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Spatial causal inference in the presence of unmeasured confounding and interference

Georgia Papadogeorgou, Srijata Samanta

TL;DR

This work addresses causal inference with spatial data by uniting causal reasoning with spatial statistics and introducing spatial causal graphs to reveal how spatial confounding and interference can be entangled. It proposes a Bayesian hierarchical framework that jointly models exposure and outcome on a spatial network, incorporating an exposure-mapping structure and unmeasured spatial confounding, and proves identifiability of causal effects under a ring-graph configuration when certain parameters are nonzero. Through simulations, it demonstrates that the proposed approach significantly reduces bias in local and interference effect estimates compared with standard methods, particularly in the presence of spatial dependence and unmeasured confounding. Applied to US county data on SO$_2$ emissions and cardiovascular mortality, the method yields exposure–response estimates comparable to OLS when unmeasured confounding is mild but provides bias protection and uncertainty quantification when spatial confounding is present, underscoring its practical value for policy-relevant spatial causal questions. The framework advances spatial causal inference by explicitly modeling interference, local and neighborhood confounding, and exposure dependence, with implications for environmental epidemiology and broader spatial outcomes research; it also outlines directions for non-linear extensions and atmospheric interference modeling.

Abstract

This manuscript unites causal inference and spatial statistics, presenting novel insights for causal inference in spatial data analysis, and drawing from tools in spatial statistics to estimate causal effects. We introduce spatial causal graphs to highlight that spatial confounding and interference can be entangled, in that investigating the presence of one can lead to wrongful conclusions in the presence of the other. Moreover, we show that spatial dependence in the exposure variable can render standard analyses invalid. To remedy these issues, we propose a Bayesian parametric approach based on tools commonly-used in spatial statistics. This approach simultaneously accounts for interference and mitigates bias from local and neighborhood unmeasured spatial confounding. From a Bayesian perspective, we show that incorporating an exposure model is necessary. Under a specific model formulation, we prove that all parameters are identifiable including the causal effects, even in the presence of unmeasured confounding. We illustrate the approach with a simulation study. We evaluate the effect of local and neighboring sulfur dioxide emissions from power plants on county-level cardiovascular mortality from observational spatial data in the United States, where unmeasured spatial confounding and interference might be present simultaneously.

Spatial causal inference in the presence of unmeasured confounding and interference

TL;DR

This work addresses causal inference with spatial data by uniting causal reasoning with spatial statistics and introducing spatial causal graphs to reveal how spatial confounding and interference can be entangled. It proposes a Bayesian hierarchical framework that jointly models exposure and outcome on a spatial network, incorporating an exposure-mapping structure and unmeasured spatial confounding, and proves identifiability of causal effects under a ring-graph configuration when certain parameters are nonzero. Through simulations, it demonstrates that the proposed approach significantly reduces bias in local and interference effect estimates compared with standard methods, particularly in the presence of spatial dependence and unmeasured confounding. Applied to US county data on SO emissions and cardiovascular mortality, the method yields exposure–response estimates comparable to OLS when unmeasured confounding is mild but provides bias protection and uncertainty quantification when spatial confounding is present, underscoring its practical value for policy-relevant spatial causal questions. The framework advances spatial causal inference by explicitly modeling interference, local and neighborhood confounding, and exposure dependence, with implications for environmental epidemiology and broader spatial outcomes research; it also outlines directions for non-linear extensions and atmospheric interference modeling.

