Many scientific questions in biomedical, environmental, and psychological research involve understanding the effects of multiple factors on outcomes. While factorial experiments are ideal for this purpose, randomized controlled treatment assignment is generally infeasible in many empirical studies. Therefore, investigators must rely on observational data, where drawing reliable causal inferences for multiple factors remains challenging. As the number of treatment combinations grows exponentially with the number of factors, some treatment combinations can be rare or missing by chance in observed data, further complicating factorial effects estimation. To address these challenges, we propose a novel weighting method tailored to observational studies with multiple factors. Our approach uses weighted observational data to emulate a randomized factorial experiment, enabling simultaneous estimation of the effects of multiple factors and their interactions. Our investigations reveal a crucial nuance: achieving balance among covariates, as in single-factor scenarios, is necessary but insufficient for unbiasedly estimating factorial effects; balancing the factors is also essential in multi-factor settings. Moreover, we extend our weighting method to handle missing treatment combinations in observed data. Finally, we study the asymptotic behavior of the new weighting estimators and propose a consistent variance estimator, providing reliable inferences on factorial effects in observational studies.

翻译：在生物医学、环境及心理学研究中，许多科学问题涉及理解多种因素对结果的影响。虽然析因实验是解决此类问题的理想方法，但在许多实证研究中，随机化控制处理分配通常难以实现。因此，研究者必须依赖观察性数据，而在其中对多因素进行可靠的因果推断仍然具有挑战性。随着处理组合的数量随因素数量呈指数增长，某些处理组合在观测数据中可能因偶然性而稀少或缺失，这进一步增加了析因效应估计的复杂性。为应对这些挑战，我们提出了一种专为多因素观察性研究设计的新型加权方法。该方法通过加权观察数据来模拟随机析因实验，从而能够同时估计多因素及其交互作用的影响。我们的研究发现了一个关键细节：与单因素情形类似，实现协变量的平衡对于无偏估计析因效应是必要的，但在多因素情境下并不充分；平衡因素本身同样至关重要。此外，我们将该加权方法扩展至处理观测数据中缺失的处理组合。最后，我们研究了新加权估计量的渐近性质，并提出了一致方差估计量，从而为观察性研究中的析因效应提供了可靠的统计推断。