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Causal Effect of Functional Treatment

Ruoxu Tan, Wei Huang, Zheng Zhang, Guosheng Yin

TL;DR

The paper tackles causal inference with a functional treatment by introducing three estimators—functional stabilized weight (FSW), outcome regression (OR), and doubly robust (DR)—under an unconfoundedness framework. It adopts a functional linear model for the average dose-response functional (ADRF) $E\{Y^*(z)\}=a+\int b(t)z(t)\,dt$ and develops both nonparametric and semi-parametric estimation strategies, including a backfitting approach for the OR and a DR combination that remains consistent if either component is correct. The authors establish asymptotic properties under comprehensive regularity conditions, derive convergence rates for the nonparametric weight and the slope function, and discuss how tuning parameters are selected via cross-validation. Through extensive simulations and a real EEG application, the methods demonstrate robustness to model misspecification and practical value in biomedical settings, notably showing the ADRF slope and its implications for treatment effects across the functional domain. Overall, the work advances causal analysis with functional treatments by providing principled identification, robust estimation, and theoretical guarantees, supported by empirical demonstrations.

Abstract

We study the causal effect with a functional treatment variable, where practical applications often arise in neuroscience, biomedical sciences, etc. Previous research concerning the effect of a functional variable on an outcome is typically restricted to exploring correlation rather than causality. The generalized propensity score, which is often used to calibrate the selection bias, is not directly applicable to a functional treatment variable due to a lack of definition of probability density function for functional data. We propose three estimators for the average dose-response functional based on the functional linear model, namely, the functional stabilized weight estimator, the outcome regression estimator and the doubly robust estimator, each of which has its own merits. We study their theoretical properties, which are corroborated through extensive numerical experiments. A real data application on electroencephalography data and disease severity demonstrates the practical value of our methods.

Causal Effect of Functional Treatment

TL;DR

The paper tackles causal inference with a functional treatment by introducing three estimators—functional stabilized weight (FSW), outcome regression (OR), and doubly robust (DR)—under an unconfoundedness framework. It adopts a functional linear model for the average dose-response functional (ADRF) and develops both nonparametric and semi-parametric estimation strategies, including a backfitting approach for the OR and a DR combination that remains consistent if either component is correct. The authors establish asymptotic properties under comprehensive regularity conditions, derive convergence rates for the nonparametric weight and the slope function, and discuss how tuning parameters are selected via cross-validation. Through extensive simulations and a real EEG application, the methods demonstrate robustness to model misspecification and practical value in biomedical settings, notably showing the ADRF slope and its implications for treatment effects across the functional domain. Overall, the work advances causal analysis with functional treatments by providing principled identification, robust estimation, and theoretical guarantees, supported by empirical demonstrations.

Abstract

We study the causal effect with a functional treatment variable, where practical applications often arise in neuroscience, biomedical sciences, etc. Previous research concerning the effect of a functional variable on an outcome is typically restricted to exploring correlation rather than causality. The generalized propensity score, which is often used to calibrate the selection bias, is not directly applicable to a functional treatment variable due to a lack of definition of probability density function for functional data. We propose three estimators for the average dose-response functional based on the functional linear model, namely, the functional stabilized weight estimator, the outcome regression estimator and the doubly robust estimator, each of which has its own merits. We study their theoretical properties, which are corroborated through extensive numerical experiments. A real data application on electroencephalography data and disease severity demonstrates the practical value of our methods.
Paper Structure (19 sections, 8 theorems, 106 equations, 4 figures, 1 table)

This paper contains 19 sections, 8 theorems, 106 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Identification of the Average Dose-Response Functional
  3. Assumption 1
  4. Models and Estimation Methods
  5. Functional Stabilized Weight Estimator
  6. Outcome Regression Estimator and Doubly Robust Estimator
  7. Asymptotic Properties
  8. Condition A
  9. Condition B
  10. Condition C
  11. Selection of Tuning Parameters
  12. Simulation Study
  13. Real Data Analysis
  14. Discussion
  15. Proof of Proposition \ref{['prop1']}
  16. ...and 4 more sections

Key Result

Proposition 1

For any fixed $z\in L^2(\mathcal{T})$, holds for all integrable functions $v(X)$ if and only if $\pi(z,X)=\pi_0(z,X)$ a.s.

Figures (4)

  • Figure 1: A random subsample of 20 frontal asymmetric curves $Z(\cdot)$ on a given frequency domain from the EEG dataset.
  • Figure 2: True curve (----), first (--- --- ---), second (--- $\cdot$ ---) and third ($\cdot$$\cdot$$\cdot$) quartile estimated slope functions under model (i) with $n=200$ and 500 for all methods.
  • Figure 3: True curve (----), first (--- --- ---), second (--- $\cdot$ ---) and third ($\cdot$$\cdot$$\cdot$) quartile estimated slope functions under model (iv) with $n=200$ and 500 for all methods.
  • Figure 4: Left: the estimated slope functions using FSW (---), OR (-- --), DR (-- $\cdot$ --), PCW ($\cdots$) and FLR ($\circ$); middle: the frontal asymmetry curves corresponding to the smallest ($\circ$), median ($+$) and largest ($*$) average values over 4 to 15 Hz; right: the estimated $E\{Y^*(Z)\}$ of the three selected curves with marks corresponding to the middle panel.

Theorems & Definitions (8)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4