Usualmente a análise de sobrevivência considera a modelagem da função da taxa de falha ou função de risco. Uma alternativa a essa visão é estudar a vida residual, que em alguns casos é mais intuitiva do que a função de risco. A vida residual é o tempo de sobrevida adicional de um indivíduo que sobreviveu até um dado instante t0. Este trabalho descreve técnicas semiparamétricas e não paramétricas para estimar a média e a mediana de vida residual em uma população, testes para igualdade dessas medidas em duas populações e também modelos de regressão. Tais técnicas já foram testadas anteriormente em dados com baixa presença de censura; aqui elas são aplicadas a um conjunto de dados de pacientes com insuficiência cardíaca que possui uma alta quantidade de observações censuradas. / Usually, survival analysis is based on the modeling of the hazard function. One alternative approach is to consider the residual life, which would be more intuitive than the hazard function. Residual lifetime is the remaining survival time of a person given he or she survived a given time point t0. We describe semiparametric and non-parametric techniques for mean and median residual life estimation in a one-sample population, as well as tests for two-sample cases and regression models. Such techniques were previously tested for moderate censored data; here we apply them to heart-failure patients data with a high rate of censoring.

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This dissertation consists of five independent projects. In each project, a novel statistical method was developed to address a practical problem encountered in genomic contexts. For example, we considered testing for constant nonparametric effects in a general semiparametric regression model in genetic epidemiology; analyzed the relationship between covariates in the secondary analysis of case-control data; performed model selection in joint modeling of paired functional data; and assessed the prediction ability of genes in gene expression data generated by the CodeLink System from GE. In the first project in Chapter II we considered the problem of testing for constant nonparametric effects in a general semiparametric regression model when there is the potential for interaction between the parametrically and nonparametrically modeled variables. We derived a generalized likelihood ratio test for this hypothesis, showed how to implement it, and gave evidence that it can improve statistical power when compared to standard partially linear models. The second project in Chapter III addressed the issue of score testing for the independence of X and Y in the second analysis of case-control data. The semiparametric efficient approaches can be used to construct semiparametric score tests, but they suffer from a lack of robustness to the assumed model for Y given X. We showed how to adjust the semiparametric score test to make its level/Type I error correct even if the assumed model for Y given X is incorrect, and thus the test is robust. The third project in Chapter IV took up the issue of estimation of a regression function when Y given X follows a homoscedastic regression model. We showed how to estimate the regression parameters in a rare disease case even if the assumed model for Y given X is incorrect, and thus the estimates are model-robust. In the fourth project in Chapter V we developed novel AIC and BIC-type methods for estimating the smoothing parameters in a joint model of paired, hierarchical sparse functional data, and showed in our numerical work that they are many times faster than 10-fold crossvalidation while at the same time giving results that are remarkably close to the crossvalidated estimates. In the fifth project in Chapter VI we introduced a practical permutation test that uses cross-validated genetic predictors to determine if the list of genes in question has “good” prediction ability. It avoids overfitting by using cross-validation to derive the genetic predictor and determines if the count of genes that give “good” prediction could have been obtained by chance. This test was then used to explore gene expression of colonic tissue and exfoliated colonocytes in the fecal stream to discover similarities between the two.

Semiparametric models play important roles in the field of biological statistics. In this dissertation, two types of semiparametic models are to be studied. One is the partially linear model, where the parametric part is a linear function. We are to investigate the two common estimation methods for the partially linear models when the data is correlated — longitudinal or clustered. The other is a semiparametric model where a latent covariate is incorporated in a mixed effects model. We will propose a semiparametric approach for estimation of this model and apply it to the study on colon carcinogenesis. First, we study the profilekernel and backfitting methods in partially linear models for clustered/longitudinal data. For independent data, despite the potential rootn inconsistency of the backfitting estimator noted by Rice (1986), the two estimators have the same asymptotic variance matrix as shown by Opsomer and Ruppert (1999). In this work, theoretical comparisons of the two estimators for multivariate responses are investigated. We show that, for correlated data, backfitting often produces a larger asymptotic variance than the profilekernel method; that is, in addition to its bias problem, the backfitting estimator does not have the same asymptotic efficiency as the profilekernel estimator when data is correlated. Consequently, the common practice of using the backfitting method to compute profilekernel estimates is no longer advised. We illustrate this in detail by following Zeger and Diggle (1994), Lin and Carroll (2001) with a working independence covariance structure for nonparametric estimation and a correlated covariance structure for parametric estimation. Numerical performance of the two estimators is investigated through a simulation study. Their application to an ophthalmology dataset is also described. Next, we study a mixed effects model where the main response and covariate variables are linked through the positions where they are measured. But for technical reasons, they are not measured at the same positions. We propose a semiparametric approach for this misaligned measurements problem and derive the asymptotic properties of the semiparametric estimators under reasonable conditions. An application of the semiparametric method to a colon carcinogenesis study is provided. We find that, as compared with the corn oil supplemented diet, fish oil supplemented diet tends to inhibit the increment of bcl2 (oncogene) gene expression in rats when the amount of DNA damage increases, and thus promotes apoptosis.

