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- All Subjects: Statistics
- All Subjects: Machine Learning
- Creators: Reiser, Mark R.
- Creators: Montgomery, Douglas C.
Description
By the von Neumann min-max theorem, a two person zero sum game with finitely many pure strategies has a unique value for each player (summing to zero) and each player has a non-empty set of optimal mixed strategies. If the payoffs are independent, identically distributed (iid) uniform (0,1) random variables, then with probability one, both players have unique optimal mixed strategies utilizing the same number of pure strategies with positive probability (Jonasson 2004). The pure strategies with positive probability in the unique optimal mixed strategies are called saddle squares. In 1957, Goldman evaluated the probability of a saddle point (a 1 by 1 saddle square), which was rediscovered by many authors including Thorp (1979). Thorp gave two proofs of the probability of a saddle point, one using combinatorics and one using a beta integral. In 1965, Falk and Thrall investigated the integrals required for the probabilities of a 2 by 2 saddle square for 2 × n and m × 2 games with iid uniform (0,1) payoffs, but they were not able to evaluate the integrals. This dissertation generalizes Thorp's beta integral proof of Goldman's probability of a saddle point, establishing an integral formula for the probability that a m × n game with iid uniform (0,1) payoffs has a k by k saddle square (k ≤ m,n). Additionally, the probabilities of a 2 by 2 and a 3 by 3 saddle square for a 3 × 3 game with iid uniform(0,1) payoffs are found. For these, the 14 integrals observed by Falk and Thrall are dissected into 38 disjoint domains, and the integrals are evaluated using the basic properties of the dilogarithm function. The final results for the probabilities of a 2 by 2 and a 3 by 3 saddle square in a 3 × 3 game are linear combinations of 1, π2, and ln(2) with rational coefficients.
ContributorsManley, Michael (Author) / Kadell, Kevin W. J. (Thesis advisor) / Kao, Ming-Hung (Committee member) / Lanchier, Nicolas (Committee member) / Lohr, Sharon (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2011
Description
Mostly, manufacturing tolerance charts are used these days for manufacturing tolerance transfer but these have the limitation of being one dimensional only. Some research has been undertaken for the three dimensional geometric tolerances but it is too theoretical and yet to be ready for operator level usage. In this research, a new three dimensional model for tolerance transfer in manufacturing process planning is presented that is user friendly in the sense that it is built upon the Coordinate Measuring Machine (CMM) readings that are readily available in any decent manufacturing facility. This model can take care of datum reference change between non orthogonal datums (squeezed datums), non-linearly oriented datums (twisted datums) etc. Graph theoretic approach based upon ACIS, C++ and MFC is laid out to facilitate its implementation for automation of the model. A totally new approach to determining dimensions and tolerances for the manufacturing process plan is also presented. Secondly, a new statistical model for the statistical tolerance analysis based upon joint probability distribution of the trivariate normal distributed variables is presented. 4-D probability Maps have been developed in which the probability value of a point in space is represented by the size of the marker and the associated color. Points inside the part map represent the pass percentage for parts manufactured. The effect of refinement with form and orientation tolerance is highlighted by calculating the change in pass percentage with the pass percentage for size tolerance only. Delaunay triangulation and ray tracing algorithms have been used to automate the process of identifying the points inside and outside the part map. Proof of concept software has been implemented to demonstrate this model and to determine pass percentages for various cases. The model is further extended to assemblies by employing convolution algorithms on two trivariate statistical distributions to arrive at the statistical distribution of the assembly. Map generated by using Minkowski Sum techniques on the individual part maps is superimposed on the probability point cloud resulting from convolution. Delaunay triangulation and ray tracing algorithms are employed to determine the assembleability percentages for the assembly.
