Statistical models are often used for the prediction of future random variables. There are two types of prediction, point prediction and probabilistic prediction. The prediction accuracy is quantified by performance measures, which are typically based on loss functions. We study the estimators of these performance measures, the prediction error and performance scores, for point and probabilistic predictors, respectively. The focus of this thesis is to assess the prediction performance of survival models that analyze censored survival times. To accommodate censoring, we extend the inverse probability censoring weighting (IPCW) method, thus arbitrary loss functions can be handled. We also develop confidence interval procedures for these performance measures. We compare model-based, apparent loss based and cross-validation estimators of prediction error under model misspecification and variable selection, for absolute relative error loss (in chapter 3) and misclassification error loss (in chapter 4). Simulation results indicate that cross-validation procedures typically produce reliable point estimates and confidence intervals, whereas model-based estimates are often sensitive to model misspecification. The methods are illustrated for two medical contexts in chapter 5. The apparent loss based and cross-validation estimators of performance scores for probabilistic predictor are discussed and illustrated with an example in chapter 6. We also make connections for performance.

Being able to explore a relationship between two life events is of great interest to scientists from different disciplines. Some issues of particular concern are, for example, the connection between smoking cessation and pregnancy (Thompson and Pantoja-Galicia 2003), the interrelation between entry into marriage for individuals in a consensual union and first pregnancy (Blossfeld and Mills 2003), and the association between job loss and divorce (Charles and Stephens 2004, Huang 2003 and Yeung and Hofferth 1998). Establishing causation in observational studies is seldom possible. Nevertheless, if one of two events tends to precede the other closely in time, a causal interpretation of an association between these events can be more plausible. The role of longitudinal surveys is crucial, then, since they allow sequences of events for individuals to be observed. Thompson and Pantoja-Galicia (2003) discuss in this context several notions of temporal association and ordering, and propose an approach to investigate a possible relationship between two lifetime events. In longitudinal surveys individuals might be asked questions of particular interest about two specific lifetime events. Therefore the joint distribution might be advantageous for answering questions of particular importance. In follow-up studies, however, it is possible that interval censored data may arise due to several reasons. For example, actual dates of events might not have been recorded, or are missing, for a subset of (or all) the sampled population, and can be established only to within specified intervals. Along with the notions of temporal association and ordering, Thompson and Pantoja-Galicia (2003) also discuss the concept of one type of event "triggering" another. In addition they outline the construction of tests for these temporal relationships. The aim of this thesis is to implement some of these notions using interval censored data from longitudinal complex surveys. Therefore, we present some proposed tools that may be used for this purpose. This dissertation is divided in five chapters, the first chapter presents a notion of a temporal relationship along with a formal nonparametric test. The mechanisms of right censoring, interval censoring and left truncation are also overviewed. Issues on complex surveys designs are discussed at the end of this chapter. For the remaining chapters of the thesis, we note that the corresponding formal nonparametric test requires estimation of a joint density, therefore in the second chapter a nonparametric approach for bivariate density estimation with interval censored survey data is provided. The third chapter is devoted to model shorter term triggering using complex survey bivariate data. The semiparametric models in Chapter 3 consider both noncensoring and interval censoring situations. The fourth chapter presents some applications using data from the National Population Health Survey and the Survey of Labour and Income Dynamics from Statistics Canada. An overall discussion is included in the fifth chapter and topics for future research are also addressed in this last chapter.

Statistical models are often used for the prediction of future random variables. There are two types of prediction, point prediction and probabilistic prediction. The prediction accuracy is quantified by performance measures, which are typically based on loss functions. We study the estimators of these performance measures, the prediction error and performance scores, for point and probabilistic predictors, respectively. The focus of this thesis is to assess the prediction performance of survival models that analyze censored survival times. To accommodate censoring, we extend the inverse probability censoring weighting (IPCW) method, thus arbitrary loss functions can be handled. We also develop confidence interval procedures for these performance measures. We compare model-based, apparent loss based and cross-validation estimators of prediction error under model misspecification and variable selection, for absolute relative error loss (in chapter 3) and misclassification error loss (in chapter 4). Simulation results indicate that cross-validation procedures typically produce reliable point estimates and confidence intervals, whereas model-based estimates are often sensitive to model misspecification. The methods are illustrated for two medical contexts in chapter 5. The apparent loss based and cross-validation estimators of performance scores for probabilistic predictor are discussed and illustrated with an example in chapter 6. We also make connections for performance.

We first show theoretically and in simulation how power varies as a function of SNP correlation structure with currently-implemented gene-based testing methods. We propose alternative testing methods whose power does not vary with the correlation structure. We then propose hypothesis tests for detecting prevalence-incidence bias in case-control studies, a bias perhaps overrepresented in GWAS due to currently used study designs. Lastly, we hypothesize how different incentive structures used to keep clinical trial participants in studies may interact with a background of dependent censoring and result in variation in the bias of the Kaplan-Meier survival curve estimator.

In this thesis, we consider the problem of constructing an additive risk model based on the right censored survival data to predict the survival times of the cancer patients, especially when the dimension of the covariates is much larger than the sample size. For microarray Gene Expression data, the number of gene expression levels is far greater than the number of samples. Such ¡°small n, large p¡± problems have attracted researchers to investigate the association between cancer patient survival times and gene expression profiles for recent few years. We apply Partial Least Squares to reduce the dimension of the covariates and get the corresponding latent variables (components), and these components are used as new regressors to fit the extensional additive risk model. Also we employ the time dependent AUC curve (area under the Receiver Operating Characteristic (ROC) curve) to assess how well the model predicts the survival time. Finally, this approach is illustrated by re-analysis of the well known AML data set and breast cancer data set. The results show that the model fits both of the data sets very well.