Many important scientific questions can not be studied properly using a single measurement as a response. For example, many phenotypes of interest in recent clinical research may be difficult to characterize due to their inherent complexity. It may be difficult to determine the presence or absence of disease based on a single measurement, or even a few measurements, or the phenotype may only be defined based on a series of symptoms. Similarly, a set of related phenotypes or measurements may be studied together in order to detect a shared etiology. In this work, we propose methods for studying complex phenotypes of these types, where the phenotype may be characterized either longitudinally or by a diverse set of continuous, discrete, or not fully observed components. In chapter 1, we seek to identify predictors that are related to multiple components of diverse outcomes. We take up specifically the question of identifying a multiple regulator, where we seek a genetic marker that is associated with multiple biomarkers for autoimmune disease. To do this, we propose sparse multiple regulation testing (SMRT) both to estimate the relationship between a set of predictors and diverse outcomes and to provide a testing framework in which to identify which predictors are associated with multiple elements of the outcomes, while controlling error rates. In chapter 2, we seek to identify risk profiles or risk scores for diverse outcomes, where a risk profile is a linear combination of predictors. The risk profiles will be chosen to be highly correlated to latent traits underlying the outcomes. To do this, we propose semiparametric canonical correlation analysis (sCCA), an updated version of the classical canonical correlation analysis. In chapter 3, the scientific question of interest pertains directly to the progression of disease over time. We provide a testing framework in which to detect the association between a set of genetic markers and the progression of disease in the context of a GWAS. To test for this association while allowing for highly nonlinear longitudinal progression of disease, we propose functional principal variance component (FPVC) testing.

This thesis contributes to the research field of causal inference, where the effect of a treatment on an outcome is of interest is concerned. Many such effects cannot be estimated through randomised experiments. For example, the effect of higher education on future income needs to be estimated using observational data. In the estimation, assumptions are made to make individuals that get higher education comparable with those not getting higher education, to make the effect estimable. Another assumption often made in causal inference (both in randomised an nonrandomised studies) is that the treatment received by one individual has no effect on the outcome of others. If this assumption is not met, the meaning of the causal effect of the treatment may be unclear. In the first paper the effect of college choice on income is investigated using Swedish register data, by comparing graduates from old and new Swedish universities. A semiparametric method of estimation is used, thereby relaxing functional assumptions for the data. One assumption often made in causal inference in observational studies is that individuals in different treatment groups are comparable, given that a set of pretreatment variables have been adjusted for in the analysis. This so called unconfoundedness assumption is in principle not possible to test and, therefore, in the second paper we propose a Bayesian sensitivity analysis of the unconfoundedness assumption. This analysis is then performed on the results from the first paper. In the third paper of the thesis, we study profile likelihood as a tool for semiparametric estimation of a causal effect of a treatment. A semiparametric version of the Bayesian sensitivity analysis of the unconfoundedness assumption proposed in Paper II is also performed using profile likelihood. The last paper of the thesis is concerned with the estimation of direct and indirect causal effects of a treatment where interference between units is present, i.e., where the treatment of one individual affects the outcome of other individuals. We give unbiased estimators of these direct and indirect effects for situations where treatment probabilities vary between individuals. We also illustrate in a simulation study how direct and indirect causal effects can be estimated when treatment probabilities need to be estimated using background information on individuals.

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Many statistical models, like measurement error models, a general class of survival models, and a mixture data model with random censoring, are semiparametric where interest lies in estimating finite-dimensional parameters in the presence of infinite-dimensional nuisance parameters. Developing efficient estimators for the parameters of interest in these models is important because such estimators provide better inferences. For a general regression model with measurement error, we utilize semiparametric theory to develop an unprecedented estimation procedure which delivers consistent estimators even when the model error and latent variable distributions are misspecified. Until now, root-$n$ consistent estimators for this setting were not attainable except for special cases, like a polynomial relationship between the response and mismeasured variables. Through simulation studies and a nutrition study application, we demonstrate that our method outperforms existing methods which ignore measurement error or require a correct model error distribution. In randomized clinical trials, scientists often compare two-sample survival data with a log-rank test. The two groups typically have nonproportional hazards, however, and using a log rank test results in substantial power loss. To ameliorate this issue and improve model efficiency, we propose a model-free strategy of incorporating auxiliary covariates in a general class of survival models. Our approach produces an unbiased, asymptotically normal estimator with significant efficiency gains over current methods. Lastly, we apply semiparametric theory to mixture data models common in kin-cohort designs of Huntington's disease where interest lies in comparing the estimated age-at-death distributions for disease gene carriers and non-carriers. The distribution of the observed, possibly censored, outcome is a mixture of the genotype-specific distributions where the mixing proportions are computed based on the genotypes which are independent of the trait outcomes. Current methods for such data include a Cox proportional hazards model which is susceptible to model misspecification, and two types of nonparametric maximum likelihood estimators which are either inefficient or inconsistent. Using semiparametric theory, we propose an inverse probability weighting estimator (IPW), a nonparametrically imputed estimator and an optimal augmented IPW estimator which provide more reasonable estimates for the age-at-death distributions, and are not susceptible to model misspecification nor poor efficiencies.

Usualmente a análise de sobrevivência considera a modelagem da função da taxa de falha ou função de risco. Uma alternativa a essa visão é estudar a vida residual, que em alguns casos é mais intuitiva do que a função de risco. A vida residual é o tempo de sobrevida adicional de um indivíduo que sobreviveu até um dado instante t0. Este trabalho descreve técnicas semiparamétricas e não paramétricas para estimar a média e a mediana de vida residual em uma população, testes para igualdade dessas medidas em duas populações e também modelos de regressão. Tais técnicas já foram testadas anteriormente em dados com baixa presença de censura; aqui elas são aplicadas a um conjunto de dados de pacientes com insuficiência cardíaca que possui uma alta quantidade de observações censuradas. / Usually, survival analysis is based on the modeling of the hazard function. One alternative approach is to consider the residual life, which would be more intuitive than the hazard function. Residual lifetime is the remaining survival time of a person given he or she survived a given time point t0. We describe semiparametric and non-parametric techniques for mean and median residual life estimation in a one-sample population, as well as tests for two-sample cases and regression models. Such techniques were previously tested for moderate censored data; here we apply them to heart-failure patients data with a high rate of censoring.