AI 中文总结
研究在聚类观测设计下，线性回归模型中遗漏变量偏差敏感性分析问题。通过对单位级回归和聚类聚合回归的分析，发现敏感性分析结论因分析级别而异，提出用Pearson偏$\eta$校正以恢复等效性，并给出相关建议。
AI 中文摘要
Cinelli和Hazlett（2020）为线性回归模型开发了一种敏感性分析方法，该方法根据两个部分$R^2$参数对遗漏变量偏差进行参数化，分别捕获遗漏混杂因素在处理和结果中解释的残差变化。此方法常用于在更高级别聚合时分配处理的单位级数据回归，如聚类观测设计。本文表明，尽管单位级回归和适当加权的聚类聚合回归在估计处理效果上数值等效，但敏感性分析程序根据所选分析级别会得出不同结论。结果 - 混杂因素部分$R^2$反映了组间和组内变异，但后者与遗漏变量偏差无关，因为它按构造不能由组级混杂因素解释。使用Pearson偏$\eta$对单位级回归的稳健性值和极端情况分析进行直接校正可恢复这两种方法之间的等效性。本文最后提醒在以单位级协变量为基准时要谨慎，并建议始终将这些协变量的聚类级平均值作为回归变量（Mundlak，1978）。
英文摘要
Cinelli and Hazlett (2020) develops a sensitivity analysis method for the linear regression model that parameterizes omitted variable bias in terms of two partial $R^2$ parameters capturing the residual variation explained by an omitted confounder in the treatment and outcome respectively. This method is often applied to regressions fit to unit-level data when treatment is assigned at a higher level of aggregation -- as in clustered observational designs. This paper shows that despite the numerical equivalence of the unit-level regression and an appropriately weighted cluster-aggregated regression for estimating the treatment effect, the sensitivity analysis procedure yields different conclusions depending on the chosen level of analysis. The outcome-confounder partial $R^2$ reflects both between- and within- group variation but the latter is irrelevant to omitted variable bias as it by construction cannot be explained by a group-level confounder. Straightforward corrections to the robustness value and the extreme scenario analysis from the unit-level regression using Pearson's partial-$η$ recover equivalence between these two approaches. The paper concludes with a point of caution when benchmarking against unit-level covariates and recommends always including cluster-level averages of these covariates as regressors (Mundlak, 1978).