Abstract

This manuscript unites causal inference and spatial statistics, presenting novel insights for causal inference in spatial data analysis, and drawing from tools in spatial statistics to estimate causal effects. We introduce spatial causal graphs to highlight that spatial confounding and interference can be entangled, in that investigating the presence of one can lead to wrongful conclusions in the presence of the other. Moreover, we show that spatial dependence in the exposure variable can render standard analyses invalid. To remedy these issues, we propose a Bayesian parametric approach based on tools commonly-used in spatial statistics. This approach simultaneously accounts for interference and mitigates bias from local and neighborhood unmeasured spatial confounding. From a Bayesian perspective, we show that incorporating an exposure model is necessary. Under a specific model formulation, we prove that all parameters are identifiable including the causal effects, even in the presence of unmeasured confounding. We illustrate the approach with a simulation study. We evaluate the effect of local and neighboring sulfur dioxide emissions from power plants on county-level cardiovascular mortality from observational spatial data in the United States, where unmeasured spatial confounding and interference might be present simultaneously.
Paper Structure (56 sections, 9 theorems, 49 equations, 20 figures, 5 tables)

This paper contains 56 sections, 9 theorems, 49 equations, 20 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Evaluating the effect of SO$_2$ emissions from power plants on cardiovascular mortality
  3. Causal diagrams and identifiability of estimands with paired spatial data
  4. Causal estimands for paired spatial data with a binary treatment
  5. The observed pair data
  6. Identifiability of estimands in the presence of statistical and causal dependencies
  7. Causal inference with spatial dependencies on a spatial network
  8. Local and interference effects under an exposure mapping
  9. Ignorability in the presence of unmeasured spatial covariates
  10. Bayesian inference of local and interference effects with spatial dependencies and unmeasured spatial confounding
  11. The role of the treatment assignment mechanism in spatial settings
  12. Exposure-confounder assumptions
  13. A model on potential outcomes and identifiability in the presence of unmeasured spatial confounding
  14. Prior distributions for confounding adjustment
  15. Simulations
  16. ...and 41 more sections

Key Result

Theorem 1

Consider the setting of eq:linear_sem for units organized on a ring graph with $n$ nodes and without measured covariates. Using data $(\bm Z, \bm Y)$ we can identify whether or not $\rho\phi_U = 0$. If $\rho \phi_U \neq 0$, and for $\beta_U = 1$, we have that all model parameters $(\beta_Z, \beta_{\

Figures (20)

  • Figure 1: County-level neighborhood exposure and outcome in the evaluation of local and interference effects of SO$_2$ emissions from power plants on cardiovascular mortality. Local values of SO$_2$ emissions for each county are shown in the Supplement.
  • Figure 2: Causal and statistical dependencies depicted at the pair- and at the unit-level.
  • Figure 3: Graphical representation of spatial confounding and interference with a spatially correlated covariate $\bm U = (U_1, U_2)$, a spatial exposure $\bm Z = (Z_1, Z_2)$, and outcome $\bm Y = (Y_1, Y_2)$.
  • Figure 4: Estimated exposure-response (ER) curve for SO$_2$ emissions and cardiovascular mortality rate. The line corresponds to the posterior mean, and the bands to the pointwise 95% credible intervals. The color reflects whether the proposed approach (blue) or OLS that ignores unmeasured spatial confounding is considered (orange). Left: Estimated ER curve as a function of local emissions when neighborhood emissions are set to the 40$^{th}$ quantile of their observed distribution. Center: Estimated ER curve as a function of neighborhood emissions when local emissions are set to the 40$^{th}$ quantile of their observed distribution. Right: Posterior mean and 95% credible interval for the correlation parameter between the unmeasured spatial variable $U$ and the exposure $Z$ in \ref{['ass:UZ_normal']}.
  • Figure :
  • ...and 15 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Proposition S.1: Identifiability of local and interference effects under direct spatial confounding
  • proof : Proof of \ref{['supp_prop:direct_identify']}
  • Proposition S.2: Identifiability of local and interference effects under direct and indirect spatial confounding
  • proof : Proof of \ref{['supp_prop:general_spatial_conf_identify']}
  • Proposition S.3
  • proof : Proof of \ref{['supp_prop:interf_nospat_identif_local']}
  • Proposition S.4
  • proof : Proof of \ref{['supp_prop:interf_identif_local']}
  • Remark S.1
  • ...and 9 more