Adjusting for an unmeasured confounder is generally an intractable problem, but in the spatial setting it may be possible under certain conditions. In this paper, we derive necessary conditions on the coherence between the treatment variable of interest and the unmeasured confounder that ensure the causal effect of the treatment is estimable. We specify our model and assumptions in the spectral domain to allow for different degrees of confounding at different spatial resolutions. The key assumption that ensures identifiability is that confounding present at global scales dissipates at local scales. We show that this assumption in the spectral domain is equivalent to adjusting for global-scale confounding in the spatial domain by adding a spatially smoothed version of the treatment variable to the mean of the response variable. Within this general framework, we propose a sequence of confounder adjustment methods that range from parametric adjustments based on the Matern coherence function to more robust semi-parametric methods that use smoothing splines. These ideas are applied to areal and geostatistical data for both simulated and real datasets
翻译:调整一个无法计量的混淆器通常是一个棘手的问题,但在空间环境中,在某些条件下是可能的。在本文中,我们为处理利益变数与确保处理因果关系的未计量混结器之间的一致性创造了必要的条件,以确保处理结果的因果关系是可以估计的。我们在光谱域中指定了我们的模型和假设,以便在不同空间分辨率上进行不同程度的混杂。确保可识别性的关键假设是,全球尺度上的混杂现象在地方尺度上消失。我们表明,光谱域中的这一假设相当于调整空间域中全球规模混杂的假设,方法是在反应变数的平均值中增加一个空间上平滑的处理变数版本。在此总体框架内,我们提出了一系列调和调整方法,从基于数学一致性功能的对等调整到使用滑动样条纹的更稳健的半参数方法。这些想法适用于模拟和真实数据集的自然和地理统计数据。