Longitudinal studies, where data on study subjects are collected over time, is increasingly involving multivariate longitudinal responses. Frequently, the heterogeneity observed in a multivariate longitudinal response can be attributed to underlying unobserved disease states in addition to any between-subject differences. We propose modeling such disease states using a hidden Markov model (HMM) approach and expand upon previous work, which incorporated random effects into HMMs for the analysis of univariate longitudinal data, to the setting of a multivariate longitudinal response. Multivariate longitudinal data are modeled jointly using separate but correlated random effects between longitudinal responses of mixed data types in addition to a shared underlying hidden process. We use a computationally efficient Bayesian approach via Markov chain Monte Carlo (MCMC) to fit such models. We apply this methodology to bivariate longitudinal response data from a smoking cessation clinical trial. Under these models, we examine how to incorporate a treatment effect on the disease states, as well as develop methods to classify observations by disease state and to attempt to understand patient dropout. Simulation studies were performed to evaluate the properties of such models and their applications under a variety of realistic situations.

In practice, data are frequently incomplete in one way or another. It can be a significant challenge to make valid inferences about the parameters of interest in this situation. In this thesis, three problems involving such data are addressed. The first two problems involve interval-censored life history data with mismeasured covariates. Data of this type are incomplete in two ways. First, the exact event times are unknown due to censoring. Second, the true covariate is missing for most, if not all, individuals. This work focuses primarily on the impact of covariate measurement error in progressive multi-state models with data arising from panel (i.e., interval-censored) observation. These types of problems arise frequently in clinical settings (e.g. when disease progression is of interest and patient information is collected during irregularly spaced clinic visits). Two and three state models are considered in this thesis. This work is motivated by a research program on psoriatic arthritis (PsA) where the effects of error-prone covariates on rates of disease progression are of interest and patient information is collected at clinic visits (Gladman et al. 1995; Bond et al. 2006). Information regarding the error distributions were available based on results from a separate study conducted to evaluate the reliability of clinical measurements that are used in PsA treatment and follow-up (Gladman et al. 2004). The asymptotic bias of covariate effects obtained ignoring error in covariates is investigated and shown to be substantial in some settings. In a series of simulation studies, the performance of corrected likelihood methods and methods based on a simulation-extrapolation (SIMEX) algorithm (Cook \& Stefanski 1994) were investigated to address covariate measurement error. The methods implemented were shown to result in much smaller empirical biases and empirical coverage probabilities which were closer to the nominal levels. The third problem considered involves an extreme case of interval censoring known as current status data. Current status data arise when individuals are observed only at a single point in time and it is then determined whether they have experienced the event of interest. To complicate matters, in the problem considered here, an unknown proportion of the population will never experience the event of interest. Again, this type of data is incomplete in two ways. One assessment is made on each individual to determine whether or not an event has occurred. Therefore, the exact event times are unknown for those who will eventually experience the event. In addition, whether or not the individuals will ever experience the event is unknown for those who have not experienced the event by the assessment time. This problem was motivated by a series of orthopedic trials looking at the effect of blood thinners in hip and knee replacement surgeries. These blood thinners can cause a negative serological response in some patients. This response was the outcome of interest and the only available information regarding it was the seroconversion time under current status observation. In this thesis, latent class models with parametric, nonparametric and piecewise constant forms of the seroconversion time distribution are described. They account for the fact that only a proportion of the population will experience the event of interest. Estimators based on an EM algorithm were evaluated via simulation and the orthopedic surgery data were analyzed based on this methodology.

