Topics in statistical analysis with missing data

回归模型被广泛用于研究响应变量和一组协变量之间的关联。标准推理方法要求没有选择偏倚，被选对象的变量被充分观察，没有测量误差。然而，在实践中，许多研究涉及一些选择偏差、缺失或错误测量数据的组合，这些数据要么是由设计引起的，要么是偶然的。采用复杂的抽样设计，包括两阶段或多阶段设计（TMD）和部分问卷设计（PQD），在降低数据收集成本的同时提高数据质量；这样的设计产生的数据在单调模式（例如，TMD）或非单调模式（例如，PQD）中随机丢失某些变量。偶然性缺失可能发生在任何研究中，包括设计好的实验，其中缺失通常不是随机的和非单调的模式。该研究计划的两个目标是：(i)开发和应用数据随机缺失（非随机）和任意模式缺失的回归模型的统计方法；（ii）研究和应用两阶段和多阶段研究的最佳抽样策略，以及最大限度地利用观测数据中的信息的PQD。

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.