Browsing by Author "Dr. Sujit K. Ghosh, Committee Chair"

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    Bayesian Analysis of Circular Data Using Wrapped Distributions
    (2003-01-27) Ravindran, Palanikumar; Dr. John Monahan, Committee Member; Dr. Sastry Pantula, Committee Member; Dr. Peter Bloomfield, Committee Member; Dr. Sujit K. Ghosh, Committee Chair
    Circular data arise in a number of different areas such as geological, meteorological, biological and industrial sciences. We cannot use standard statistical techniques to model circular data, due to the circular geometry of the sample space. One of the common methods used to analyze such data is the wrapping approach. Using the wrapping approach, we assume that, by wrapping a probability distribution from the real line onto the circle, we obtain the probability distribution for circular data. This approach creates a vast class of probability distributions that are flexible to account for different features of circular data. However, the likelihood-based inference for such distributions can be very complicated and computationally intensive. The EM algorithm used to compute the MLE is feasible, but is computationally unsatisfactory. Instead, we use Markov Chain Monte Carlo (MCMC) methods with a data augmentation step, to overcome such computational difficulties. Given a probability distribution on the circle, we assume that the original distribution was distributed on the real line, and then wrapped onto the circle. If we can "unwrap" the distribution off the circle and obtain a distribution on the real line, then the standard statistical techniques for data on the real line can be used. Our proposed methods are flexible and computationally efficient to fit a wide class of wrapped distributions. Furthermore, we can easily compute the usual summary statistics. We present extensive simulation studies to validate the performance of our method. We apply our method to several real data sets and compare our results to parameter estimates available in the literature. We find that the Wrapped Double Exponential family produces robust parameter estimates with good frequentist coverage probability. We extend our method to the regression model. As an example, we analyze the association between ozone data and wind direction. A major contribution of this dissertation is to illustrate a technique to interpret the circular regression coefficients in terms of the linear regression model setup. Regression diagnostics can be developed after augmenting wrapping numbers to the circular data (refer Section 3.5). We extend our method to fit time-correlated data. We can compute other statistics such as circular autocorrelation functions and their standard errors very easily. We use the Wrapped Normal model to analyze the hourly wind directions, which is an example of the time series circular data.
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    Bayesian Imputation Methods to Measure Quality of Life
    (2003-04-23) Umbach, Amy Therese; Dr. John F. Monahan, Committee Member; Dr. William H. Swallow, Committee Member; Dr. Sujit K. Ghosh, Committee Chair; Dr. David A. Dickey, Committee Member
    The most widely used general health outcomes measure is the SF-36 Health Status questionnaire. The SF-36 is a 36 item general health survey which evaluates eight dimensions of health. This questionnaire is therapeutic non-specific. Often times, an analysis is done to determine if a subject's quality of life is better on one drug than another. This can be beneficial to the patient when selecting a drug and to the company when marketing a drug. The SF-36 form is often used in clinical trials. One problem that is often encountered during a clinical trial is missing data. The industry standard for dealing with missing data of this type might not be the best. The industry standard of evaluating SF-36 data converts the data into eight score functions and treats the score functions as continuous data, even though they are discrete. We take a Bayesian perspective to obtain parameter estimates based on the posterior distribution of the model parameters. We employ Gibbs sampling to obtain simulation-based estimates. One of the practical advantages of our proposed method is that the MCMC method can be implemeted using WinBUGS. WinBUGS is a windows-based software package that is specialized for implementing MCMC-based analysis of full probability models. It allows the user to easily construct models and is available on the World Wide Web. In this thesis, we begin by presenting background information for modeling SF-36 health survey data. We then develop the method of estimating missing responses in quality of life data, taking into account the ordering in the data. We present two simulation studies to validate our proposed method. This method is applied to data from a clinical trial conducted by GlaxoSmithKline Pharmaceutical company. The trial is an open-label, multinational, parallel group study to evaluate the impact of oral Naratriptan 2.5mg on migraines. It has been observed that people in different countries respond differently to the SF-36 questionnaire. In order to account for these differences, we conclude this thesis by fitting an ordinal response model with varying cut-points. One benefit of this type of model is that it allows one to compare treatments across countries.
