Diagnostics such as Moran's index and approximate profile likelihood-based estimators (APLE) for Gaussian spatial autoregressive models are widely used in exploratory data analysis to assess the strength of spatial dependence. Yet, although Moran's index is often applied to regression residuals, and APLE is typically formulated for raw outcomes, neither is explicitly constructed as an estimator of residual spatial dependence after adjustment for large-scale trends and covariates. We propose RESAPLE, a one-step approximate restricted maximum likelihood (REML) estimator of the spatial error model's spatial dependence parameter $ρ$, constructed from REML residuals. Because RESAPLE is a Rayleigh coefficient, it retains the interpretability and diagnostic convenience of exploratory indices, while also providing a computationally inexpensive and accurate estimator of $ρ$ for moderate dependence. We show that for small to medium sample sizes and adequately specified trend models, RESAPLE is a better estimator of, and test statistic for, residual spatial dependence relative to existing alternatives including Moran's index and the APLE across a wide range of practical settings. The theory we develop also yields a diagnostic for spatial weight selection, providing guidance towards resolving a common point of ambiguity in spatial data analysis. We illustrate the method using simulations on both regular and highly irregular lattices with a case study using American Community Survey tract-level data.

翻译：在探索性数据分析中，诸如莫兰指数和高斯空间自回归模型的近似轮廓似然估计器（APLE）等诊断方法被广泛用于评估空间依赖性的强度。然而，尽管莫兰指数常应用于回归残差，而APLE通常针对原始结果构建，但两者均未明确设计为在调整大尺度趋势和协变量后对残差空间依赖性的估计器。我们提出了RESAPLE，一种基于受限最大似然（REML）残差构建的空间误差模型空间依赖性参数$ρ$的一步近似受限最大似然（REML）估计器。由于RESAPLE是一个瑞利系数，它保留了探索性指数的可解释性和诊断便利性，同时为中等依赖性提供了计算成本低廉且准确的$ρ$估计器。我们证明，在中小样本量及趋势模型充分设定的条件下，相较于包括莫兰指数和APLE在内的现有替代方法，RESAPLE在广泛的实际场景中能更好地估计和检验残差空间依赖性。我们发展的理论还产生了一种用于空间权重选择的诊断方法，为解决空间数据分析中常见的模糊点提供了指导。我们通过模拟研究（在规则和高度不规则格网上）以及一个使用美国社区调查区块级数据的案例研究来阐释该方法。