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Identifiability of Treatment Effects with Unobserved Spatially Varying Confounders

Tommy Tang, Xinran Li, Bo Li

Abstract

The study of causal effects in the presence of unmeasured spatially varying confounders has garnered increasing attention. However, a general framework for identifiability, which is critical for reliable causal inference from observational data, has yet to be advanced. In this paper, we study a linear model with various parametric model assumptions on the covariance structure between the unmeasured confounder and the exposure of interest. We establish identifiability of the treatment effect for many commonly 20 used spatial models for both discrete and continuous data, under mild conditions on the structure of observation locations and the exposure-confounder association. We also emphasize models or scenarios where identifiability may not hold, under which statistical inference should be conducted with caution.

Identifiability of Treatment Effects with Unobserved Spatially Varying Confounders

Abstract

The study of causal effects in the presence of unmeasured spatially varying confounders has garnered increasing attention. However, a general framework for identifiability, which is critical for reliable causal inference from observational data, has yet to be advanced. In this paper, we study a linear model with various parametric model assumptions on the covariance structure between the unmeasured confounder and the exposure of interest. We establish identifiability of the treatment effect for many commonly 20 used spatial models for both discrete and continuous data, under mild conditions on the structure of observation locations and the exposure-confounder association. We also emphasize models or scenarios where identifiability may not hold, under which statistical inference should be conducted with caution.
Paper Structure (38 sections, 36 theorems, 190 equations, 1 figure)

This paper contains 38 sections, 36 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Spatial Model with Unmeasured Confounder
  3. Conditional Autoregressive Model
  4. Schnell & Papadogeorgou's Conditional Autoregressive Model
  5. Leroux Conditional Autoregressive Model
  6. Models for Geostatistical Data
  7. Linear Model of Coregionalization
  8. Bivariate Spatial Covariance Models
  9. Parsimonious Matérn Covariance Models
  10. Technical details for § \ref{['sec:model']}
  11. Simplification of the linear structrual model
  12. Representation for the observed data distribution
  13. Technical details for § \ref{['sec:car-papadogeorgu']}
  14. Proof of Theorem \ref{['thm:car-identifiability']}
  15. Proof of Corollary \ref{['cor:car-identifiability-binary']}
  16. ...and 23 more sections

Key Result

Theorem 1

Consider $n$ spatial locations with a proximity matrix $W$. Let $\mathcal{M}_1,...,\mathcal{M}_B$ be the disjoint connected components, which form a partition of all the locations. Assume that the data generating process follows eq:simp-model and eq:Sigma_SP. If $\varphi_U \neq 0$ and there exists o

Figures (1)

  • Figure 1: Examples of neighborhood structures among 6 locations. The condition on $W$ specified in Corollary \ref{['cor:car-identifiability-binary']} is violated in (a) and (b), but satisfied in (c) and (d).

Theorems & Definitions (75)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1
  • Theorem 5
  • Theorem 6
  • Lemma A1
  • proof : Proof of Lemma \ref{['lem:cond_Y_Z']}(i)
  • ...and 65 more