Disentangling spatial interference and spatial confounding biases in causal inference

Technical Summary: Disentangling Spatial Interference and Spatial Confounding Biases in Causal Inference

Problem Statement
Observational spatial data analysis faces two critical challenges that compromise the precision of causal estimates: spatial interference and spatial confounding. Spatial interference occurs when the treatment or exposure at one location affects the outcome at another, violating the Stable Unit Treatment Value Assumption (SUTVA). Spatial confounding arises when unmeasured variables with spatial structure influence both the treatment and the outcome.

The literature currently suffers from significant ambiguity regarding the definition of spatial confounding, often conflating distinct phenomena such as omitted variable bias, random effect collinearity, and concurvity. Furthermore, existing methods for quantifying bias typically rely on restrictive assumptions, specifically that treatments and confounders follow Normal distributions. There is a notable gap in the literature regarding the quantification of bias induced by indirect spatial confounding (where an unmeasured variable at one location affects outcomes at others) and the simultaneous presence of spatial interference and confounding. Additionally, the impact of non-Normal distributions (e.g., Poisson) on these biases remains largely unexplored.

Methodology
The authors employ a rigorous analytical and simulation-based approach to disentangle these biases:

1. Directed Acyclic Graph (DAG) Framework: The paper utilizes DAGs to provide a conceptual clarification of spatial confounding. It distinguishes between:

   • Causal Spatial Confounding: Where an unmeasured spatial variable $U$ causally affects both treatment $A$ and outcome $Y$ (satisfying temporal order).
   • Non-Causal Spatial Association: Where $U$ and $A$ are associated but not causally linked.
   • Direct vs. Indirect Confounding: Direct confounding occurs when $U_i$ affects $A_i$ and $Y_i$. Indirect confounding occurs when $U_i$ affects $A_j$ and/or $Y_j$ (where $i \neq j$), often mediated through spatial interference mechanisms.

2. Analytical Bias Derivation:

   • The authors derive analytical expressions for bias under various models ($M_1$ to $M_6$), ranging from simple linear models to those incorporating spatial interference ($\tilde{A}$), direct confounding ($U$), and indirect confounding ($\tilde{U}$).
   • Spatial Interference Bias: Derived using Ordinary Least Squares (OLS) for non-spatial settings and Generalized Least Squares (GLS) for spatial settings. The bias is shown to be a function of the treatment distribution, interference strength ($\beta_{\tilde{a}}$), and the spatial weight matrix ($\Psi$).
   • Non-Normal Distributions: The paper relaxes the Normality assumption by deriving bias expressions for Poisson-distributed treatments and confounders. Using properties of the sum of independent Poisson variables, they derive the conditional distribution of the confounder given the treatment to quantify the resulting bias.
   • Indirect Confounding Bias: Analytical expressions are derived for the bias induced by indirect spatial confounding ($\beta_{\tilde{u}}$), demonstrating that it is distinct from direct confounding bias.

3. Simulation Studies:

   • Extensive simulations (1,000–10,000 iterations) were conducted to evaluate Root Mean Square Error (RMSE), bias, and coverage probability.
   • Scenarios varied by:
     • Presence/absence of interference and confounding (direct and indirect).
     • Treatment and confounder distributions (Normal, Poisson, Binary).
     • Weight matrix construction: k-Nearest Neighbors (k-NN) and Distance-based (thresholded) weights.
     • Relative magnitude of treatment vs. interference effects.

4. Real-World Application:

   • The framework was applied to Manchester climate data (January 2018) analyzing the effect of Air Temperature (treatment) on Land Surface Temperature (outcome), with Precipitation as a confounder.
   • Models were fitted accounting for various combinations of interference, direct, and indirect confounding to compare estimates.

Key Contributions

1. Conceptual Clarification: The paper provides a novel DAG-based distinction between causal spatial confounding and other spatial associations, and explicitly separates direct and indirect spatial confounding, which are often treated as a single phenomenon in existing literature.
2. Quantification of Indirect Bias: This is the first work (to the authors' knowledge) to analytically derive and quantify the bias induced specifically by indirect spatial confounding and spatial interference in causal inference.
3. Relaxation of Distributional Assumptions: The study derives bias expressions for Poisson-distributed treatments and confounders, demonstrating that bias magnitudes and signs change significantly when Normality assumptions are violated.
4. Weight Matrix Sensitivity: The analysis reveals that the choice of spatial weights (k-NN vs. distance-based) critically determines the magnitude of bias. For instance, bias tends to increase with the number of neighbors in k-NN schemes but decreases with larger distance thresholds in distance-based schemes.
5. Disentanglement of Phenomena: The study proves that neglecting spatial interference, direct confounding, or indirect confounding leads to biased treatment effect estimates, and that these three phenomena are distinct and must be modeled separately.

Results

• Interference Bias: When spatial interference is present but unaccounted for, bias and RMSE are highest, and coverage probabilities drop significantly. The bias is sensitive to the weight matrix choice; k-NN weights generally produced higher bias than distance-based weights in non-spatial settings, though patterns varied in spatial settings.
• Distributional Impact: Bias magnitudes differed significantly across Normal, Poisson, and Binary distributions. Notably, the bias was highest for Normal distributions and lowest for Binary distributions in the simulations, contradicting the assumption that bias is invariant to distribution.
• Direct vs. Indirect Confounding: Simulation results confirmed that direct and indirect spatial confounding produce different bias profiles. In some scenarios (e.g., using k-NN weights), accounting for only direct confounding while ignoring indirect confounding resulted in higher bias than accounting for both.
• Real Data Application: In the Manchester climate study, estimates varied significantly depending on which phenomena were included. The model accounting for all three (interference, direct, and indirect confounding) yielded the best fit (lowest AIC), and estimates differed substantially from models ignoring any single component.

Significance and Claims
The authors claim that their work addresses a critical gap in causal inference for spatial data by:

• Providing clarity on misconceptions surrounding spatial confounding via DAGs.
• Demonstrating that indirect spatial confounding is a distinct source of bias that cannot be assumed to be identical to direct confounding.
• Showing that standard Normality assumptions are insufficient for accurate bias quantification, particularly for count data (Poisson).
• Establishing that the choice of spatial weights and the distribution of variables are key determinants of bias magnitude.

The paper concludes that if spatial interference, direct confounding, or indirect confounding are present but neglected, the resulting treatment effect estimates will be biased, leading to unreliable causal inferences. The authors propose a simple bias-correction algorithm based on their derived analytical expressions for spatial interference, though they acknowledge that a full parametric framework for simultaneous adjustment of all three phenomena remains a subject for future development.