A Semi-parametric Approach for Analyzing Longitudinal Measurements with Non-ignorable Missingness Using Regression Spline

In longitudinal studies with missingness, shared parameter models (SPM) provide appropriate framework for the joint modeling of the measurements and missingness process. These models use a set of random effects to account for the interdependence between two processes. Sometimes the longitudinal responses may not be fitted well by using a linear model and some non-parametric methods have to be used. Also, parametric assumptions are typically made for the random effects distribution, and violation of those may affect the parameter estimates and standard errors. To overcome these problems, we propose a semi-parametric model for the joint modelling of longitudinal markers and a missing not at random mechanism. In this model, because of the flexibility in nonparametric regression models, the relationship between the response variables and the covariates has been modeled by semi-parametric mixed effect model. Also, we do not assume any parametric assumption for the random effects distribution and we allow it to be unspecified. The parameter estimations are made using a vertex exchange method. In order to evaluate the performance of the proposed model, we compare SPM using regression spline (Spline-SPM) and semi-parametric SPM (SpSPM) models. We also conduct a simulation study with different parametric assumptions for the random effects distribution. A real example from a recent HIV study is analyzed for illustration of the proposed approach

Multiple Imputation and Quantile Regression Methods for Biomarker Data subject to Detection Limits

Biomarkers are increasingly used in biomedical studies to better understand the natural history and development of a disease, identify the patients at high-risk and guide the therapeutic strategies for intervention. However, the measurement of these markers is often limited by the sensitivity of the given assay, resulting in data that are censored either at the lower limit or upper limit of detection. Ignoring censoring issue in any analysis may lead to the biased results. For a regression analysis where multiple censored biomarkers are included as predictors, we develop multiple imputation methods based on Gibbs sampling approach. The simulation study shows that our method significantly reduces the estimation bias as compared to the other simple imputation methods when the correlation between markers is high or the censoring proportion is high. The likelihood based mean regression for repeatedly measured biomarkers often assume a multivariate normal distribution that may not hold for biomarker data even after transformations. We consider a robust alternative, median regression, for censored longitudinal data. We develop an estimating equation approach that can incorporate the serial correlations between repeated measurements. We conduct simulation studies to evaluate the proposed estimators and compare median regression model with the mixed models under various specifications of distributions and covariance structures. Missing data is a common problem with longitudinal study. Under the assumptions that the missing pattern is monotonic and the missingness may only depend on the observed data, we propose a weighted estimating equation approach for the censored quantile regression models. The contribution of each individual to the estimating equation is weighted by the inverse probability of dropout at the given occasion. The resultant regression estimators are consistent when the dropout process is correctly specified. The performance of our estimating procedure is evaluated via simulation study. We illustrate all the proposed methods using the biomarker data of the Genetic and Inflammatory Markers of Sepsis (GenIMS) study. Appropriate handling of censored data in biomarker analysis is of public health importance because it will improve the understanding of the biological mechanisms of the underlying disease and aid in the successful development of future effective treatments

Some Aspects of Change Point Analysis

In the first part of the dissertation, we propose two tests with the purpose of detecting change point in a sequence of independent random variables. Both the consistency and rate of convergence of the estimated change point are established. We then extend the application of the proposed test in the field of multiple change points detection problem. Simulation studies and real data analysis are given to examine the performance of our proposed methods. In the second part of the dissertation, we propose a procedure for detecting multiple change points in a mean-shift model, where the number of change points is allowed to increase with the sample size. A theoretic justification for our new method is also given. The proposed procedure is implemented in an algorithm which, compared to two popular methods via simulation studies, demonstrates satisfactory performance in terms of accuracy, stability and computational complexity. Finally, we apply our new algorithm to analyze two real data examples. In the third part of the dissertation, our research is motivated by HIV viral dynamic studies. We jointly model HIV viral dynamics, CD4 process with measurement errors and change point model, and estimate the model parameters simultaneously via the Monte Carlo EM approach and hierarchical likelihood approximation approach. These approaches are illustrated in a real data example. Simulation results show that both of these two methods perform well and are much better than the commonly used naive method