This research is motivated by a drug discovery problem -- the AIDS anti-viral database from the National Cancer Institute. The objective of the study is to develop effective statistical methods to model the relationship between the chemical structure of a compound and its activity against the HIV-1 virus. And as a result, the structure-activity model can be used to predict the activity of new compounds and thus helps identify those active chemical compounds that can be used as drug candidates. Since active compounds are generally rare in a compound library, we recognize the drug discovery problem as an application of the so-called statistical detection problem. In a typical statistical detection problem, we have data {Xi,Yi}, where Xi is the predictor vector of the ith observation and Yi={0,1} is its class label. The objective of a statistical detection problem is to identify class-1 observations, which are extremely rare. Besides drug discovery problem, other applications of statistical detection include direct marketing and fraud detection. We propose a computationally efficient detection method called LAGO, which stands for "locally adjusted GO estimator". The original idea is inspired by an ancient game known today as "GO". The construction of LAGO consists of two steps. In the first step, we estimate the density of class 1 with an adaptive bandwidth kernel density estimator. The kernel functions are located at and only at the class-1 observations. The bandwidth of the kernel function centered at a certain class-1 observation is calculated as the average distance between this class-1 observation and its K-nearest class-0 neighbors. In the second step, we adjust the density estimated in the first step locally according to the density of class 0. It can be shown that the amount of adjustment in the second step is approximately inversely proportional to the bandwidth calculated in the first step. Application to the NCI data demonstrates that LAGO is superior to methods such as K nearest neighbors and support vector machines. One drawback of the existing LAGO is that it only provides a point estimate of a test point's possibility of being class 1, ignoring the uncertainty of the model. In the second part of this thesis, we present a Bayesian framework for LAGO, referred to as BLAGO. This Bayesian approach enables quantification of uncertainty. Non-informative priors are adopted. The posterior distribution is calculated over a grid of (K, alpha) pairs by integrating out beta0 and beta1 using the Laplace approximation, where K and alpha are two parameters to construct the LAGO score. The parameters beta0, beta1 are the coefficients of the logistic transformation that converts the LAGO score to the probability scale. BLAGO provides proper probabilistic predictions that have support on (0,1) and captures uncertainty of the predictions as well. By avoiding Markov chain Monte Carlo algorithms and using the Laplace approximation, BLAGO is computationally very efficient. Without the need of cross-validation, BLAGO is even more computationally efficient than LAGO.

Estimating functions have been widely used for parameter estimation in various statistical problems. Regular estimating functions produce parameter estimators which have desirable properties, such as consistency and asymptotic normality. In quasi-likelihood inference, an important example of estimating functions, correct specification of the first two moments of the underlying distribution leads to the information unbiasedness, which states that two forms of the information matrix: the negative sensitivity matrix (negative expectation of the first order derivative of an estimating function) and the variability matrix (variance of an estimating function) are equal, or in other words, the analogue of the Fisher information is equivalent to the Godambe information. Consequently, the information unbiasedness indicates that the model-based covariance matrix estimator and sandwich covariance matrix estimator are equivalent. By comparing the model-based and sandwich variance estimators, we propose information ratio (IR) statistics for testing model misspecification of variance/covariance structure under correctly specified mean structure, in the context of linear regression models, generalized linear regression models and generalized estimating equations. Asymptotic properties of the IR statistics are discussed. In addition, through intensive simulation studies, we show that the IR statistics are powerful in various applications: test for heteroscedasticity in linear regression models, test for overdispersion in count data, and test for misspecified variance function and/or misspecified working correlation structure. Moreover, the IR statistics appear more powerful than the classical information matrix test proposed by White (1982). In the literature, model selection criteria have been intensively discussed, but almost all of them target choosing the optimal mean structure. In this thesis, two model selection procedures are proposed for selecting the optimal variance/covariance structure among a collection of candidate structures. One is based on a sequence of the IR tests for all the competing variance/covariance structures. The other is based on an ``information discrepancy criterion" (IDC), which provides a measurement of discrepancy between the negative sensitivity matrix and the variability matrix. In fact, this IDC characterizes the relative efficiency loss when using a certain candidate variance/covariance structure, compared with the true but unknown structure. Through simulation studies and analyses of two data sets, it is shown that the two proposed model selection methods both have a high rate of detecting the true/optimal variance/covariance structure. In particular, since the IDC magnifies the difference among the competing structures, it is highly sensitive to detect the most appropriate variance/covariance structure.

