RESAPLE: An Approximate One-Step Restricted Likelihood Estimator of Spatial Dependence for Exploratory Spatial Analysis

Abstract

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.

Figures (12)

• Figure 1: Overall workflow for using RESAPLE as a residual-space diagnostic for spatial dependence, in ESDA.
• Figure 2: Schematic RESAPLE scatterplot. The slope equals $\hat{\rho}_{RESAPLE}$. Unit contributions are $C_i=\tilde{x}_i\tilde{y}_i$ with leverage proportional to $\tilde{x}_i^2$.
• Figure 3: RMSE against $\rho$. $K=2000$ MC replicates. Faceted by $n$ (horizontal) and $p$ (vertical). For each design point, for each of the 20 values of $\rho$ (from $0$ to $0.95$), we count and annotate how many times RESAPLE attains the best RMSE.
• Figure 4: Simulation results of the RESAPLE, MAPLE, and APLE (residual) sampling standard deviation represented by $\sqrt{\text{RMSE}^2 - \text{Bias}^2}$ versus spatial autocorrelation $\rho$ (from $0$ to $0.95$) calculated on $K=2000$ MC replicates for (a) queen contiguity and (b) B07 contiguity. The plots are faceted by sample size $n$ (horizontal) and number of covariates $p$ (vertical), and are annotated with how many times RESAPLE had the lowest sampling variability.
• Figure 5: Power curves for (a) queen contiguity and (b) B07 contiguity across design points outlined in Table \ref{['tab:sim1-design']}. The test size $\alpha = 0.05$ is dashed in black horizontally on the plots.

Theorems & Definitions (27)

• Lemma 2.1
• proof
• Lemma 3.1
• Remark 3.2
• Theorem 3.3
• Corollary 3.4
• Remark 4.1
• Theorem 5.1
• proof
• Lemma B.1