One problem of interest is to relate genes to survival outcomes of patients for the purpose of building regression models to predict future patients¡¯ survival based on their gene expression data. Applying semeparametric additive risk model of survival analysis, this thesis proposes a new approach to conduct the analysis of gene expression data with the focus on model¡¯s predictive ability. The method modifies the correlation principal component regression to handle the censoring problem of survival data. Also, we employ the time dependent AUC and RMSEP to assess how well the model predicts the survival time. Furthermore, the proposed method is able to identify significant genes which are related to the disease. Finally, this proposed approach is illustrated by simulation data set, the diffuse large B-cell lymphoma (DLBCL) data set, and breast cancer data set. The results show that the model fits both of the data sets very well.

Use of progression-free survival in the evaluation of clinical interventions is hampered by a variety of issues, including censoring patterns not addressed in the usual methods for survival analysis. Progression can be right-censored before survival or interval-censored between inspection times. Current practice calls for imputing events to their time of detection. Such an approach is prone to bias, underestimates standard errors and makes inefficient use of the data at hand. Moreover a composite outcome prevents inference about the actual treatment effect on the risk of progression. This thesis develops semiparametric and sieve maximum likelihood estimators to more formally analyze progression-related endpoints. For the special case where death rarely precedes progression, a Cox-Aalen model is proposed for regression analysis of time-to-progression under intermittent inspection. The general setting considering both progression and survival is examined with a Markov Cox-type illness-death model under various censoring schemes. All of the resulting estimators globally converge to the truth slower than the parametric rate, but their finite-dimensional components are asymptotically efficient. Numerical studies suggest that the new methods perform better than their imputation-based alternatives under moderate to large samples having higher rates of censoring.

Cross-sectional surveys are often used in epidemiological studies to identify subjects with a disease. When estimating the survival function from onset of disease, this sampling mechanism introduces bias, which must be accounted for. If the onset times of the disease are assumed to be coming from a stationary Poisson process, this bias, which is caused by the sampling of prevalent rather than incident cases, is termed length-bias. A one-sample Kolomogorov-Smirnov type of goodness-of-fit test for right-censored length-biased data is proposed and investigated with Weibull, log-normal and log-logistic models. Algorithms detailing how to efficiently generate right-censored length-biased survival data of these parametric forms are given. Simulation is employed to assess the effects of sample size and censoring on the power of the test. Finally, the test is used to evaluate the goodness-of-fit using length-biased survival data of patients with dementia from the Canadian Study of Health and Aging.

Nesta tese, apresenta-se um modelo de confiabilidade estatística para aplicação em dados de vida de um produto, buscando classificar três modos de falhas distintos associados à ocorrência de falhas prematuras, aleatórias e por desgaste. A ocorrência dos três modos de falhas segue os princípios de aplicação dos modelos teóricos por riscos concorrentes e seccionais. O modelo proposto utiliza duas distribuições de Weibull, com dois e três parâmetros, e uma distribuição exponencial. A distribuição de Weibull com dois parâmetros tem por objetivo representar os modos de falhas prematuras: a distribuição de Weibull com três parâmetros busca capturar os modos de falhas por desgaste; a distribuição exponencial mede a ocorrência de falhas aleatórias decorrentes de uso operacional de um produto. Considera-se que falhas prematuras e por desgaste ocorram seqüencialmente, enquanto falhas aleatórias ocorram de forma concorrente às falhas prematuras e por desgaste tão logo o produto seja colocado em operação. Para dimensionar o número de ocorrências vinculadas aos três modos de falhas são utilizados dados coletados durante o período de garantia e pós-garantia. Os dados de garantia são registros históricos do produtor e os dados da pós-garantia referem-se a informações obtidas de especialistas, já que dados após a garantia apresentam elevado nível de censura. Equações de confiabilidade e estimadores de máxima verossimilhança são apresentados para definir o perfil e os parâmetros do modelo proposto. Um estudo de caso com dados coletados de um equipamento elétrico-eletrônico subsidia a aplicação do modelo enquanto que um teste estatístico de ajuste de dados é utilizado para validar o referido modelo. / This thesis presents a reliability model for product life data presenting three different failure modes, associated with early, random and wear-out failures. The model is based on theoretical concepts related to competing risk and sectional models. The proposed model is structured based on two Weibull distributions, with two and three parameters, and one exponential distribution. The Weibull distribution with two parameters is aimed at modeling early failure modes; the Weibull distribution with three parameters models wear-out failure modes; the exponential distribution models random failures due to operational use. It is considered that early and wearout failures take place one after the other while random failures occur at the same time as early and wear-out failures as soon as the product starts operating. To measure each period related to the three failure modes, data from warranty and post-warranty periods are used. Warranty data are historical records; post-warranty data are gathered from experts, and are aimed at decreasing the degree of censoring in the data. Once the model is defined, reliability figures and maximum likelihood estimators are derived. Real data obtained from warranty claims on electricelectronic equipments are used to illustrate the developments proposed and a goodness-of-fit test is used to validate the performance of this model.

We present a theoretical approach to the statistical analysis of the dependence of the gap time length between consecutive recurrent events, on a set of explanatory random variables and in the presence of right censoring. The dependence is expressed through regression-like and overdispersion parameters, estimated via estimating functions and equations. The mean and variance of the length of each gap time, conditioned on the observed history of prior events and other covariates, are known functions of parameters and covariates, and are part of the estimating functions. Under certain conditions on censoring, we construct normalized estimating functions that are asymptotically unbiased and contain only observed data. We then use modern mathematical techniques to prove the existence, consistency and asymptotic normality of a sequence of estimators of the parameters. Simulations support our theoretical results.