Topics in statistical analysis with missing data

回归模型广泛用于研究响应变量和一组协变量之间的关联。标准的推理方法要求不存在选择偏差，并且所选受试者的变量得到充分观察，没有测量误差。然而，在实践中，许多研究涉及选择偏差、缺失或误测数据的某些组合，这些都是由设计或偶然造成的。采用复杂抽样设计，包括两阶段或多阶段设计（TMD）和部分问卷设计（PQD），降低数据收集成本，同时提高数据质量；此类设计生成的数据中某些变量以单调模式（例如，TMD）或非单调模式（例如，PQD）随机丢失。偶然缺失可能发生在任何研究中，包括设计的实验，其中缺失通常不是随机的并且以非单调模式出现。拟议研究计划的两个目标是：（i）开发和应用数据随机丢失（非随机）和任意模式数据丢失的回归模型的统计方法，以及（ii）调查和应用两阶段和多阶段研究的最佳抽样策略以及最大化观测数据信息的 PQD。 (i) 下探讨的模型将包括广义线性模型、Cox 比例风险模型和部分线性模型，其中某些变量（响应或协变量）缺失，并且辅助信息有时可用。我们最近在该领域基于参数和半参数模型的研究对于响应和协变量缺失问题取得了一些有希望的结果。参数工作回归模型（Chen and Chen，2000；Lawless and Kalbfleisch，2011；Zhao et al.，2013）可以有效地利用不完整观测和辅助变量的信息来提高估计效率，并对随机情况下的缺失和某些非随机情况下的缺失产生无偏估计。可以开发参数和半参数程序（Chen et al., 2011）来提高常用多重插补方法的一致性，并进一步增强多重插补方法的应用。半参数最大似然估计可以通过分段非参数模型（Zhao，2009）来处理一般的缺失数据问题，用于具有缺失值的变量和观测或构造的辅助变量的联合分布（Breslow 等，2009）。 (ii) 下的主题将与 (i) 下考虑的估计方法联合研究。我们的理论和数值研究（Zhao 等人，2012）比较了在某些情况下最优的平衡设计（Breslow 和 Cain，1988）与最小化正态模型感兴趣回归参数的全参数最大似然估计方差的抽样设计，表明两阶段研究的更有效抽样设计是可能的，并且可以扩展到多阶段研究和 PQD，并且正态模型的最优设计策略可以扩展到更一般的设置和一般来说，它们会产生更好的结果。在拟议的研究中，我们将通过分析和数值方法研究更一般设置中的多相研究和 PQD 的设计策略。如果通用优化设计策略能够成功开发和实施，它们将广泛应用于涉及数据收集和分析的自然科学和工程的所有领域。

Regression models are widely used to study the association between a response variable and a set of covariates. Standard inference methods require that there is no selection bias and that the variables of selected subjects are fully observed with no measurement errors. However, in practice, many studies involve some combinations of selection bias, missing or mismeasured data, which are caused either by design or happenstance. Complex sampling designs, including the two-phase or multi-phase design (TMD) and the partial questionnaire design (PQD), are used to reduce the cost of data collection while at the same time improve data quality; such designs produce data with certain variables missing at random in a monotone pattern (e.g., TMD) or nonmonotone patterns (e.g., PQD). Missing by happenstance may occur in any study, including designed experiments, where the missingness is often not at random and in nonmonotone patterns. The two objectives of the proposed research program are: (i) development and application of statistical methods for regression models with data missing (not) at random and data missing in arbitrary patterns, and (ii) investigation and application optimal sampling strategies for the two-phase and multi-phase studies and the PQD which maximize the information in observed data. Models explored under (i) will include the generalized linear models, Cox proportional hazards model and partial linear models where certain variables (response or covariates) are missing and auxiliary information is sometimes available. Our recent research in this area based on parametric and semiparametric models has several promising results for both response and covariates missing problems. Parametric working regression models (Chen and Chen, 2000; Lawless and Kalbfleisch, 2011; Zhao et al., 2013) can efficiently use the information from incomplete observations and auxiliary variables to improve estimation efficiency and produce unbiased estimates for missing at random case and certain missing not at random case. Both parametric and semiparametric procedures (Chen et al., 2011) can be developed to improve consistency of the commonly used multiple imputation methods and further enhance the application of the multiple imputation methods. Semiparametric maximum likelihood estimation can be developed to deal with the general missing data problems through a piece-wise nonparametric model (Zhao, 2009) for the joint distribution of the variables with missing values and auxiliary variables either observed or constructed (Breslow et al., 2009). Topics under (ii) will be investigated jointly with the estimation methods considered under (i). Both our theoretical and numerical studies (Zhao et al., 2012) in comparing the balance design, which is optimal in certain cases (Breslow and Cain, 1988), with sampling designs which minimize the variance of the fully parametric maximum likelihood estimator of a regression parameter of interest for normal models indicate that more efficient sampling designs for two-phase studies are possible and can be extended to the multi-phase studies and the PQD, and the optimal design strategies for normal models can be extended to more general settings and they produce better results in general. In the proposed research we will investigate design strategies for the multi-phase studies and the PQD in more general settings through both analytical and numerical methods. If the general optimal design strategies can be successfully developed and implemented they will be widely used in all fields of nature sciences and engineering involving data collections and analyses.