Sensitivity analysis for the unconfoundedness assumption is crucial in observational studies. For this purpose, the marginal sensitivity model (MSM) gained popularity recently due to its good interpretability and mathematical properties. However, as a quantification of confounding strength, the $L^{\infty}$-bound it puts on the logit difference between the observed and full data propensity scores may render the analysis conservative. In this article, we propose a new sensitivity model that restricts the $L^2$-norm of the propensity score ratio, requiring only the average strength of unmeasured confounding to be bounded. By characterizing sensitivity analysis as an optimization problem, we derive closed-form sharp bounds of the average potential outcomes under our model. We propose efficient one-step estimators for these bounds based on the corresponding efficient influence functions. Additionally, we apply multiplier bootstrap to construct simultaneous confidence bands to cover the sensitivity curve that consists of bounds at different sensitivity parameters. Through a real-data study, we illustrate how the new $L^2$-sensitivity analysis can improve calibration using observed confounders and provide tighter bounds when the unmeasured confounder is additionally assumed to be independent of the measured confounders and only have an additive effect on the potential outcomes.

翻译：对于无混淆假设的敏感性分析在观测研究中至关重要。为此，边际敏感性模型（MSM）因其良好的可解释性和数学性质而近期广受青睐。然而，作为混杂强度的量化指标，该模型对观测倾向得分与完整数据倾向得分之间的对数比值差施加的$L^{\infty}$界可能使分析结果过于保守。本文提出一种新的敏感性模型，该模型限制倾向得分比值的$L^2$范数，仅要求未测量混杂的平均强度有界。通过将敏感性分析刻画为优化问题，我们推导出该模型下平均潜在结果的闭式紧界。基于对应的有效影响函数，我们提出了这些边界的有效一步估计量。此外，我们采用乘子自助法构建同时置信带，以覆盖由不同敏感性参数下的边界构成的敏感性曲线。通过一项真实数据研究，我们展示了新的$L^2$敏感性分析如何利用观测混杂变量改进校准，并在额外假设未测量混杂变量与测量混杂变量独立且仅对潜在结果产生可加效应时提供更紧的边界。