Causal Inference for Preprocessed Outcomes with an Application to Functional Connectivity

Technical Summary: Causal Inference for Preprocessed Outcomes with an Application to Functional Connectivity

Problem Statement
In biomedical research, particularly in neuroimaging (e.g., resting-state fMRI), repeated measurements within subjects are routinely processed to remove artifacts (e.g., motion, physiological noise) before inter-subject statistical analysis. This intra-subject processing generates "derived outcomes" that serve as proxies for unobservable scientific quantities (e.g., functional connectivity). While widely practiced, the impact of this preprocessing on inter-subject causal inference has not been systematically studied. Existing semiparametric frameworks for causal inference (e.g., doubly robust estimators) assume outcomes are directly observed or that derived outcomes arise solely from estimation errors, ignoring the additional uncertainty introduced by the preprocessing step itself. Furthermore, standard methods often rely on rigid linear nuisance regression and volume scrubbing (removing data points), which can introduce selection bias by excluding subjects with high motion—a group often correlated with behavioral issues. There is a lack of a principled framework that allows for flexible, data-driven intra-subject processing (including machine learning) while maintaining valid causal inference at the inter-subject level.

Methodology
The authors propose a semiparametric framework for causal inference under a hierarchical setting involving intra-subject processing followed by inter-subject analysis.

1. Hierarchical Model and Derived Outcomes:

   • The framework models a mediation structure where treatment $A$ affects an outcome $Y$ both directly and indirectly through a mediator $M$ (e.g., motion traits).
   • The true subject-level outcome $Y(a, m)$ is unobservable. Instead, it is defined as a functional $g$ of residuals obtained after regressing nuisance variables $H$ (e.g., motion parameters) from raw responses $X$ using an intra-subject processing function $f$.
   • The observed "derived outcome" $\hat{Y}$ is constructed using fitted functions $\hat{f}$ and $\hat{g}$. The framework explicitly accounts for the bias introduced by estimating $f$ and $g$.

2. Identification and Assumptions:

   • Identification of causal parameters (Natural Direct Effect, Natural Indirect Effect, Average Treatment Effect) relies on standard causal assumptions (Consistency, Positivity, Conditional Randomization, Cross-world Independence) extended to the full set of intra-subject measurements ($X, H$).
   • A key assumption is Asymptotic Unbiasedness: The derived outcome must converge to the true potential outcome as the number of intra-subject time points $T \to \infty$.

3. Multiply Robust Estimation:

   • The authors develop a Multiply Robust Augmented Inverse Probability Weighted (AIPW) estimator.
   • Rate Conditions: Theoretical guarantees require specific convergence rates for both intra-subject processing functions ($\hat{f}$) and inter-subject nuisance models (propensity scores, outcome regressions).
   • Cross-Product Structure: For outcomes defined by cross-products of residuals (e.g., correlations in functional connectivity), the framework allows the intra-subject processing functions to converge at rates slower than $T^{-1/2}$ (specifically $O(T^{-\alpha_0})$ where $n^{1/2}T^{-2\alpha_0} = o(1)$). This relaxation permits the use of flexible machine learning algorithms for nuisance regression.
   • Multiple Robustness: The estimator is consistent if any one of three combinations of nuisance models is correctly specified: (i) both propensity score models, (ii) the outcome regression and the treatment propensity score, or (iii) the outcome regression and the sequential regression term.

4. Inference and Multiple Testing:

   • The estimator admits an asymptotic linear expansion, allowing for asymptotic normality.
   • For high-dimensional inference (e.g., testing thousands of brain connections), the authors employ a step-down procedure combined with a Gaussian multiplier bootstrap. This controls the exceedance rate of the False Discovery Proportion (FDP), offering stronger error control than standard False Discovery Rate (FDR) methods.

5. Implementation:

   • The framework utilizes the Super Learner ensemble method to estimate both intra-subject nuisance functions and inter-subject models, minimizing model misspecification.
   • Cross-fitting is used to handle empirical process terms and avoid Donsker conditions.

Key Contributions

• Theoretical Framework: This is the first work to provide a theoretical framework for causal inference after intra-subject processing, explicitly modeling the propagation of estimation error from the intra-subject stage to the inter-subject causal estimates.
• Multiply Robust Estimators: The development of estimators that remain consistent under flexible machine learning for both nuisance regression (intra-subject) and causal modeling (inter-subject), provided specific rate conditions are met.
• Relaxed Convergence Rates: The demonstration that for cross-product type outcomes, intra-subject processing functions need not converge at the parametric rate, enabling the use of complex, non-linear machine learning models for artifact removal.
• High-Dimensional Inference: The application of a step-down procedure to control FDP exceedance rates in the context of derived outcomes.

Results

• Simulations: In simulations with varying sample sizes ($n=150, 300$) and time points ($T=300, 600$), the proposed SL+AIPW method (Super Learner for processing + AIPW for inference) demonstrated:
  • Low Bias: Significantly lower bias compared to standard linear regression methods, which suffered from substantial bias due to unmodeled motion mediation.
  • Valid Type-I Error: The proposed method maintained Type-I error rates close to the nominal level (0.05), whereas linear methods without mediation adjustment showed severe inflation (up to ~0.85).
  • Power: The proposed method achieved reasonable statistical power (exceeding 0.8 for $n=300$), outperforming linear methods with mediation adjustment which suffered drastic power loss.
  • Motion Correction: The Super Learner approach effectively removed motion artifacts without the need for "scrubbing" (data removal), preserving more data points and avoiding selection bias.
• Application (ASD rs-fMRI): Applied to data from 128 children with Autism Spectrum Disorder (ASD) to estimate the effect of stimulant medication on functional connectivity (FC).
  • Data Preservation: Unlike the "36p + Scrubbing" approach, which excluded 75% of participants (leaving only 32), the proposed method retained all 128 subjects.
  • Findings: The proposed method identified fewer spurious significant associations compared to the unadjusted linear model. While no effects remained significant after strict multiple testing correction in the primary analysis, an exploratory post-hoc analysis suggested potential normalizing effects of stimulants on connectivity between posterior and anterior default mode network regions. The results highlighted that motion acts as a mediator that can attenuate observed treatment effects, reinforcing the necessity of the mediation framework.

Significance and Claims
The paper claims to fill a critical gap in biomedical causal inference by providing a rigorous framework for settings where outcomes are derived from processed intra-subject data. The authors emphasize that their approach:

1. Enables Adaptive Learning: It supports the use of modern machine learning for nuisance regression (e.g., denoising fMRI data) without sacrificing the validity of causal inference, a capability lacking in existing doubly robust theory.
2. Reduces Selection Bias: By avoiding the need for data scrubbing and participant exclusion, the method mitigates the selection bias inherent in removing high-motion subjects, who often differ systematically in clinical characteristics.
3. Provides Robustness: The multiply robust property ensures that causal estimates remain valid even if some nuisance models are misspecified, provided the rate conditions are met.

The authors conclude that while the framework is theoretically sound and performs well in simulations and a real-world application, it relies on the consistency of intra-subject processing estimates. They note limitations regarding the modeling of positive definite correlation matrices in functional connectivity and suggest future work on modeling dependence structures and extending the framework to longitudinal or subgroup-specific analyses.