With the booming of big complex data, various Statistical methods and Data Science techniques have been developed to retrieve valuable information from them.

The progress is slower with survival data due to the additional difficulty from censoring and truncation. Except for a few straightforward extensions, most modern learning methods have been absent in survival analysis for years since their invention. The theory on the survival version of those methods also falls further behind. There is a strong demand on computational efficient and theoretical reliable methods for big complex data with

time-to-event outcomes

in various Health related fields where immense resource has been poured into.

This thesis is devoted to incorporating censoring and truncation to state-of-art Statistical methodology and theory, to promote the evolution of survival analysis and support Medical research with up-to-date tools. In Chapter 1, I study the mixture cure-rate model with left truncation and right-censoring. We propose a Nonparametric Maximum Likelihood Estimation (NPMLE) approach to effectively handle the truncation issue. We adopt an efficient and stable EM algorithm. We are able to give a closed form variance estimator giving rise to valid inference. In Chapter 2, I study the estimation and inference for the Fine-Gray competing risks model with high-dimensional covariates. We develop confidence intervals based on a one-step bias-correction to an initial regularized estimator. We lay down a methodological and theoretical framework for the one-step bias-corrected estimator with the partial likelihood. In Chapter 3, I study the inference on treatment effect with censored time-to-event outcome while adjusting for high-dimensional covariates. We propose an orthogonal score method to construct honest confidence intervals for the treatment effect. With a slight modification, we obtain a doubly robust estimator extremely tolerant to both estimation inconsistency and volatility. All the methods in aforementioned chapters are tested through extensive numerical experiments

and applied on real data with authentic medical interests.

Aerobic glycolysis (Warburg effect) is accompanied by significant alterations in cellular redox state and constitutes one of the hallmarks of cancer cell metabolism.  Label-free multi-photon microscopy (MPM) methods based on two-photon excited fluorescence (TPEF) have been used extensively to form high-resolution images of redox state in cells and tissues based on intrinsic NADH and FAD+ fluorescence.  However, changes in cellular redox alone are insufficient to fully characterize cancer metabolism and predict invasive potential.  We demonstrate that label-free MPM measurements of TPEF-derived redox state (optical redox ratio, ORR = FAD+/(FAD + NADH)) combined with coherent Raman imaging of lipid formation can be used to quantitatively characterize cancer cells and their relative invasive potential.  In addition, we confirm, using coherent Raman and deuterium labeling methods, that glucose is a significant source for the cellular synthesis of lipid biomass in glycolytic breast cancer cells.  Live cell metabolism was imaged in 3D models of primary mammary epithelial cells (PME) and 2 cancer cell lines, T47D and MDA-MB-231. While we observed overlap in the distribution of the optical redox ratio between these different cell lines, the combination of ORR and lipid volume fraction derived from coherent Raman signals provided complementary independent measures and clear separation. Furthermore, we observed an increase in both lipid synthesis and consumption rates in E2-treated T47D cancer cells cultured in deuterated glucose by tracking the formation and disappearance of deuterated lipids.  These results suggest that due to the relatively wide range of ORR values that reflect the natural diversity of breast cancer cellular redox states, the addition of lipid signatures obtained from coherent Raman imaging can improve our ability to characterize and understand key metabolic features that are hallmarks of the disease.