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.

It is of interest that researchers study competing risks in which subjects may fail from any one of K causes. Comparing any two competing risks with covariate effects is very important in medical studies. This thesis develops omnibus tests for comparing cause-specific hazard rates and cumulative incidence functions at specified covariate levels. In the thesis, the omnibus tests are derived under the additive risk model, that is an alternative to the proportional hazard model, with by a weighted difference of estimates of cumulative cause-specific hazard rates. Simultaneous confidence bands for the difference of two conditional cumulative incidence functions are also constructed. A simulation procedure is used to sample from the null distribution of the test process in which the graphical and numerical techniques are used to detect the significant difference in the risks. A melanoma data set is used for the purpose of illustration.

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.

In the dissertation, we study the statistical evaluation of treatment comparisons by evaluating the relative comparison of survival experiences between two treatment groups. We construct confidence interval and simultaneous confidence bands for the ratio and odds ratio of two survival functions through both parametric and nonparametric approaches.We first construct empirical likelihood confidence interval and simultaneous confidence bands for the odds ratio of two survival functions to address small sample efficacy and sufficiency. The empirical log-likelihood ratio is developed, and the corresponding asymptotic distribution is derived. Simulation studies show that the proposed empirical likelihood band has outperformed the normal approximation band in small sample size cases in the sense that it yields closer coverage probabilities to chosen nominal levels.Furthermore, in order to incorporate prognostic factors for the adjustment of survival functions in the comparison, we construct simultaneous confidence bands for the ratio and odds ratio of survival functions based on both the Cox model and the additive risk model. We develop simultaneous confidence bands by approximating the limiting distribution of cumulative hazard functions by zero-mean Gaussian processes whose distributions can be generated through Monte Carlo simulations. Simulation studies are conducted to evaluate the performance for proposed models. Real applications on published clinical trial data sets are also studied for further illustration purposes.In the end, the population attributable fraction function is studied to measure the impact of risk factors on disease incidence in the population. We develop semiparametric estimation of attributable fraction functions for cohort studies with potentially censored event time under the additive risk model.

In the dissertation, we study the statistical evaluation of treatment comparisons by evaluating the relative comparison of survival experiences between two treatment groups. We construct confidence interval and simultaneous confidence bands for the ratio and odds ratio of two survival functions through both parametric and nonparametric approaches.We first construct empirical likelihood confidence interval and simultaneous confidence bands for the odds ratio of two survival functions to address small sample efficacy and sufficiency. The empirical log-likelihood ratio is developed, and the corresponding asymptotic distribution is derived. Simulation studies show that the proposed empirical likelihood band has outperformed the normal approximation band in small sample size cases in the sense that it yields closer coverage probabilities to chosen nominal levels.Furthermore, in order to incorporate prognostic factors for the adjustment of survival functions in the comparison, we construct simultaneous confidence bands for the ratio and odds ratio of survival functions based on both the Cox model and the additive risk model. We develop simultaneous confidence bands by approximating the limiting distribution of cumulative hazard functions by zero-mean Gaussian processes whose distributions can be generated through Monte Carlo simulations. Simulation studies are conducted to evaluate the performance for proposed models. Real applications on published clinical trial data sets are also studied for further illustration purposes.In the end, the population attributable fraction function is studied to measure the impact of risk factors on disease incidence in the population. We develop semiparametric estimation of attributable fraction functions for cohort studies with potentially censored event time under the additive risk model.

Cox proportional hazard model is often used to estimate the effect of covariates on hazard for censored event times. In this thesis we study the semiparametric models of additive risk for censored data. In this model the hazard is given as a sum of unknown baseline hazard function and a product of covariates and coefficients. Further the general additive-multiplicative model is assumed. In this model the effect of a covariate can be either multiplicative, additive or both at the same time. We focuse on determining the effect of a covariate in the general model. This model can be used to test for the multiplicative or addtive effect of a covariate on the hazard.