Theory and Methodology for Semiparametric Linear Models with Censored Data

与著名的Cox模型不同，直接通过某种变换对生存时间进行建模越来越受到实践者的青睐，因为它假设了响应变量和协变量之间的简单关系，具有易于解释的参数。作为特例，用对数变换失效时间的半参数加速失效时间模型在过去的十年中得到了广泛的研究。这些半参数线性变换模型面临的挑战包括：半参数有效估计、具有更现实条件的渐近理论可能导致生存时间预测的良好性质、结果相关加权估计方法的随机积分公式中的可测性问题以及高维数据分析。研究人员提出了新的方法来解决半参数线性变换模型中出现的问题。渐近理论将用现代经验过程理论来证明。所有这些方法的数值实现要么基于那些成熟的离散估计函数算法，要么基于光滑目标函数的牛顿-拉夫森方法，其中无限维参数由平滑估计器逼近。为了提高预测能力，对于高维数据问题，考虑了更灵活的变换。为了获得同时的变量选择和生存时间预测，将研究惩罚方法。本项目中考虑的统计模型在生物学、医学、健康研究和工程等广泛的学科中具有重要的应用。这项拟议的研究特别受到女性生殖生活阶段的多队列研究和密歇根州卵巢癌和肺癌研究的推动，在多队列研究中，绝经年龄的预测是主要的兴趣所在，密歇根州的卵巢癌和肺癌研究利用基因表达数据寻找相关基因和良好的模型来预测患者的生存。它将提供更有效地使用数据和产生更准确预测的方法。它还将允许研究人员在研究生的高级生存分析和半参数模型课程中添加更全面的统计结果。拟议的研究活动将激励研究生成为能够从事基础统计研究的独立研究人员。

Instead of modeling the hazard function for censored survival data as the famous Cox model does, modeling the survival time directly by certain transformation becomes increasingly appealing to practitioners because it postulates a simple relationship between the response variable and covariates with easily interpretable parameters. As a special example, the semiparametric accelerated failure time model that transforms the failure time by logarithm has been studied extensively in the past decade. Existing challenges for those semiparametric linear transformation models include semiparametric efficient estimation, asymptotic theory with more realistic conditions that may lead to good properties for survival time prediction, a measurability issue in the stochastic integral formulation for the outcome-dependent weighted estimating methods, and high-dimensional data analysis. The investigator proposes new methods to tackle those emerging issues in the semiparametric linear transformation models. Asymptotic theories will be proved by using the modern empirical process theory. Numerical implementations of all the proposed methods will be based on either those well developed algorithms for discrete estimating functions or the Newton-Raphson method for smooth objective functions in which the infinite-dimensional parameter is approximated by a smoothing estimator. To enhance the predicting ability, more flexible transformations are considered for problems with high-dimensional data. Penalized method will be investigated in order to obtain simultaneous variable selection and survival time prediction.Statistical models considered in this project have important applications in a wide spectrum of disciplines such as biology, medicine, health studies, and engineering. The proposed research is particularly motivated by the multi-cohort study for the women's reproductive life staging in which the prediction of age at menopause is of major interest, and by the Michigan ovary cancer and lung cancer studies that look for relevant genes and good models for predicting patients' survival using gene expression data. It will provide methods that use data more efficiently and yield more precise prediction. It will also allow the investigator to add more thorough statistical results to the courses of advanced survival analysis and semiparametric models for graduate students. The proposed research activities will motivate graduate students to become independent researchers who are able to engage in fundamental statistical research.