Browsing by Author "Dr. Daowen Zhang, Committee Member"

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Adjustment for Measurement Error
(2009-11-02) Elliott, Laine E; Dr. Anastasios Tsiatis, Committee Member; Dr. Daowen Zhang, Committee Member; Dr. Marie Davidian, Committee Co-Chair; Dr. Len Stefanski, Committee Co-Chair
A variety of complications arise when imperfect measurements, W, are observed in place of a true variable of interest, X. In the context of linear and non-linear regression models where X is a covariate, regression parameter estimators obtained when W is substituted for X may be substantially biased. Many strategies for correcting for measurement error depend on the specific modeling or regression context and can be intractable in highly non-linear models. In addition, previous methods often assume that the measurement error is normally distributed. In our work, we focus on re-creating the distribution of X from the observed W, either as the primary quantity of interest or as a means to improving parameter estimation. We obtain estimators of X for which the first M sample moments are unbiased for the corresponding moments of X. We investigate the benefit of substituting these estimates in density estimation, logistic regression and survival models. We compare this moment adjusted imputation (MAI) approach to existing alternatives in applications with normally distributed measurement error. We identify an important case of chi-square measurement error and propose a variety of methods to adjust for it, including a version of MAI. We find that MAI is often superior and has the advantage that once the estimates of X are obtained, they can be substituted in any model, including complicated non-linear models.
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.
Nonparametric and semiparametric inference about ROC curves
(2008-07-06) Gu, Jiezhun; Dr. Daowen Zhang, Committee Member; Dr. Subhashis Ghosal, Committee Chair; Dr. Bibhuti Bhattacharyya, Committee Member; Dr. Montserrat Fuentes, Committee Member
Optimal Two-stage designs in Phase-II Clinical Trials.
(2006-08-17) Banerjee, Anindita; Dr. Dennis Boos, Committee Member; Dr. Daowen Zhang, Committee Member; Dr. Marie Davidian, Committee Member; Dr. Anastasios A. Tsiatis, Committee Chair
Two-stage designs have been widely used in phase II clinical trials. Such designs are desirable because they allow a decision to be made on whether a treatment is effective or not after the accumulation of the data at the end of each stage. Optimal fixed two-stage designs, where the sample size at each stage is fixed in advance, were proposed by Simon when the primary outcome is a binary response. We propose an adaptive two-stage design which allows the sample size at the second stage to depend on the results at the first stage. Using a Bayesian decision theoretic construct, we derive optimal adaptive two-stage designs. The optimality criterion is to minimize the expected sample size under the null hypothesis value. We further explore optimal adaptive designs that minimize the expected sample size at the alternative hypothesis, at a probability mid-point between the null and alternative hypotheses and a weighted combination of the null, alternative and mid-point value. We also construct an envelope function that gives the lowest expected sample size for any possible value of the response probability. The different designs are compared to Simon's design as well as the envelope function. The designs that minimize the expected sample size at the mid-point between the null and alternative hypotheses and the design that minimizes a weighted average of the response probabilities are closer to the envelope function. Results show that these designs perform better across a range of the response probability values, and generally surpass Simon's design.
Sparse Estimation and Inference for Censored Median Regression
(2009-07-20) Shows, Justin Hall; Dr. Wenbin Lu, Committee Chair; Dr. Hao Helen Zhang, Committee Co-Chair; Dr. Dennis Boos, Committee Member; Dr. Daowen Zhang, Committee Member
Censored median regression models have been shown to be useful for analyzing a variety of censored survival data with the robustness property. We study sparse estimation and inference of censored median regression. The new method minimizes an inverse censoring probability weighted least absolute deviation subject to the adaptive LASSO penalty. We show that, with a proper choice of the tuning parameter, the proposed estimator has nice theoretical properties such as root-n consistency and asymptotic normality. The estimator can also identify the underlying sparse model consistently. We propose using a resampling method to estimate the variance of the proposed estimator. Furthermore, the new procedure enjoys great advantages in computation, since its entire solution path can be obtained efficiently. Also, the method can be extended to multivariate survival data, where there is a natural or artificial clustering structure. The performance of our estimator is evaluated by extensive simulations and two real data applications.