Missing covariate data occurs often in regression analysis, which frequently arises in the health and social sciences as well as in survey sampling. We study methods for the analysis of a nonignorable covariate-missing data problem in an assumed conditional mean function when some covariates are completely observed but other covariates are missing for some subjects. We adopt the semiparametric perspective of Bartlett et al. (Improving upon the efficiency of complete case analysis when covariates are MNAR. Biostatistics 2014;15:719-30) on regression analyses with nonignorable missing covariates, in which they have introduced the use of two working models, the working probability model of missingness and the working conditional score model. In this paper, we study an empirical likelihood approach to nonignorable covariate-missing data problems with the objective of effectively utilizing the two working models in the analysis of covariate-missing data. We propose a unified approach to constructing a system of unbiased estimating equations, where there are more equations than unknown parameters of interest. One useful feature of these unbiased estimating equations is that they naturally incorporate the incomplete data into the data analysis, making it possible to seek efficient estimation of the parameter of interest even when the working regression function is not specified to be the optimal regression function. We apply the general methodology of empirical likelihood to optimally combine these unbiased estimating equations. We propose three maximum empirical likelihood estimators of the underlying regression parameters and compare their efficiencies with other existing competitors. We present a simulation study to compare the finite-sample performance of various methods with respect to bias, efficiency, and robustness to model misspecification. The proposed empirical likelihood method is also illustrated by an analysis of a data set from the US National Health and Nutrition Examination Survey (NHANES).

中文翻译：

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不可忽略的协变量缺失数据问题中的经验似然。

协变量数据丢失经常发生在回归分析中，这在健康和社会科学以及调查抽样中经常出现。我们研究了在假设条件均值函数中无法忽略的协变量缺失数据问题的分析方法，其中某些协变量已完全观察到，但某些受试者缺少其他协变量。我们采用Bartlett等人的半参数观点。（提高协变量为MNAR时完整案例分析的效率。Biostatistics 2014; 15：719-30）针对具有不可忽略的缺失协变量的回归分析，其中他们引入了两种工作模型的使用：缺失的工作概率模型和缺失的工作概率模型。工作条件评分模型。在本文中，我们研究了经验似然方法来解决不可忽略的协变量缺失数据问题，目的是有效地利用两个工作模型来分析协变量缺失数据。我们提出了一种统一的方法来构造一个无偏估计方程组，在该系统中，存在比感兴趣的未知参数更多的方程。这些无偏估计方程的一个有用特征是，它们自然会将不完整的数据合并到数据分析中，从而即使未将工作回归函数指定为最佳回归函数，也可以寻求对目标参数的有效估计。我们应用经验似然的一般方法来最佳地组合这些无偏估计方程。我们提出了基础回归参数的三个最大经验似然估计，并将其效率与其他现有竞争者进行比较。我们提供了一项仿真研究，以比较各种方法在偏误，效率和鲁棒性方面对有限方法的性能进行建模。还通过对美国国家健康和营养检查调查（NHANES）的数据集进行分析来说明所提出的经验似然法。