Causal inference in connected populations is non-trivial, because the treatment assignments of units can affect the outcomes of other units via treatment and outcome spillover. Since outcome spillover induces dependence among outcomes, closed-form expressions for causal effects and convergence rates for causal effect estimators are challenging and unavailable. We make three contributions. First, we provide closed-form expressions for causal effects under treatment and outcome spillover without making assumptions about the joint probability law of treatment assignments, outcomes, and connections beyond linearity of conditional expectations of outcomes and the standard assumptions of ignorability and positivity. The main results permit complex dependence among outcomes and connections. Second, we show that ignoring dependence among outcomes due to outcome spillover can induce asymptotic bias in causal effect estimators. Third, we establish convergence rates for causal effect estimators by controlling dependence and characterizing a high-probability subset of data that addresses collinearity issues.

翻译：在关联群体中进行因果推断并非易事，因为个体的处理分配可能通过处理效应与结果溢出影响其他个体的结果。由于结果溢出会导致结果间产生依赖性，因果效应的闭式表达式及其估计量的收敛速率难以推导且尚未得到解决。本文作出三项贡献。首先，我们在处理效应与结果溢出并存的情况下，仅基于结果条件期望的线性假设以及可忽略性与正值性标准假设，无需对处理分配、结果及关联关系的联合概率分布作额外限定，推导出因果效应的闭式表达式。主要结论允许结果与关联关系之间存在复杂依赖性。其次，我们证明忽略结果溢出导致的结果间依赖性可能引发因果效应估计量的渐近偏误。第三，通过控制依赖性并构建解决共线性问题的高概率数据子集，我们确立了因果效应估计量的收敛速率。