In practice, data are frequently incomplete in one way or another. It can be a significant challenge to make valid inferences about the parameters of interest in this situation. In this thesis, three problems involving such data are addressed. The first two problems involve interval-censored life history data with mismeasured covariates. Data of this type are incomplete in two ways. First, the exact event times are unknown due to censoring. Second, the true covariate is missing for most, if not all, individuals. This work focuses primarily on the impact of covariate measurement error in progressive multi-state models with data arising from panel (i.e., interval-censored) observation. These types of problems arise frequently in clinical settings (e.g. when disease progression is of interest and patient information is collected during irregularly spaced clinic visits). Two and three state models are considered in this thesis. This work is motivated by a research program on psoriatic arthritis (PsA) where the effects of error-prone covariates on rates of disease progression are of interest and patient information is collected at clinic visits (Gladman et al. 1995; Bond et al. 2006). Information regarding the error distributions were available based on results from a separate study conducted to evaluate the reliability of clinical measurements that are used in PsA treatment and follow-up (Gladman et al. 2004). The asymptotic bias of covariate effects obtained ignoring error in covariates is investigated and shown to be substantial in some settings. In a series of simulation studies, the performance of corrected likelihood methods and methods based on a simulation-extrapolation (SIMEX) algorithm (Cook \& Stefanski 1994) were investigated to address covariate measurement error. The methods implemented were shown to result in much smaller empirical biases and empirical coverage probabilities which were closer to the nominal levels. The third problem considered involves an extreme case of interval censoring known as current status data. Current status data arise when individuals are observed only at a single point in time and it is then determined whether they have experienced the event of interest. To complicate matters, in the problem considered here, an unknown proportion of the population will never experience the event of interest. Again, this type of data is incomplete in two ways. One assessment is made on each individual to determine whether or not an event has occurred. Therefore, the exact event times are unknown for those who will eventually experience the event. In addition, whether or not the individuals will ever experience the event is unknown for those who have not experienced the event by the assessment time. This problem was motivated by a series of orthopedic trials looking at the effect of blood thinners in hip and knee replacement surgeries. These blood thinners can cause a negative serological response in some patients. This response was the outcome of interest and the only available information regarding it was the seroconversion time under current status observation. In this thesis, latent class models with parametric, nonparametric and piecewise constant forms of the seroconversion time distribution are described. They account for the fact that only a proportion of the population will experience the event of interest. Estimators based on an EM algorithm were evaluated via simulation and the orthopedic surgery data were analyzed based on this methodology.

This research is motivated by a drug discovery problem -- the AIDS anti-viral database from the National Cancer Institute. The objective of the study is to develop effective statistical methods to model the relationship between the chemical structure of a compound and its activity against the HIV-1 virus. And as a result, the structure-activity model can be used to predict the activity of new compounds and thus helps identify those active chemical compounds that can be used as drug candidates. Since active compounds are generally rare in a compound library, we recognize the drug discovery problem as an application of the so-called statistical detection problem. In a typical statistical detection problem, we have data {Xi,Yi}, where Xi is the predictor vector of the ith observation and Yi={0,1} is its class label. The objective of a statistical detection problem is to identify class-1 observations, which are extremely rare. Besides drug discovery problem, other applications of statistical detection include direct marketing and fraud detection. We propose a computationally efficient detection method called LAGO, which stands for "locally adjusted GO estimator". The original idea is inspired by an ancient game known today as "GO". The construction of LAGO consists of two steps. In the first step, we estimate the density of class 1 with an adaptive bandwidth kernel density estimator. The kernel functions are located at and only at the class-1 observations. The bandwidth of the kernel function centered at a certain class-1 observation is calculated as the average distance between this class-1 observation and its K-nearest class-0 neighbors. In the second step, we adjust the density estimated in the first step locally according to the density of class 0. It can be shown that the amount of adjustment in the second step is approximately inversely proportional to the bandwidth calculated in the first step. Application to the NCI data demonstrates that LAGO is superior to methods such as K nearest neighbors and support vector machines. One drawback of the existing LAGO is that it only provides a point estimate of a test point's possibility of being class 1, ignoring the uncertainty of the model. In the second part of this thesis, we present a Bayesian framework for LAGO, referred to as BLAGO. This Bayesian approach enables quantification of uncertainty. Non-informative priors are adopted. The posterior distribution is calculated over a grid of (K, alpha) pairs by integrating out beta0 and beta1 using the Laplace approximation, where K and alpha are two parameters to construct the LAGO score. The parameters beta0, beta1 are the coefficients of the logistic transformation that converts the LAGO score to the probability scale. BLAGO provides proper probabilistic predictions that have support on (0,1) and captures uncertainty of the predictions as well. By avoiding Markov chain Monte Carlo algorithms and using the Laplace approximation, BLAGO is computationally very efficient. Without the need of cross-validation, BLAGO is even more computationally efficient than LAGO.

