This paper proposes a new approach to address the problem of unmeasured confounding in spatial designs. Spatial confounding occurs when some confounding variables are unobserved and not included in the model, leading to distorted inferential results about the effect of an exposure on an outcome. We show the relationship existing between the confounding bias of a non-spatial model and that of a semi-parametric model that includes a basis matrix to represent the unmeasured confounder conditional on the exposure. This relationship holds for any basis expansion, however it is shown that using the semi-parametric approach guarantees a reduction in the confounding bias only under certain circumstances, which are related to the spatial structures of the exposure and the unmeasured confounder, the type of basis expansion utilized, and the regularization mechanism. To adjust for spatial confounding, and therefore try to recover the effect of interest, we propose a Bayesian semi-parametric regression model, where an expansion matrix of principal spline basis functions is used to approximate the unobserved factor, and spike-and-slab priors are imposed on the respective expansion coefficients in order to select the most important bases. From the results of an extensive simulation study, we conclude that our proposal is able to reduce the confounding bias with respect to the non-spatial model, and it also seems more robust to bias amplification than competing approaches.

翻译：本文提出了一种新方法来解决空间设计中的未测量混杂问题。当一些混杂变量未被观测且未纳入模型时，会出现空间混杂，导致暴露对结果效应的推断结果失真。我们展示了非空间模型的混杂偏倚与包含基矩阵以表示暴露条件下的未测量混杂变量的半参数模型之间的关联。这种关联适用于任何基展开，但研究表明，仅在某些情况下（这些情况与暴露和未测量混杂变量的空间结构、所用基展开类型及正则化机制有关），使用半参数方法才能保证减少混杂偏倚。为了调整空间混杂并尝试恢复目标效应，我们提出了一种贝叶斯半参数回归模型，其中使用主样条基函数的扩展矩阵来逼近未观测因子，并对相应的扩展系数施加尖峰-板先验以选择最重要的基函数。通过广泛的模拟研究结果，我们得出结论：本方法能有效减少相对于非空间模型的混杂偏倚，并且相比竞争方法，在偏倚放大方面表现出更强的稳健性。