Ecological 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 one-step 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 analyze 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.

翻译：生态空间区域模型面临众所周知的且具有挑战性的空间混杂问题。该问题使得区分观测协变量与空间随机效应的影响变得困难。尽管已有研究及多种方法被提出以解决此问题，但找到明确的解决方案仍难以实现。本文提出一种一步版空间+方法，该方法将协变量分为两个部分：一个部分捕捉大尺度空间依赖性，另一部分则处理小尺度依赖性。该方法无需单独为协变量拟合空间模型。我们将此方法应用于分析印度北方邦针对女性的两种犯罪形式（强奸与嫁妆致死）及其与社会人口协变量的关系。为评估新方法的性能，我们在不同空间混杂场景下进行了广泛的模拟研究。结果表明，所提方法能够提供固定效应及不同响应变量间后验相关性的可靠估计。