ContributorsKhan, M Nadeem Shafi (Author) / Phelan, Patrick E (Thesis advisor) / Montgomery, Douglas C. (Committee member) / Farin, Gerald (Committee member) / Roberts, Chell (Committee member) / Henderson, Mark (Committee member) / Arizona State University (Publisher)
Created2011
Description
It is common in the analysis of data to provide a goodness-of-fit test to assess the performance of a model. In the analysis of contingency tables, goodness-of-fit statistics are frequently employed when modeling social science, educational or psychological data where the interest is often directed at investigating the association among multi-categorical variables. Pearson's chi-squared statistic is well-known in goodness-of-fit testing, but it is sometimes considered to produce an omnibus test as it gives little guidance to the source of poor fit once the null hypothesis is rejected. However, its components can provide powerful directional tests. In this dissertation, orthogonal components are used to develop goodness-of-fit tests for models fit to the counts obtained from the cross-classification of multi-category dependent variables. Ordinal categories are assumed. Orthogonal components defined on marginals are obtained when analyzing multi-dimensional contingency tables through the use of the QR decomposition. A subset of these orthogonal components can be used to construct limited-information tests that allow one to identify the source of lack-of-fit and provide an increase in power compared to Pearson's test. These tests can address the adverse effects presented when data are sparse. The tests rely on the set of first- and second-order marginals jointly, the set of second-order marginals only, and the random forest method, a popular algorithm for modeling large complex data sets. The performance of these tests is compared to the likelihood ratio test as well as to tests based on orthogonal polynomial components. The derived goodness-of-fit tests are evaluated with studies for detecting two- and three-way associations that are not accounted for by a categorical variable factor model with a single latent variable. In addition the tests are used to investigate the case when the model misspecification involves parameter constraints for large and sparse contingency tables. The methodology proposed here is applied to data from the 38th round of the State Survey conducted by the Institute for Public Policy and Michigan State University Social Research (2005) . The results illustrate the use of the proposed techniques in the context of a sparse data set.
ContributorsMilovanovic, Jelena (Author) / Young, Dennis (Thesis advisor) / Reiser, Mark R. (Thesis advisor) / Wilson, Jeffrey (Committee member) / Eubank, Randall (Committee member) / Yang, Yan (Committee member) / Arizona State University (Publisher)
Created2011
Description
Many longitudinal studies, especially in clinical trials, suffer from missing data issues. Most estimation procedures assume that the missing values are ignorable or missing at random (MAR). However, this assumption leads to unrealistic simplification and is implausible for many cases. For example, an investigator is examining the effect of treatment on depression. Subjects are scheduled with doctors on a regular basis and asked questions about recent emotional situations. Patients who are experiencing severe depression are more likely to miss an appointment and leave the data missing for that particular visit. Data that are not missing at random may produce bias in results if the missing mechanism is not taken into account. In other words, the missing mechanism is related to the unobserved responses. Data are said to be non-ignorable missing if the probabilities of missingness depend on quantities that might not be included in the model. Classical pattern-mixture models for non-ignorable missing values are widely used for longitudinal data analysis because they do not require explicit specification of the missing mechanism, with the data stratified according to a variety of missing patterns and a model specified for each stratum. However, this usually results in under-identifiability, because of the need to estimate many stratum-specific parameters even though the eventual interest is usually on the marginal parameters. Pattern mixture models have the drawback that a large sample is usually required. In this thesis, two studies are presented. The first study is motivated by an open problem from pattern mixture models. Simulation studies from this part show that information in the missing data indicators can be well summarized by a simple continuous latent structure, indicating that a large number of missing data patterns may be accounted by a simple latent factor. Simulation findings that are obtained in the first study lead to a novel model, a continuous latent factor model (CLFM). The second study develops CLFM which is utilized for modeling the joint distribution of missing values and longitudinal outcomes. The proposed CLFM model is feasible even for small sample size applications. The detailed estimation theory, including estimating techniques from both frequentist and Bayesian perspectives is presented. Model performance and evaluation are studied through designed simulations and three applications. Simulation and application settings change from correctly-specified missing data mechanism to mis-specified mechanism and include different sample sizes from longitudinal studies. Among three applications, an AIDS study includes non-ignorable missing values; the Peabody Picture Vocabulary Test data have no indication on missing data mechanism and it will be applied to a sensitivity analysis; the Growth of Language and Early Literacy Skills in Preschoolers with Developmental Speech and Language Impairment study, however, has full complete data and will be used to conduct a robust analysis. The CLFM model is shown to provide more precise estimators, specifically on intercept and slope related parameters, compared with Roy's latent class model and the classic linear mixed model. This advantage will be more obvious when a small sample size is the case, where Roy's model experiences challenges on estimation convergence. The proposed CLFM model is also robust when missing data are ignorable as demonstrated through a study on Growth of Language and Early Literacy Skills in Preschoolers.
ContributorsZhang, Jun (Author) / Reiser, Mark R. (Thesis advisor) / Barber, Jarrett (Thesis advisor) / Kao, Ming-Hung (Committee member) / Wilson, Jeffrey (Committee member) / St Louis, Robert D. (Committee member) / Arizona State University (Publisher)
Created2013
Description
Nowadays product reliability becomes the top concern of the manufacturers and customers always prefer the products with good performances under long period. In order to estimate the lifetime of the product, accelerated life testing (ALT) is introduced because most of the products can last years even decades. Much research has been done in the ALT area and optimal design for ALT is a major topic. This dissertation consists of three main studies. First, a methodology of finding optimal design for ALT with right censoring and interval censoring have been developed and it employs the proportional hazard (PH) model and generalized linear model (GLM) to simplify the computational process. A sensitivity study is also given to show the effects brought by parameters to the designs. Second, an extended version of I-optimal design for ALT is discussed and then a dual-objective design criterion is defined and showed with several examples. Also in order to evaluate different candidate designs, several graphical tools are developed. Finally, when there are more than one models available, different model checking designs are discussed.
