Two-phase sampling designs are frequently employed in epidemiological studies and large-scale health surveys. In such designs, certain variables are exclusively collected within a second-phase random subsample of the initial first-phase sample, often due to factors such as high costs, response burden, or constraints on data collection or measurement assessment. Consequently, second-phase sample estimators can be inefficient due to the diminished sample size. Model-assisted calibration methods have been widely used to improve the efficiency of second-phase estimators. However, none of the existing methods have considered the complexities arising from the intricate sample designs present in both first- and second-phase samples in regression analyses. This paper proposes to calibrate the sample weights for the second-phase subsample to the weighted first-phase sample based on influence functions of regression coefficients for a prediction of the covariate of interest, which can be computed for the entire first-phase sample. We establish the consistency of the proposed calibration estimation and provide variance estimation. Empirical evidence underscores the robustness of calibration on influence functions compared to the imputation method, which can be sensitive to misspecified prediction models for the variable only collected in the second phase. Examples using data from the National Health and Nutrition Examination Survey are provided.

翻译：两阶段抽样设计常用于流行病学研究和大型健康调查。在此类设计中，某些变量仅在初始第一阶段样本的随机第二阶子样本中收集，这通常是由于高成本、响应负担或数据收集/测量评估限制等因素。因此，第二阶样本估计量可能因样本量减少而效率低下。模型辅助校准方法已被广泛用于提高第二阶估计量的效率。然而，现有方法均未考虑回归分析中第一阶段和第二阶样本复杂设计带来的复杂性问题。本文提出基于回归系数影响函数（可对整个第一阶段样本计算）将第二阶子样本的样本权重校准为加权第一阶段样本，以预测感兴趣协变量。我们建立了所提校准估计的一致性，并提供了方差估计。实证证据表明，与插补方法相比，基于影响函数的校准方法具有鲁棒性——插补方法可能对仅收集于第二阶的变量的错误指定预测模型敏感。文章还提供了使用国家健康与营养调查数据的实例。