Estimating functions have been widely used for parameter estimation in various statistical problems. Regular estimating functions produce parameter estimators which have desirable properties, such as consistency and asymptotic normality. In quasi-likelihood inference, an important example of estimating functions, correct specification of the first two moments of the underlying distribution leads to the information unbiasedness, which states that two forms of the information matrix: the negative sensitivity matrix (negative expectation of the first order derivative of an estimating function) and the variability matrix (variance of an estimating function) are equal, or in other words, the analogue of the Fisher information is equivalent to the Godambe information. Consequently, the information unbiasedness indicates that the model-based covariance matrix estimator and sandwich covariance matrix estimator are equivalent. By comparing the model-based and sandwich variance estimators, we propose information ratio (IR) statistics for testing model misspecification of variance/covariance structure under correctly specified mean structure, in the context of linear regression models, generalized linear regression models and generalized estimating equations. Asymptotic properties of the IR statistics are discussed. In addition, through intensive simulation studies, we show that the IR statistics are powerful in various applications: test for heteroscedasticity in linear regression models, test for overdispersion in count data, and test for misspecified variance function and/or misspecified working correlation structure. Moreover, the IR statistics appear more powerful than the classical information matrix test proposed by White (1982). In the literature, model selection criteria have been intensively discussed, but almost all of them target choosing the optimal mean structure. In this thesis, two model selection procedures are proposed for selecting the optimal variance/covariance structure among a collection of candidate structures. One is based on a sequence of the IR tests for all the competing variance/covariance structures. The other is based on an ``information discrepancy criterion" (IDC), which provides a measurement of discrepancy between the negative sensitivity matrix and the variability matrix. In fact, this IDC characterizes the relative efficiency loss when using a certain candidate variance/covariance structure, compared with the true but unknown structure. Through simulation studies and analyses of two data sets, it is shown that the two proposed model selection methods both have a high rate of detecting the true/optimal variance/covariance structure. In particular, since the IDC magnifies the difference among the competing structures, it is highly sensitive to detect the most appropriate variance/covariance structure.