Joint modeling of longitudinal and survival data with missing and left-censored time-varying covariates

We propose a joint model for longitudinal and survival data with time-varying covariates subject to detection limits and intermittent missingness at random (MAR). The model is motivated by data from the Multicenter AIDS Cohort Study (MACS), in which HIV+ subjects have viral load and CD4 cell count measured at repeated visits along with survival data. We model the longitudinal component using a normal linear mixed model, modeling the trajectory of CD4 cell count by regressing on viral load and other covariates. The viral load data are subject to both left-censoring due to detection limits (17%) and intermittent missingness (27%). The survival component of the joint model is a Cox model with time-dependent covariates for death due to AIDS. The longitudinal and survival models are linked using the trajectory function of the linear mixed model. A Bayesian analysis is conducted on the MACS data using the proposed joint model. The proposed method is shown to improve the precision of estimates when compared to alternative methods

Joint modeling of longitudinal and survival outcomes using generalized estimating equations

Indiana University-Purdue University Indianapolis (IUPUI)Joint models for longitudinal and time-to-event data has been introduced to study the association between repeatedly measured exposures and the risk of an event. The use of joint models allows a survival outcome to depend on some characteristic functions from the longitudinal measures. Current estimation methods include a two-stage approach, Bayesian and maximum likelihood estimation (MLEs) methods. The twostage method is computationally straightforward but often yields biased estimates. Bayesian and MLE methods rely on the joint likelihood of longitudinal and survival outcomes and can be computationally intensive. In this work, we propose a joint generalized estimating equation framework using an inverse intensity weighting approach for parameter estimation from joint models. The proposed method can be used to longitudinal outcomes from the exponential family of distributions and is computationally e cient. The performance of the proposed method is evaluated in simulation studies. The proposed method is used in an aging cohort to determine the relationship between longitudinal biomarkers and the risk of coronary artery disease

Bayesian modeling and inference for asymmetric responses with applications

University of Minnesota Ph.D. dissertation. July 2017. Major: Biostatistics. Advisors: Dipankar Bandyopadhyay, Lynn Eberly. 1 computer file (PDF); x, 129 pages.Analysis of asymmetric data poses several unique challenges. In this thesis, we propose a series of parametric models under the Bayesian hierarchical framework to account for asymmetry (arising from non-Gaussianity, tail behavior, etc) in both continuous and discrete response data. First, we model continuous asymmetric responses assuming normal random errors by using a dynamic linear model discretized from a differential equation which absorbs the asymmetry from the data generation mechanism. We then extend the skew-normal/independent parametric family to accommodate spatial clustering and non-random missingness observed in asymmetric continuous responses, and demonstrate its utility in obtaining precise parameter estimates and prediction in presence of skewness and thick-tails. Finally, under a latent variable formulation, we use a generalized extreme value (GEV) link to model multivariate asymmetric spatially-correlated binary responses that also exhibit non-random missingness, and show how this proposal improves inference over other popular alternative link functions in terms of bias and prediction. We assess our proposed method via simulation studies and two real data analyses on public health. Using simulated data, we investigate the performance of the proposed method to accurately accommodate asymmetry along with other data features such as spatial dependency and non-random missingness simultaneously, leading to precise posterior parameter estimates. Regarding data illustrations, we first validate the efficiency in using differential equations to handle skewed exposure assessment responses derived from an occupational hygiene study. Furthermore, we also conduct efficient risk evaluation of various covariates on periodontal disease responses from a dataset on oral epidemiology. The results from our investigation re-establishes the significance of moving away from the normality assumption and instead consider pragmatic distributional assumptions on the random model terms for efficient Bayesian parameter estimation under a unified framework with a variety of data complexities not earlier considered in the two aforementioned areas of public health research

Joint Modelling Inference for Longitudinal and Time To Event Data with Application to Biomarkers in Medical and Clinical Studies

