Causal Inference for Preprocessed Outcomes with an Application to Functional Connectivity

TL;DR

The paper tackles causal inference when outcomes are derived from intra-subject processing, a common step in biomedical imaging. It introduces a semiparametric, two-level framework that reframes outcomes as derived $\hat{\mathbf{Y}}(a,m)$ and identifies causal parameters $\bm{\psi}(a,a')$ under standard causal assumptions, using a multiply robust AIPW estimator with cross-fitting. To handle high-dimensional settings, it develops a Gaussian multiplier bootstrap-based simultaneous inference with FDP control and a step-down augmentation. Simulations show the proposed approach reduces bias and maintains nominal error rates across varying sample sizes and time points, and its application to ASD rs-fMRI suggests motion-related mediation can attenuate treatment effects, with potential direct effects emerging under flexible modeling. Overall, the framework enables principled, data-adaptive inference in studies where intra-subject processing yields derived outcomes.

Abstract

In biomedical research, repeated measurements within each subject are often processed to remove artifacts and unwanted sources of variation. The resulting data are used to construct derived outcomes that act as proxies for scientific outcomes that are not directly observable. Although intra-subject processing is widely used, its impact on inter-subject statistical inference has not been systematically studied, and a principled framework for causal analysis in this setting is lacking. In this article, we propose a semiparametric framework for causal inference with derived outcomes obtained after intra-subject processing. This framework applies to settings with a modular structure, where intra-subject analyses are conducted independently across subjects and are followed by inter-subject analyses based on parameters from the intra-subject stage. We develop multiply robust estimators of causal parameters under rate conditions on both intra-subject and inter-subject models, which allows the use of flexible machine learning. We specialize the framework to a mediation setting and focus on the natural direct effect. For high dimensional inference, we employ a step-down procedure that controls the exceedance rate of the false discovery proportion. Simulation studies demonstrate the superior performance of the proposed approach. We apply our method to estimate the impact of stimulant medication on brain connectivity in children with autism spectrum disorder.

Figures (4)

• Figure 1: Panel A: The general diagram of causal mediation analysis with derived outcomes in the presence of intra-subject nuisance. Treatment is denoted by $A \in \{0,1\}$. $\mathbf{U}_D \in \mathbb{R}^{l_D}$ and $\mathbf{U}_I \in \mathbb{R}^{l_I}$ are latent factors along the direct and indirect treatment–outcome pathways respectively. The observed mediator $M \in \mathbb{R}$ is generated from the indirect latent factor $\mathbf{U}_I$. Intra-subject repeated measurements of responses $\mathbf{X}_1,\cdots,\mathbf{X}_T \in \mathbb{R}^V$ and nuisance $\mathbf{H}_1,\cdots,\mathbf{H}_T \in \mathbb{R}^p$ are influenced by both $\mathbf{U}_D$ and $M$. The vector of derived outcomes $\mathbf{Y} \in \mathbb{R}^J$ is constructed from $\mathbf{X}$ and $\mathbf{H}$. Panel B: The diagram for causal inference in functional connectivity studies. $\mathbf{U}_M \in \mathbb{R}^{l_M}$ and $\mathbf{U}_B \in \mathbb{R}^{l_B}$ are latent factors representing motion-related and non-motion-related brain wiring, respectively. Non-motion neural activity $\mathbf{S}_{B} \in \mathbb{R}^{l_{sb}}$ is generated by $\mathbf{U}_B$, while motion-related activity $\mathbf{S}_{M} \in \mathbb{R}^{l_{sm}}$ is generated by $\mathbf{U}_M$. Both $\mathbf{S}_{B}$ and $\mathbf{S}_{M}$ are unobserved. $\mathbf{S}_{M}$ further generates the observed motion trait $M$, often represented as mean framewise displacement. Both $\mathbf{S}_{B}$ and $M$ influence the fMRI response $\mathbf{X}_t$ and transient movement variables $\mathbf{H}_t$ at each scan time $t=1,\cdots,T$. These are then used to construct the $J$-dimensional derived outcomes of functional connectivity $\mathbf{Y}$.
• Figure 2: Example of intra-subject motion regression from a single simulation. The black line shows the simulated motion-free neural signal. In the first panel, the red line represents the motion-contaminated signal before correction. In the second panel, the green line (12p) shows residuals from linear regression with 3 main terms, 3 temporal derivatives, and 6 quadratic terms. In the third panel, the blue line (12p + Scrubbing) applies the same regression after removing high-motion time points, with censored points marked in pink. In the final panel, the purple line (SL) shows residuals obtained from Super Learner nuisance regression.
• Figure 3: The stimulant treatment effects on FC between Scheafer 100 and 19 MNI subcortical brain regions using "36p + Linear" (A), "36p Scrub + Linear M" (B), and our proposed method (C). D, E, F: The circle plots of significant treatment effects estimations from three methods.
• Figure 4: The brain map shows the Z-statistics for stimulant treatment effects on the FC between the seed region "17networks_LH_DefaultA_pCunPCC_1" and all other regions in the Schaefer-100 parcellation, estimated using "36p + Linear" (A), "36p + Linear M" (B), and our proposed method (C).

Theorems & Definitions (9)

• Definition 1
• Example 1
• Definition 2
• Theorem 1
• Definition 3
• Lemma 1
• Theorem 2
• Corollary 1
• Corollary 2