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Causal Inference in Biomedical Imaging via Functional Linear Structural Equation Models

Ting Li, Ethan Fan, Tengfei Li, Hongtu Zhu

TL;DR

This work addresses causal inference for the effect of endogenous, infinite-dimensional functional imaging exposures on scalar clinical outcomes under unobserved confounding. The authors formulate a Functional Linear Structural Equation Model ($FLSEM$) with scalar covariates and functional instruments, establishing identifiability through an injective integral operator $\mathcal{K}$ and Mercer's decomposition of the kernel $K(s,t)$. Estimation combines a three-step, $L_0$-penalized, RKHS-based approach with a Functional Group Support Detection and Root Finding ($\text{FGSDAR}$) algorithm, implemented via a region-based divide-and-conquer scheme, and a nullity test for the functional coefficient $B(t)$ with a derived null distribution. Theoretical guarantees include selection consistency, nonasymptotic error bounds for both scalar and functional components, and a practical $S_n$ statistic approximated by a scaled $\chi^2$ distribution. Empirical evaluation in simulations and a UK Biobank application demonstrates robust performance in detecting causal relationships between genetics, brain imaging, and cognition, offering a scalable framework for causal imaging genetics research.

Abstract

Understanding the causal effects of organ-specific features from medical imaging on clinical outcomes is essential for biomedical research and patient care. We propose a novel Functional Linear Structural Equation Model (FLSEM) to capture the relationships among clinical outcomes, functional imaging exposures, and scalar covariates like genetics, sex, and age. Traditional methods struggle with the infinite-dimensional nature of exposures and complex covariates. Our FLSEM overcomes these challenges by establishing identifiable conditions using scalar instrumental variables. We develop the Functional Group Support Detection and Root Finding (FGS-DAR) algorithm for efficient variable selection, supported by rigorous theoretical guarantees, including selection consistency and accurate parameter estimation. We further propose a test statistic to test the nullity of the functional coefficient, establishing its null limit distribution. Our approach is validated through extensive simulations and applied to UK Biobank data, demonstrating robust performance in detecting causal relationships from medical imaging.

Causal Inference in Biomedical Imaging via Functional Linear Structural Equation Models

TL;DR

This work addresses causal inference for the effect of endogenous, infinite-dimensional functional imaging exposures on scalar clinical outcomes under unobserved confounding. The authors formulate a Functional Linear Structural Equation Model () with scalar covariates and functional instruments, establishing identifiability through an injective integral operator and Mercer's decomposition of the kernel . Estimation combines a three-step, -penalized, RKHS-based approach with a Functional Group Support Detection and Root Finding () algorithm, implemented via a region-based divide-and-conquer scheme, and a nullity test for the functional coefficient with a derived null distribution. Theoretical guarantees include selection consistency, nonasymptotic error bounds for both scalar and functional components, and a practical statistic approximated by a scaled distribution. Empirical evaluation in simulations and a UK Biobank application demonstrates robust performance in detecting causal relationships between genetics, brain imaging, and cognition, offering a scalable framework for causal imaging genetics research.

Abstract

Understanding the causal effects of organ-specific features from medical imaging on clinical outcomes is essential for biomedical research and patient care. We propose a novel Functional Linear Structural Equation Model (FLSEM) to capture the relationships among clinical outcomes, functional imaging exposures, and scalar covariates like genetics, sex, and age. Traditional methods struggle with the infinite-dimensional nature of exposures and complex covariates. Our FLSEM overcomes these challenges by establishing identifiable conditions using scalar instrumental variables. We develop the Functional Group Support Detection and Root Finding (FGS-DAR) algorithm for efficient variable selection, supported by rigorous theoretical guarantees, including selection consistency and accurate parameter estimation. We further propose a test statistic to test the nullity of the functional coefficient, establishing its null limit distribution. Our approach is validated through extensive simulations and applied to UK Biobank data, demonstrating robust performance in detecting causal relationships from medical imaging.
Paper Structure (12 sections, 6 theorems, 18 equations, 3 figures, 4 tables, 1 algorithm)

This paper contains 12 sections, 6 theorems, 18 equations, 3 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Identifiability Conditions for FLSEM
  4. Identifiability Condition for General $f$
  5. Identifiability Condition for linear $f$
  6. Estimation
  7. Theoretical Properties
  8. Simulation Studies
  9. Real Data Analysis of UK Biobank
  10. Discussion
  11. Acknowledgment
  12. Author Contributions

Key Result

Proposition 1

Consider the reduced form eq:reduced form, if the operator ${\mathcal{K} }$ is injective, and the null space of the operator ${\mathcal{K} }$ only contains 0 such that $\mathcal{N} ( \mathcal{K} )=\{0\}$, then $\bm{B}(t)$ is identifiable.

Figures (3)

  • Figure 1: (a) Directed acyclic graph showing the causal genetic-imaging-clinical (CGIC) pathway that links from genetic factors to organ imaging measures to clinical outcomes confounded with environmental factors (e.g., lifestyle factors) and possible unobserved confounders. (b) Four groups of mixed covariates.
  • Figure 2: Empirical rejection of proposed testing method, "Ignore endogeneity" indicates using the observed functional data for testing.
  • Figure 3: Panel (a) shows the positions of SNPs selected by FLSEM and PFLM; Panel (b) displays the positive and negative fMRI effect estimates on fluid intelligence from both methods.

Theorems & Definitions (10)

  • Proposition 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Example 1
  • Example 2
  • Example 3
  • Example 4