ContributorsYang, Tao (Author) / Pan, Rong (Thesis advisor) / Montgomery, Douglas C. (Committee member) / Borror, Connie (Committee member) / Rigdon, Steve (Committee member) / Arizona State University (Publisher)
Created2013
Description
Parallel Monte Carlo applications require the pseudorandom numbers used on each processor to be independent in a probabilistic sense. The TestU01 software package is the standard testing suite for detecting stream dependence and other properties that make certain pseudorandom generators ineffective in parallel (as well as serial) settings. TestU01 employs two basic schemes for testing parallel generated streams. The first applies serial tests to the individual streams and then tests the resulting P-values for uniformity. The second turns all the parallel generated streams into one long vector and then applies serial tests to the resulting concatenated stream. Various forms of stream dependence can be missed by each approach because neither one fully addresses the multivariate nature of the accumulated data when generators are run in parallel. This dissertation identifies these potential faults in the parallel testing methodologies of TestU01 and investigates two different methods to better detect inter-stream dependencies: correlation motivated multivariate tests and vector time series based tests. These methods have been implemented in an extension to TestU01 built in C++ and the unique aspects of this extension are discussed. A variety of different generation scenarios are then examined using the TestU01 suite in concert with the extension. This enhanced software package is found to better detect certain forms of inter-stream dependencies than the original TestU01 suites of tests.
ContributorsIsmay, Chester (Author) / Eubank, Randall (Thesis advisor) / Young, Dennis (Committee member) / Kao, Ming-Hung (Committee member) / Lanchier, Nicolas (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2013
Description
Coarsely grouped counts or frequencies are commonly used in the behavioral sciences. Grouped count and grouped frequency (GCGF) that are used as outcome variables often violate the assumptions of linear regression as well as models designed for categorical outcomes; there is no analytic model that is designed specifically to accommodate GCGF outcomes. The purpose of this dissertation was to compare the statistical performance of four regression models (linear regression, Poisson regression, ordinal logistic regression, and beta regression) that can be used when the outcome is a GCGF variable. A simulation study was used to determine the power, type I error, and confidence interval (CI) coverage rates for these models under different conditions. Mean structure, variance structure, effect size, continuous or binary predictor, and sample size were included in the factorial design. Mean structures reflected either a linear relationship or an exponential relationship between the predictor and the outcome. Variance structures reflected homoscedastic (as in linear regression), heteroscedastic (monotonically increasing) or heteroscedastic (increasing then decreasing) variance. Small to medium, large, and very large effect sizes were examined. Sample sizes were 100, 200, 500, and 1000. Results of the simulation study showed that ordinal logistic regression produced type I error, statistical power, and CI coverage rates that were consistently within acceptable limits. Linear regression produced type I error and statistical power that were within acceptable limits, but CI coverage was too low for several conditions important to the analysis of counts and frequencies. Poisson regression and beta regression displayed inflated type I error, low statistical power, and low CI coverage rates for nearly all conditions. All models produced unbiased estimates of the regression coefficient. Based on the statistical performance of the four models, ordinal logistic regression seems to be the preferred method for analyzing GCGF outcomes. Linear regression also performed well, but CI coverage was too low for conditions with an exponential mean structure and/or heteroscedastic variance. Some aspects of model prediction, such as model fit, were not assessed here; more research is necessary to determine which statistical model best captures the unique properties of GCGF outcomes.