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    Modeling Mean Residual Life Function Using Scale Mixtures
    (2008-07-23) Liu, Shufang; Dr. Sujit K. Ghosh, Committee Chair; Dr. Wenbin Lu, Committee Member; Dr. Dennis D. Boos, Committee Member; Dr. Daowen Zhang, Committee Member
    The mean residual life function (mrlf) of a subject is defined as the expected remaining (residual) lifetime of the subject given that the subject has survived up to a given time point. It is well known that under mild regularity conditions, an mrlf determines the probability distribution uniquely. Therefore, the mrlf can be used to formulate a statistical model just as it is done with the survival and hazard functions. In practice, the advantage of the mrlf over the more widely used hazard function lies in its interpretation in many applications where the primary goal is often to characterize the remaining life expectancy of a subject instead of the instantaneous failure rate. In this thesis, we first develop a smooth nonparametric estimator of the mean residual life function based on a set of right censored observations. The proposed smooth estimator is obtained by a scale mixture of the empirical estimate of the mrlf. The large sample properties of the estimator are established. A simulation study shows that the proposed scale mixture mean residual life function is more efficient in terms of having lower mean squared error (MSE) than some of the existing estimators available in the literature. Further, as the scale mixture mean residual life function has a closed analytical form, it is computationally less demanding for data with a very large sample size compared to other smooth estimators of the mrlf. Thus the scale mixture estimator of the mean residual life function turns out to be both statistically and computationally more efficient. The scale mixture framework is then extended to the regression model that allows of fixed covariates. The commonly used regression models for the mrlf, such as the proportional mean residual life (PMRL) model and the linear mean residual life (LMRL) model, have limited applications due to ad-hoc restriction on the parameter space. The regression model that we propose does not have any constraint. It turns out that the proposed proportional scaled mean residual life (PSMRL) model is equivalent to the accelerated failure time (AFT) model. We use full likelihood by nonparametrically estimating the baseline mrlf using the smooth scale mixture estimator that we developed earlier. The regression parameters are estimated using an iterative procedure. A simulation study is carried out to assess the properties of the estimates of the regression parameters. We illustrate our regression model by applying it to the well-known Veteran's Administration lung cancer data. Finally, we incorporate time-dependent covariates into our scale mixture framework by extending the AFT model (or the PSMRL model) using a nonparametric mixture of Weibull distributions. A nonparametric Bayesian approach with the Markov Chain Monte Carlo (MCMC) algorithm is used to make the statistical inference for the regression parameter. Unlike the approaches in the literature, our Bayesian approach is not based on the parametric choice of the functions of time for time-dependent covariates and hence does not suffer from the problem of deterministic bias. Our Bayesian approach is also computationally less demanding and more stable compared to the approaches in the literature. A simulation study is carried out to assess the sampling properties of the estimates of the regression parameters. The application of our Bayesian approach to the TUMOR data demonstrates the effectiveness of our approach.
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    Novel Statistical Approaches to Assessing the Risk of QT Prolongation and Sample Size Calculations in 'thorough QT/QTc studies'
    (2009-04-15) Anand, Suraj P.; Dr. Sharon C. Murray, Committee Member; Dr. Sujit K. Ghosh, Committee Chair; Dr. Dennis D. Boos, Committee Member; Dr. Jung-Ying Tzeng, Committee Member; Dr. Wenbin Lu, Committee Member
    ANAND, SURAJ P. Novel Statistical Approaches to Assessing the Risk of QT Prolongation and Sample Size Calculations in ‘thorough QT/QTc studies’. (Under the direction of Professor S. K. Ghosh). The ICH E14 guidelines mandate performing a ‘thorough QT/QTc study’ on any non-antiarrythmic drug, to assess its potential effect on cardiac repolarization, as detected by QT prolongation, before it can be approved and marketed. The standard way of analyzing a thorough QT (TQT) study to assess a drug for its potential for QT prolongation is to construct a 90% two-sided (or a 95% one-sided) confidence interval (CI), for the difference in baseline-corrected mean QTc (heart-rate corrected version of QT) between drug and placebo at each time point, and to conclude non-inferiority if the upper limit for each CI is less than 10 ms. The ICH E14 guidelines define a negative thorough QT study as one in which the upper 95% CI for the maximum time-matched mean effect of the drug as compared to placebo is less than 10 ms. A Monte Carlo simulation-based Bayesian approach is proposed to resolve this problem by constructing a posterior credible interval for the maximum difference parameter. While an interval estimation-based approach may be a way to address the QT prolongation problem, it does not necessarily confirm to the actual intent of the ICH E14 guidelines, which is to establish that the mean effect of the drug is less than 5 ms. Also proposed is a novel Bayesian approach that attempts to directly calculate the probability that the mean effect is no larger than 5 ms, thereby, providing a direct measure of evidence of whether the drug prolongs mean QTc beyond the tolerable threshold of 5 ms. Performance of the proposed approaches has been assessed using simulated data, and illustrations of the methods have been provided through real data sets obtained from TQT studies conducted at GlaxoSmithKline (GSK). Both these proposed methods as well as the other methods for analyzing QTc data are based on multivariate normal models, with common covariance structure for both drug and placebo. Such modeling assumptions may be violated and when the sample sizes are small the statistical inference can be sensitive to such stringent assumptions. A flexible class of parametric models is proposed to address the above-mentioned limitations of the currently used models. A Bayesian methodology is used for data analysis, and model comparisons are performed using the deviance information criterion (DIC). Superior performance of the proposed models over the currently used models is illustrated through a real data set obtained from a GSK-conducted TQT study. Both the proposed methods for analyzing QT data can be extended to this flexible class of models. Another major aspect of TQT studies is the sample size determination. Costs involved in conducting such studies are substantial and hence sample size calculations play a very important role in ensuring a small but adequate TQT study. A variety of methods have been proposed to perform sample size calculations under the frequentist paradigm. Such methods have a limited scope and usually apply in the context of linear mixed models, with some assumed covariance structure for the observations. A sample size determination method, using the proposed novel Bayesian method involving estimation of the probability of concluding a thorough QT study negative, is provided, which would ensure that the total error rate in the context of declaring a TQT study negative is restricted to a desired low level. This method does not rely on any restrictive covariance assumptions.