Spatial areal models encounter the well-known and challenging problem of spatial confounding. This issue makes it arduous to distinguish between the impacts of observed covariates and spatial random effects. Despite previous research and various proposed methods to tackle this problem, finding a definitive solution remains elusive. In this paper, we propose a simplified version of the spatial+ approach that involves dividing the covariate into two components. One component captures large-scale spatial dependence, while the other accounts for short-scale dependence. This approach eliminates the need to separately fit spatial models for the covariates. We apply this method to analyse two forms of crimes against women, namely rapes and dowry deaths, in Uttar Pradesh, India, exploring their relationship with socio-demographic covariates. To evaluate the performance of the new approach, we conduct extensive simulation studies under different spatial confounding scenarios. The results demonstrate that the proposed method provides reliable estimates of fixed effects and posterior correlations between different responses.

空间区域模型遇到了众所周知且具有挑战性的空间混杂问题。这个问题使得很难区分观察到的协变量的影响和空间随机效应。尽管之前的研究和提出了各种解决这个问题的方法，但找到明确的解决方案仍然难以实现。在本文中，我们提出了空间+方法的简化版本，该方法涉及将协变量分为两个组成部分。一个分量捕获大规模空间依赖性，而另一个分量则解释短期依赖性。这种方法无需单独拟合协变量的空间模型。我们应用这种方法来分析印度北方邦的两种针对妇女的犯罪，即强奸和嫁妆死亡，探讨它们与社会人口协变量的关系。为了评估新方法的性能，我们在不同的空间混杂场景下进行了广泛的模拟研究。结果表明，所提出的方法提供了不同响应之间的固定效应和后验相关性的可靠估计。