designs were augmented with I-optimal points with two goals in mind. Augmentation with candidate points generated assuming the same underlying analysis model serves to reduce the prediction variance without greatly compromising the space-filling property of the design, while augmentation with candidate points generated assuming a different underlying analysis model can greatly reduce the impact of model misspecification during the design phase. Each of these composite designs are compared to pure space-filling and optimal designs. They typically out-perform pure space-filling designs in terms of prediction variance and alphabetic efficiency, while maintaining comparability with pure optimal designs at small sample size. This justifies them as excellent candidates for initial experimentation.
ContributorsKennedy, Kathryn (Author) / Montgomery, Douglas C. (Thesis advisor) / Johnson, Rachel T. (Thesis advisor) / Fowler, John W (Committee member) / Borror, Connie M. (Committee member) / Arizona State University (Publisher)
Created2013
Description
No-confounding designs (NC) in 16 runs for 6, 7, and 8 factors are non-regular fractional factorial designs that have been suggested as attractive alternatives to the regular minimum aberration resolution IV designs because they do not completely confound any two-factor interactions with each other. These designs allow for potential estimation of main effects and a few two-factor interactions without the need for follow-up experimentation. Analysis methods for non-regular designs is an area of ongoing research, because standard variable selection techniques such as stepwise regression may not always be the best approach. The current work investigates the use of the Dantzig selector for analyzing no-confounding designs. Through a series of examples it shows that this technique is very effective for identifying the set of active factors in no-confounding designs when there are three of four active main effects and up to two active two-factor interactions.
To evaluate the performance of Dantzig selector, a simulation study was conducted and the results based on the percentage of type II errors are analyzed. Also, another alternative for 6 factor NC design, called the Alternate No-confounding design in six factors is introduced in this study. The performance of this Alternate NC design in 6 factors is then evaluated by using Dantzig selector as an analysis method. Lastly, a section is dedicated to comparing the performance of NC-6 and Alternate NC-6 designs.
To evaluate the performance of Dantzig selector, a simulation study was conducted and the results based on the percentage of type II errors are analyzed. Also, another alternative for 6 factor NC design, called the Alternate No-confounding design in six factors is introduced in this study. The performance of this Alternate NC design in 6 factors is then evaluated by using Dantzig selector as an analysis method. Lastly, a section is dedicated to comparing the performance of NC-6 and Alternate NC-6 designs.
ContributorsKrishnamoorthy, Archana (Author) / Montgomery, Douglas C. (Thesis advisor) / Borror, Connie (Thesis advisor) / Pan, Rong (Committee member) / Arizona State University (Publisher)
Created2014
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
The Partition of Variance (POV) method is a simplistic way to identify large sources of variation in manufacturing systems. This method identifies the variance by estimating the variance of the means (between variance) and the means of the variance (within variance). The project shows that the method correctly identifies the variance source when compared to the ANOVA method. Although the variance estimators deteriorate when varying degrees of non-normality is introduced through simulation; however, the POV method is shown to be a more stable measure of variance in the aggregate. The POV method also provides non-negative, stable estimates for interaction when compared to the ANOVA method. The POV method is shown to be more stable, particularly in low sample size situations. Based on these findings, it is suggested that the POV is not a replacement for more complex analysis methods, but rather, a supplement to them. POV is ideal for preliminary analysis due to the ease of implementation, the simplicity of interpretation, and the lack of dependency on statistical analysis packages or statistical knowledge.
ContributorsLittle, David John (Author) / Borror, Connie (Thesis advisor) / Montgomery, Douglas C. (Committee member) / Broatch, Jennifer (Committee member) / Arizona State University (Publisher)
Created2015
Description
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.
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.
ContributorsPark, Anson Robert (Author) / Montgomery, Douglas C. (Thesis advisor) / Mancenido, Michelle V (Thesis advisor) / Escobedo, Adolfo R. (Committee member) / Pan, Rong (Committee member) / Arizona State University (Publisher)
Created2019
Description
Nonregular designs are a preferable alternative to regular resolution four designs because they avoid confounding two-factor interactions. As a result nonregular designs can estimate and identify a few active two-factor interactions. However, due to the sometimes complex alias structure of nonregular designs, standard screening strategies can fail to identify all active effects. In this research, two-level nonregular screening designs with orthogonal main effects will be discussed. By utilizing knowledge of the alias structure, a design based model selection process for analyzing nonregular designs is proposed.
The Aliased Informed Model Selection (AIMS) strategy is a design specific approach that is compared to three generic model selection methods; stepwise regression, least absolute shrinkage and selection operator (LASSO), and the Dantzig selector. The AIMS approach substantially increases the power to detect active main effects and two-factor interactions versus the aforementioned generic methodologies. This research identifies design specific model spaces; sets of models with strong heredity, all estimable, and exhibit no model confounding. These spaces are then used in the AIMS method along with design specific aliasing rules for model selection decisions. Model spaces and alias rules are identified for three designs; 16-run no-confounding 6, 7, and 8-factor designs. The designs are demonstrated with several examples as well as simulations to show the AIMS superiority in model selection.
A final piece of the research provides a method for augmenting no-confounding designs based on a model spaces and maximum average D-efficiency. Several augmented designs are provided for different situations. A final simulation with the augmented designs shows strong results for augmenting four additional runs if time and resources permit.
The Aliased Informed Model Selection (AIMS) strategy is a design specific approach that is compared to three generic model selection methods; stepwise regression, least absolute shrinkage and selection operator (LASSO), and the Dantzig selector. The AIMS approach substantially increases the power to detect active main effects and two-factor interactions versus the aforementioned generic methodologies. This research identifies design specific model spaces; sets of models with strong heredity, all estimable, and exhibit no model confounding. These spaces are then used in the AIMS method along with design specific aliasing rules for model selection decisions. Model spaces and alias rules are identified for three designs; 16-run no-confounding 6, 7, and 8-factor designs. The designs are demonstrated with several examples as well as simulations to show the AIMS superiority in model selection.
A final piece of the research provides a method for augmenting no-confounding designs based on a model spaces and maximum average D-efficiency. Several augmented designs are provided for different situations. A final simulation with the augmented designs shows strong results for augmenting four additional runs if time and resources permit.
ContributorsMetcalfe, Carly E (Author) / Montgomery, Douglas C. (Thesis advisor) / Jones, Bradley (Committee member) / Pan, Rong (Committee member) / Pedrielli, Giulia (Committee member) / Arizona State University (Publisher)
Created2020