Correlated data frequently arise from epidemiological studies, especially familial and longitudinal studies. Longitudinal design has been used by researchers to investigate the changes of certain characteristics over time at the individual level as well as how potential factors influence the changes. Familial studies are often designed to investigate the dependence of health conditions among family members. Various models have been developed for this type of multivariate data, and a wide variety of estimation techniques have been proposed. However, data collected from observational studies are often far from perfect, as measurement error may arise from different sources such as defective measuring systems, diagnostic tests without gold references, and self-reports. Under such scenarios only rough surrogate variables are measured. Measurement error in covariates in various regression models has been discussed extensively in the literature. It is well known that naive approaches ignoring covariate error often lead to inconsistent estimators for model parameters. In this thesis, we develop inferential procedures for analyzing correlated data with response measurement error. We consider three scenarios: (i) likelihood-based inferences for generalized linear mixed models when the continuous response is subject to nonlinear measurement errors; (ii) estimating equations methods for binary responses with misclassifications; and (iii) estimating equations methods for ordinal responses when the response variable and categorical/ordinal covariates are subject to misclassifications. The first problem arises when the continuous response variable is difficult to measure. When the true response is defined as the long-term average of measurements, a single measurement is considered as an error-contaminated surrogate. We focus on generalized linear mixed models with nonlinear response error and study the induced bias in naive estimates. We propose likelihood-based methods that can yield consistent and efficient estimators for both fixed-effects and variance parameters. Results of simulation studies and analysis of a data set from the Framingham Heart Study are presented. Marginal models have been widely used for correlated binary, categorical, and ordinal data. The regression parameters characterize the marginal mean of a single outcome, without conditioning on other outcomes or unobserved random effects. The generalized estimating equations (GEE) approach, introduced by Liang and Zeger (1986), only models the first two moments of the responses with associations being treated as nuisance characteristics. For some clustered studies especially familial studies, however, the association structure may be of scientific interest. With binary data Prentice (1988) proposed additional estimating equations that allow one to model pairwise correlations. We consider marginal models for correlated binary data with misclassified responses. We develop “corrected” estimating equations approaches that can yield consistent estimators for both mean and association parameters. The idea is related to Nakamura (1990) that is originally developed for correcting bias induced by additive covariate measurement error under generalized linear models. Our approaches can also handle correlated misclassifications rather than a simple misclassification process as considered by Neuhaus (2002) for clustered binary data under generalized linear mixed models. We extend our methods and further develop marginal approaches for analysis of longitudinal ordinal data with misclassification in both responses and categorical covariates. Simulation studies show that our proposed methods perform very well under a variety of scenarios. Results from application of the proposed methods to real data are presented. Measurement error can be coupled with many other features in the data, e.g., complex survey designs, that can complicate inferential procedures. We explore combining survey weights and misclassification in ordinal covariates in logistic regression analyses. We propose an approach that incorporates survey weights into estimating equations to yield design-based unbiased estimators. In the final part of the thesis we outline some directions for future work, such as transition models and semiparametric models for longitudinal data with both incomplete observations and measurement error. Missing data is another common feature in applications. Developing novel statistical techniques for dealing with both missing data and measurement error can be beneficial.

This thesis considers a variety of statistical issues related to the design and analysis of clinical trials involving multiple lifetime events. The use of composite endpoints, multivariate survival methods with dependent censoring, and recurrent events with dependent termination are considered. Much of this work is based on problems arising in oncology research. Composite endpoints are routinely adopted in multi-centre randomized trials designed to evaluate the effect of experimental interventions in cardiovascular disease, diabetes, and cancer. Despite their widespread use, relatively little attention has been paid to the statistical properties of estimators of treatment effect based on composite endpoints. In Chapter 2 we consider this issue in the context of multivariate models for time to event data in which copula functions link marginal distributions with a proportional hazards structure. We then examine the asymptotic and empirical properties of the estimator of treatment effect arising from a Cox regression model for the time to the first event. We point out that even when the treatment effect is the same for the component events, the limiting value of the estimator based on the composite endpoint is usually inconsistent for this common value. The limiting value is determined by the degree of association between the events, the stochastic ordering of events, and the censoring distribution. Within the framework adopted, marginal methods for the analysis of multivariate failure time data yield consistent estimators of treatment effect and are therefore preferred. We illustrate the methods by application to a recent asthma study. While there is considerable potential for more powerful tests of treatment effect when marginal methods are used, it is possible that problems related to dependent censoring can arise. This happens when the occurrence of one type of event increases the risk of withdrawal from a study and hence alters the probability of observing events of other types. The purpose of Chapter 3 is to formulate a model which reflects this type of mechanism, to evaluate the effect on the asymptotic and finite sample properties of marginal estimates, and to examine the performance of estimators obtained using flexible inverse probability weighted marginal estimating equations. Data from a motivating study are used for illustration. Clinical trials are often designed to assess the effect of therapeutic interventions on occurrence of recurrent events in the presence of a dependent terminal event such as death. Statistical methods based on multistate analysis have considerable appeal in this setting since they can incorporate changes in risk with each event occurrence, a dependence between the recurrent event and the terminal event and event-dependent censoring. To date, however, there has been limited methodology for the design of trials involving recurrent and terminal events, and we addresses this in Chapter 4. Based on the asymptotic distribution of regression coefficients from a multiplicative intensity Markov regression model, we derive sample size formulae to address power requirements for both the recurrent and terminal event processes. Superiority and non-inferiority trial designs are dealt with. Simulation studies confirm that the designs satisfy the nominal power requirements in both settings, and an application to a trial evaluating the effect of a bisphosphonate on skeletal complications is given for illustration.

