We investigate spatial confounding in the presence of multivariate disease dependence. In the "analysis model perspective" of spatial confounding, adding a spatially dependent random effect can lead to significant variance inflation of the posterior distribution of the fixed effects. The "data generation perspective" views covariates as stochastic and correlated with an unobserved spatial confounder, leading to inferior statistical inference over multiple realizations. Although multiple methods have been proposed for adjusting statistical models to mitigate spatial confounding in estimating regression coefficients, the results on interactions between spatial confounding and multivariate dependence are very limited. We contribute to this domain by investigating spatial confounding from the analysis and data generation perspectives in a Bayesian coregionalized areal regression model. We derive novel results that distinguish variance inflation due to spatial confounding from inflation based on multicollinearity between predictors and provide insights into the estimation efficiency of a spatial estimator under a spatially confounded data generation model. We demonstrate favorable performance of spatial analysis compared to a non-spatial model in our simulation experiments even in the presence of spatial confounding and a misspecified spatial structure. In this regard, we align with several other authors in the defense of traditional hierarchical spatial models (Gilbert et al., 2025; Khan and Berrett, 2023; Zimmerman and Ver Hoef, 2022) and extend this defense to multivariate areal models. We analyze county-level data from the US on obesity / diabetes prevalence and diabetes-related cancer mortality, comparing the results with and without spatial random effects.

翻译：本研究探讨存在多元疾病依赖时的空间混杂问题。从"分析模型视角"看空间混杂，加入空间依赖的随机效应可能导致固定效应后验分布的显著方差膨胀。"数据生成视角"则将协变量视为随机变量，并与未观测的空间混杂因子相关，导致多次实现中的统计推断效果下降。尽管已有多种方法通过调整统计模型来缓解回归系数估计中的空间混杂，但关于空间混杂与多元依赖相互作用的研究仍十分有限。我们通过贝叶斯协同区域化回归模型，从分析和数据生成双重视角研究空间混杂问题，为该领域做出贡献。我们推导出区分空间混杂导致的方差膨胀与预测变量间多重共线性引起的膨胀的新结论，并深入解析了空间混杂数据生成模型下空间估计量的估计效率。模拟实验表明，即使在存在空间混杂和空间结构误设的情况下，空间分析相比非空间模型仍表现出优越性能。在此意义上，我们与多位学者共同捍卫传统层次空间模型（Gilbert等，2025；Khan与Berrett，2023；Zimmerman与Ver Hoef，2022），并将此辩护延伸至多元区域模型。我们分析了美国县级肥胖/糖尿病患病率与糖尿病相关癌症死亡率数据，比较了包含与不包含空间随机效应的分析结果。