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Spatial Causal Tensor Completion for Multiple Exposures and Outcomes: An Application to the Health Effects of PFAS Pollution

Xiaodan Zhou, Brian J Reich, Shu Yang

Abstract

Per- and polyfluoroalkyl substances (PFAS) are typically encountered as mixtures of distinct chemicals with distinct effects on multiple health outcomes. Estimating joint causal effects using spatially-dependent observed data is challenging. We propose a spatial causal tensor completion framework that jointly models multiple exposures and outcomes within a low-rank tensor structure, while adjusting for observed confounders and latent spatial confounders. This method combines a low-rank tensor representation to pool information across exposures and outcomes with a spectral adjustment step that incorporates graph-Laplacian eigenvectors to approximate unmeasured spatial confounders, implemented via a projected-gradient descent algorithm. This framework enables causal inference in the presence of unmeasured spatial confounding and pervasive missingness of potential outcomes. We establish theoretical guarantees for the estimator and evaluate its finite-sample performance through extensive simulations. In an application to national PFAS monitoring data, our approach yields more conservative and credible causal relationships between PFOA and PFOS exposure and 13 chronic disease outcomes compared with existing alternatives.

Spatial Causal Tensor Completion for Multiple Exposures and Outcomes: An Application to the Health Effects of PFAS Pollution

Abstract

Per- and polyfluoroalkyl substances (PFAS) are typically encountered as mixtures of distinct chemicals with distinct effects on multiple health outcomes. Estimating joint causal effects using spatially-dependent observed data is challenging. We propose a spatial causal tensor completion framework that jointly models multiple exposures and outcomes within a low-rank tensor structure, while adjusting for observed confounders and latent spatial confounders. This method combines a low-rank tensor representation to pool information across exposures and outcomes with a spectral adjustment step that incorporates graph-Laplacian eigenvectors to approximate unmeasured spatial confounders, implemented via a projected-gradient descent algorithm. This framework enables causal inference in the presence of unmeasured spatial confounding and pervasive missingness of potential outcomes. We establish theoretical guarantees for the estimator and evaluate its finite-sample performance through extensive simulations. In an application to national PFAS monitoring data, our approach yields more conservative and credible causal relationships between PFOA and PFOS exposure and 13 chronic disease outcomes compared with existing alternatives.
Paper Structure (24 sections, 1 theorem, 11 equations, 5 figures)

This paper contains 24 sections, 1 theorem, 11 equations, 5 figures.

Table of Contents

  1. Introduction
  2. PFAS Data Description
  3. Spatial Causal Tensor Model
  4. Notations and Tensor Operations
  5. Tensor Formulation of Potential Outcomes
  6. Causal Assumptions
  7. A Three-Step Estimating Procedure
  8. Step 1: Spectral adjustment for spatial confounding
  9. Step 2: Propensity score and weight estimation
  10. Step 3: Weighted Tensor Completion
  11. Spatial Projected Gradient Descent Algorithm
  12. Theoretical Properties
  13. Tensor Structural Assumptions
  14. Asymptotic Bounds
  15. Average Treatment Effect
  16. ...and 9 more sections

Key Result

Theorem 1

Under Assumptions (sutva)--(assume:incoherent), with probability at least $1 - 4\,(N+L+O)^{-2}$, our estimator satisfies: where where $a \wedge b$, $a \vee b$ denote the minimum and maximum of $a$ and $b$ respectively.

Figures (5)

  • Figure 1: Illustration of the spatial causal tensor decomposition. Left: a tensor of counterfactual outcomes, where observed entries (yellow) and missing entries (gray) are arranged by spatial units (PWS), exposures, and outcomes. Right: Tucker decomposition into spatial ($\hbox{\bf U}_1$), exposure ($\hbox{\bf U}_2$), and outcome ($\hbox{\bf U}_3$) factors with a low-rank core $\mathcal{G}$. The spatial factor $\hbox{\bf U}_1$ is further expressed as a combination of measured covariates $\hbox{\bf Z}$ and latent spatial components $\hbox{\bf S}$. The illustration omits the noise term $\mathcal{E}$ for clarity.
  • Figure 2: Conceptual diagram of PFAS health effect analysis.
  • Figure 3: Estimated odds ratios for 13 disease outcomes under PFOA, PFOS, and combined exposures across different models. Black indicates statistically significant associations; gray indicates non-significant results. Note that the x-axis scales differ across panels, so effect sizes are not directly comparable between the top and bottom rows.
  • Figure 4: Estimated marginal odds ratios for 13 disease outcomes under PFOA and PFOS exposures across different models. Black indicates statistically significant associations; gray indicates non-significant results.
  • Figure 5: Latent factors in the spatial tensor model. Top: Clustering of geographic units based on the location factor $\widehat{ \hbox{\bf U}}_1$. Bottom: Loadings in outcome factor $\widehat{ \hbox{\bf U}}_3$.

Theorems & Definitions (1)

  • Theorem 1: Frobenius Error Bound - Simplified Version