In the past couple of decades, longitudinal and survival data analysis have emerged as important and popular concepts of biostatistics and statistics for disease modelling. In recent years, these two statistics concepts have been combined to develop a joint model for longitudinal and survival data analysis. The Joint model is a simultaneous modelling application of longitudinal and survival data while taking into account a possible association between them. In this thesis, three sub-topics (Conditional score approach, estimating equation approach, and modified Cholesky decomposition approach) are utilised to model the association if the independence assumption is violated. Using the conditional score approach, the study investigated the association between longitudinal covariates and the time-to-event process to examine the within-subject measurement error that could influence estimation when the assumption of normality and mutual independence is violated. Given the assumption violation, I proposed an estimating equation approach based on the conditional score to relax parametric distributional assumptions for repeated measures of random effects. I jointly modelled the time-dependent biomarkers and event times using the Cox model with intermittent time-dependent covariates measure, in which the longitudinal model was used to characterize the biomarker underlying (unobservable) trajectory and incorporated as a latent time-dependent covariate in the survival model to predict failure times. Estimates of the parameters were obtained by a restricted maximum likelihood estimate (REML). A modified Cholesky decomposition method was used to capture the within-subject covariance for a positive definite and symmetric matrix, with the assumption that the observed data from different subjects are independent. I illustrated the proposed method by a real data set from a lung study and simulation. An extension to the joint model of longitudinal-survival data was also proposed, in which the longitudinal data has a cumulative and weighted effect on the hazard event function. Using a Bayesian parametric method, I proposed a skewed weighted probability density function to estimate the parameters. The weighted cumulative effect used enabled different longitudinal profiles to be incorporated over time in calculating the hazard ratio between the subjects. The proposed functions provide greater flexibility for modelling the association structure of different longitudinal and survival sub-model. The focus was on the association between the biomarker (serum creatinine, sCr) and the development of end-stage renal disease (ESRD). Since the effect of sCr biomarker is anticipated to be a cumulative effect, with the development of sCr biomarker over time leading to progressively higher damage of the kidney. The approach was applied a simulation for validation of the proposed metho

Joint Modelling Inference for Longitudinal and Time To Event Data with Application to Biomarkers in Medical and Clinical Studies

In the past couple of decades, longitudinal and survival data analysis have emerged as important and popular concepts of biostatistics and statistics for disease modelling. In recent years, these two statistics concepts have been combined to develop a joint model for longitudinal and survival data analysis. The Joint model is a simultaneous modelling application of longitudinal and survival data while taking into account a possible association between them. In this thesis, three sub-topics (Conditional score approach, estimating equation approach, and modified Cholesky decomposition approach) are utilised to model the association if the independence assumption is violated. Using the conditional score approach, the study investigated the association between longitudinal covariates and the time-to-event process to examine the within-subject measurement error that could influence estimation when the assumption of normality and mutual independence is violated. Given the assumption violation, I proposed an estimating equation approach based on the conditional score to relax parametric distributional assumptions for repeated measures of random effects. I jointly modelled the time-dependent biomarkers and event times using the Cox model with intermittent time-dependent covariates measure, in which the longitudinal model was used to characterize the biomarker underlying (unobservable) trajectory and incorporated as a latent time-dependent covariate in the survival model to predict failure times. Estimates of the parameters were obtained by a restricted maximum likelihood estimate (REML). A modified Cholesky decomposition method was used to capture the within-subject covariance for a positive definite and symmetric matrix, with the assumption that the observed data from different subjects are independent. I illustrated the proposed method by a real data set from a lung study and simulation. An extension to the joint model of longitudinal-survival data was also proposed, in which the longitudinal data has a cumulative and weighted effect on the hazard event function. Using a Bayesian parametric method, I proposed a skewed weighted probability density function to estimate the parameters. The weighted cumulative effect used enabled different longitudinal profiles to be incorporated over time in calculating the hazard ratio between the subjects. The proposed functions provide greater flexibility for modelling the association structure of different longitudinal and survival sub-model. The focus was on the association between the biomarker (serum creatinine, sCr) and the development of end-stage renal disease (ESRD). Since the effect of sCr biomarker is anticipated to be a cumulative effect, with the development of sCr biomarker over time leading to progressively higher damage of the kidney. The approach was applied a simulation for validation of the proposed metho