Generalized propensity score weighting for functional causal inference framework

Technical Summary: Generalized Propensity Score Weighting for Functional Causal Inference

Problem Statement
Causal inference in observational studies typically relies on propensity score methods to adjust for confounding between treatment and outcome. However, existing methodologies are designed for scalar or low-dimensional vector treatments, covariates, and outcomes. Modern applications increasingly involve variables observed repeatedly over time, which are more naturally represented as functions (functional data). Standard propensity score approaches fail in these settings because they rely on balancing finite-dimensional covariate representations, whereas functional treatments exist in infinite-dimensional spaces ($L^2(T)$). While recent work (Zhang, Xue and Wang, 2021) introduced a Functional Propensity Score (FPS) using Karhunen–Loève expansions, it suffers from two critical limitations: (1) weight estimation relies on constrained optimization with sequential tuning and proportionality constraints, creating computational burdens for large samples; and (2) the framework is restricted to scalar covariates and outcomes, failing to accommodate functional confounders or functional outcomes simultaneously.

Methodology
The authors propose a unified framework for causal inference with functional data that extends propensity score weighting to functional treatments, covariates, and outcomes.

1. Functional Representation: Functional treatments $X(t)$ and covariates are approximated via Karhunen–Loève expansions (truncated Functional Principal Component Analysis, FPCA) into finite-dimensional score vectors $A$ and $B$.
2. Dual Formulation of Weight Estimation: Instead of the constrained optimization used in prior work, the authors derive a dual formulation of the empirical likelihood balancing problem.
   • The primal problem minimizes $\sum \log(w_i)$ subject to balancing constraints (mean-zero and uncorrelated treatment scores and confounders).
   • The dual problem transforms this into a smooth, unconstrained convex minimization over a parameter vector $\theta$.
   • Weights are recovered via a softmax transformation: $w_i = \exp(-\theta^\top g_i) / \sum \exp(-\theta^\top g_j)$, where $g_i$ stacks the balancing moments.
   • This approach eliminates the need for sequential tuning or proportionality constraints, significantly improving computational scalability.
3. Extension to Functional Covariates: Functional confounders are incorporated by augmenting the balancing constraint vector with their FPC scores. This preserves the structure of the dual optimization, increasing the dimension of the optimization problem only by the number of retained components, not the sample size.
4. Extension to Functional Outcomes: For functional outcomes $Y(t)$, the authors develop a function-on-function marginal structural model. The causal effect is characterized by a bivariate surface $\mu(s,t)$. By expanding both treatment and outcome in their respective FPCA bases, the integral equation reduces to a system of independent weighted least squares problems for each outcome component.
5. Temporal Constraints: When treatment and outcome domains overlap, the authors enforce a "non-anticipativity" constraint ($\mu(s,t) = 0$ for $s > t$) using a triangular basis of piecewise linear functions, ensuring causal validity where past treatment affects future outcomes.

Key Contributions

• Unified Framework: The paper provides the first general framework for covariate-balancing causal inference that simultaneously handles functional treatments, functional covariates, and functional outcomes.
• Computational Efficiency: The proposed dual formulation yields a smooth, unconstrained optimization problem. This removes the computational bottlenecks of existing methods, making the approach scalable to large observational datasets (e.g., $n=3000$) and high-dimensional covariate spaces without the erratic performance of previous constrained solvers.
• Theoretical Extension: The work extends the Generalized Propensity Score (GPS) theory to infinite-dimensional settings via FPCA surrogates and establishes identification conditions (Strong Ignorability and Positivity) for functional marginal structural models.

Results

• Simulation Studies:
  • Bias Reduction: Across various settings (linear and nonlinear treatment-covariate/outcome relationships), the proposed weighting procedure substantially reduces bias compared to unweighted estimators and existing FPS methods.
  • Covariate Balance: The method achieves stable covariate balance (correlations $<0.1$) across varying sample sizes and covariate dimensions, whereas the existing FPS method showed erratic behavior.
  • Scalability: The proposed method demonstrated a massive reduction in computational time (approx. 1890 times faster in one setting) compared to Zhang, Xue and Wang (2021). Its computational cost remains stable as sample size ($n$) and covariate dimension ($p$) increase, unlike the existing approach which scales poorly.
• Real Data Application (UK Biobank):
  • Analysis 1 (BMI $\to$ T2D): Estimating the causal effect of midlife BMI trajectories on incident Type 2 Diabetes. The weighted analysis revealed that the effect is strongest in early midlife (ages 50–57) and becomes non-significant later, whereas unweighted analysis showed uniformly inflated effects due to confounding by baseline health status.
  • Analysis 2 (BMI $\to$ HbA1c): Estimating the causal effect surface of BMI trajectories on longitudinal HbA1c trajectories. The results indicated that higher BMI in early midlife drives the greatest increase in subsequent HbA1c, with effects attenuating for later exposure ages.

Significance and Claims
The authors claim that this framework addresses a critical gap in causal inference for modern longitudinal data. By moving from constrained to unconstrained optimization, the method makes functional causal inference computationally feasible for large-scale studies. The application to UK Biobank data demonstrates that treating exposure as a function (rather than a scalar summary) reveals nuanced, time-varying causal effects that are obscured in standard analyses. The authors note that while the method assumes strong ignorability and relies on FPCA (requiring reasonably dense data), it offers a robust tool for fields such as metabolic epidemiology, pharmacoepidemiology, and environmental epidemiology where trajectory-based exposures are common. The paper explicitly acknowledges limitations, including the need for extensions to sparse/irregular data (e.g., via PACE or small-ball probabilities) and the assumption of linear functional effects, suggesting these as directions for future research.