Correlated data frequently arise from epidemiological studies, especially familial and longitudinal studies. Longitudinal design has been used by researchers to investigate the changes of certain characteristics over time at the individual level as well as how potential factors influence the changes. Familial studies are often designed to investigate the dependence of health conditions among family members. Various models have been developed for this type of multivariate data, and a wide variety of estimation techniques have been proposed. However, data collected from observational studies are often far from perfect, as measurement error may arise from different sources such as defective measuring systems, diagnostic tests without gold references, and self-reports. Under such scenarios only rough surrogate variables are measured. Measurement error in covariates in various regression models has been discussed extensively in the literature. It is well known that naive approaches ignoring covariate error often lead to inconsistent estimators for model parameters. In this thesis, we develop inferential procedures for analyzing correlated data with response measurement error. We consider three scenarios: (i) likelihood-based inferences for generalized linear mixed models when the continuous response is subject to nonlinear measurement errors; (ii) estimating equations methods for binary responses with misclassifications; and (iii) estimating equations methods for ordinal responses when the response variable and categorical/ordinal covariates are subject to misclassifications. The first problem arises when the continuous response variable is difficult to measure. When the true response is defined as the long-term average of measurements, a single measurement is considered as an error-contaminated surrogate. We focus on generalized linear mixed models with nonlinear response error and study the induced bias in naive estimates. We propose likelihood-based methods that can yield consistent and efficient estimators for both fixed-effects and variance parameters. Results of simulation studies and analysis of a data set from the Framingham Heart Study are presented. Marginal models have been widely used for correlated binary, categorical, and ordinal data. The regression parameters characterize the marginal mean of a single outcome, without conditioning on other outcomes or unobserved random effects. The generalized estimating equations (GEE) approach, introduced by Liang and Zeger (1986), only models the first two moments of the responses with associations being treated as nuisance characteristics. For some clustered studies especially familial studies, however, the association structure may be of scientific interest. With binary data Prentice (1988) proposed additional estimating equations that allow one to model pairwise correlations. We consider marginal models for correlated binary data with misclassified responses. We develop “corrected” estimating equations approaches that can yield consistent estimators for both mean and association parameters. The idea is related to Nakamura (1990) that is originally developed for correcting bias induced by additive covariate measurement error under generalized linear models. Our approaches can also handle correlated misclassifications rather than a simple misclassification process as considered by Neuhaus (2002) for clustered binary data under generalized linear mixed models. We extend our methods and further develop marginal approaches for analysis of longitudinal ordinal data with misclassification in both responses and categorical covariates. Simulation studies show that our proposed methods perform very well under a variety of scenarios. Results from application of the proposed methods to real data are presented. Measurement error can be coupled with many other features in the data, e.g., complex survey designs, that can complicate inferential procedures. We explore combining survey weights and misclassification in ordinal covariates in logistic regression analyses. We propose an approach that incorporates survey weights into estimating equations to yield design-based unbiased estimators. In the final part of the thesis we outline some directions for future work, such as transition models and semiparametric models for longitudinal data with both incomplete observations and measurement error. Missing data is another common feature in applications. Developing novel statistical techniques for dealing with both missing data and measurement error can be beneficial.