The deconfounder was proposed as a method for estimating causal parameters in a context with multiple causes and unobserved confounding. It is based on recovery of a latent variable from the observed causes. We disentangle the causal interpretation from the statistical estimation problem and show that the deconfounder in general estimates adjusted regression target parameters. It does so by outcome regression adjusted for the recovered latent variable termed the substitute. We refer to the general algorithm, stripped of causal assumptions, as substitute adjustment. We give theoretical results to support that substitute adjustment estimates adjusted regression parameters when the regressors are conditionally independent given the latent variable. We also introduce a variant of our substitute adjustment algorithm that estimates an assumption-lean target parameter with minimal model assumptions. We then give finite sample bounds and asymptotic results supporting substitute adjustment estimation in the case where the latent variable takes values in a finite set. A simulation study illustrates finite sample properties of substitute adjustment. Our results support that when the latent variable model of the regressors hold, substitute adjustment is a viable method for adjusted regression.

翻译：去混杂因子被提出作为一种在存在多个原因和未观测混杂因素的情境下估计因果参数的方法。它基于从观测到的原因中恢复潜在变量。我们将因果解释与统计估计问题分离开来，并证明去混杂因子通常估计的是调整后的回归目标参数。这是通过使用恢复的潜在变量（称为替代变量）进行的结局回归调整实现的。我们将剥离因果假设后的通用算法称为替代调整。我们给出理论结果，支持当预测变量在给定潜在变量条件下条件独立时，替代调整能够估计调整后的回归参数。我们还引入了一种替代调整算法的变体，该变体在最小模型假设下估计一个假设精简的目标参数。接着，在潜在变量取值于有限集的情况下，我们给出有限样本界和渐近结果以支持替代调整估计。一项模拟研究展示了替代调整的有限样本性质。我们的结果表明，当预测变量的潜在变量模型成立时，替代调整是调整后回归的一种可行方法。