ContributorsCoxe, Stefany (Author) / Aiken, Leona S. (Thesis advisor) / West, Stephen G. (Thesis advisor) / Mackinnon, David P (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2012
Description
Value-added models (VAMs) are used by many states to assess contributions of individual teachers and schools to students' academic growth. The generalized persistence VAM, one of the most flexible in the literature, estimates the ``value added'' by individual teachers to their students' current and future test scores by employing a mixed model with a longitudinal database of test scores. There is concern, however, that missing values that are common in the longitudinal student scores can bias value-added assessments, especially when the models serve as a basis for personnel decisions -- such as promoting or dismissing teachers -- as they are being used in some states. Certain types of missing data require that the VAM be modeled jointly with the missingness process in order to obtain unbiased parameter estimates. This dissertation studies two problems. First, the flexibility and multimembership random effects structure of the generalized persistence model lead to computational challenges that have limited the model's availability. To this point, no methods have been developed for scalable maximum likelihood estimation of the model. An EM algorithm to compute maximum likelihood estimates efficiently is developed, making use of the sparse structure of the random effects and error covariance matrices. The algorithm is implemented in the package GPvam in R statistical software. Illustrations of the gains in computational efficiency achieved by the estimation procedure are given. Furthermore, to address the presence of potentially nonignorable missing data, a flexible correlated random effects model is developed that extends the generalized persistence model to jointly model the test scores and the missingness process, allowing the process to depend on both students and teachers. The joint model gives the ability to test the sensitivity of the VAM to the presence of nonignorable missing data. Estimation of the model is challenging due to the non-hierarchical dependence structure and the resulting intractable high-dimensional integrals. Maximum likelihood estimation of the model is performed using an EM algorithm with fully exponential Laplace approximations for the E step. The methods are illustrated with data from university calculus classes and with data from standardized test scores from an urban school district.
ContributorsKarl, Andrew (Author) / Lohr, Sharon L (Thesis advisor) / Yang, Yan (Thesis advisor) / Kao, Ming-Hung (Committee member) / Montgomery, Douglas C. (Committee member) / Wilson, Jeffrey R (Committee member) / Arizona State University (Publisher)
Created2012
Description
This dissertation presents methods for addressing research problems that currently can only adequately be solved using Quality Reliability Engineering (QRE) approaches especially accelerated life testing (ALT) of electronic printed wiring boards with applications to avionics circuit boards. The methods presented in this research are generally applicable to circuit boards, but the data generated and their analysis is for high performance avionics. Avionics equipment typically requires 20 years expected life by aircraft equipment manufacturers and therefore ALT is the only practical way of performing life test estimates. Both thermal and vibration ALT induced failure are performed and analyzed to resolve industry questions relating to the introduction of lead-free solder product and processes into high reliability avionics. In chapter 2, thermal ALT using an industry standard failure machine implementing Interconnect Stress Test (IST) that simulates circuit board life data is compared to real production failure data by likelihood ratio tests to arrive at a mechanical theory. This mechanical theory results in a statistically equivalent energy bound such that failure distributions below a specific energy level are considered to be from the same distribution thus allowing testers to quantify parameter setting in IST prior to life testing. In chapter 3, vibration ALT comparing tin-lead and lead-free circuit board solder designs involves the use of the likelihood ratio (LR) test to assess both complete failure data and S-N curves to present methods for analyzing data. Failure data is analyzed using Regression and two-way analysis of variance (ANOVA) and reconciled with the LR test results that indicating that a costly aging pre-process may be eliminated in certain cases. In chapter 4, vibration ALT for side-by-side tin-lead and lead-free solder black box designs are life tested. Commercial models from strain data do not exist at the low levels associated with life testing and need to be developed because testing performed and presented here indicate that both tin-lead and lead-free solders are similar. In addition, earlier failures due to vibration like connector failure modes will occur before solder interconnect failures.
ContributorsJuarez, Joseph Moses (Author) / Montgomery, Douglas C. (Thesis advisor) / Borror, Connie M. (Thesis advisor) / Gel, Esma (Committee member) / Mignolet, Marc (Committee member) / Pan, Rong (Committee member) / Arizona State University (Publisher)
Created2012
Description
When analyzing longitudinal data it is essential to account both for the correlation inherent from the repeated measures of the responses as well as the correlation realized on account of the feedback created between the responses at a particular time and the predictors at other times. A generalized method of moments (GMM) for estimating the coefficients in longitudinal data is presented. The appropriate and valid estimating equations associated with the time-dependent covariates are identified, thus providing substantial gains in efficiency over generalized estimating equations (GEE) with the independent working correlation. Identifying the estimating equations for computation is of utmost importance. This paper provides a technique for identifying the relevant estimating equations through a general method of moments. I develop an approach that makes use of all the valid estimating equations necessary with each time-dependent and time-independent covariate. Moreover, my approach does not assume that feedback is always present over time, or present at the same degree. I fit the GMM correlated logistic regression model in SAS with PROC IML. I examine two datasets for illustrative purposes. I look at rehospitalization in a Medicare database. I revisit data regarding the relationship between the body mass index and future morbidity among children in the Philippines. These datasets allow us to compare my results with some earlier methods of analyses.
ContributorsYin, Jianqiong (Author) / Wilson, Jeffrey Wilson (Thesis advisor) / Reiser, Mark R. (Committee member) / Kao, Ming-Hung (Committee member) / Arizona State University (Publisher)
Created2012