Causal Inference for Unobservable Multivariate Outcomes, with Applications to Brain Effective Connectivity

Abstract

Evaluating the causal effect of an intervention on multivariate outcomes is challenging when the outcomes are interdependent and derived rather than directly observed. Effective connectivity, which summarizes the directional neural communication between brain regions, is one such derived relational outcome. Estimating how external interventions affect effective connectivity introduces two layers of causal inference problems: identifying directional relationships among brain regions from high-dimensional neuroimaging time series and estimating the causal effect of the intervention on these derived relationships. Each layer introduces distinct biases. The first arises from within-outcome dependencies unrelated to the intervention; to address this, we propose a sample-splitting method for estimating meaningful, and potentially causally informative, effective connectivity measures. The second arises from confounding between the intervention and the derived outcomes; to address this, we apply inverse probability weighting methods and incorporate multiple testing when causal effects on multiple components of the outcomes are of interest. We demonstrate, through theoretical results and simulations, that the proposed methods are asymptotically valid under certain conditions with effective type-I and familywise error control. Finally, we apply the proposed methods to examine the causal effect of amyloid on effective connectivity using the resting-state fMRI data from the Alzheimer's Disease Neuroimaging Initiative database.

Figures (5)

• Figure 1: A graphical representation of two approaches for using $\widehat{\mathbf{Y}}$ in causal inference on $\mathbf{Y}$. In panel (a), $\widehat{\mathbf{Y}}$ is a function of $\widetilde{\mathbf{X}}$ that is closely related to the true, unobserved outcome $\mathbf{Y}$ (indicated by the dashed line); when a standard measure of $\mathbf{Y}$ exists, the effect on a unit change in $\mathbf{Y}$ can be estimated by the effect on a unit change in $\widehat{\mathbf{Y}}$. In panel (b), $\widehat{\mathbf{Y}}$ serves as a proxy for $\mathbf{Y}$, and the units of $\mathbf{Y}$ and $\widehat{\mathbf{Y}}$ are not necessarily comparable; the intervention $Z$ is assumed to affect $\widehat{\mathbf{Y}}$ through $\mathbf{Y}$ and the intermediate outcome $\widetilde{\mathbf{X}}$.
• Figure 2: Representations of Granger causality relationships under (a) control and (b) intervention, where the presence of a direct arrow indicates Granger causality between two time series and the arrow width reflects the strength of that relationship.
• Figure 3: Simulation results when the number of subjects $n=100$. Panel (a) shows the FWER under the global null; panel (b) shows power under alternatives with varying intervention intensities ($\delta$); panels (c) and (d) show the FDP and FDPex, respectively, across different values of $\delta$.
• Figure 4: A heat map visualizing the full matrix of estimated causal effects of amyloid positivity on effective connectivity across pairs of ROIs ($p=120$).
• Figure A1: Simulation results when the number of subjects $n=200$. Panel (a) shows the FWER under global null; panel (b) shows power under alternatives with varying intervention intensities ($\delta$); panels (c) and (d) show the FDP and FDPex, respectively, across different values of $\delta$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Remark 1
• Definition 1: Asymptotic validity of sample-splitting lunde_sample_2019
• Definition 2: Asymptotic deletion stability
• Theorem 3
• Theorem 4
• Corollary 1
• Corollary 2
• Lemma 1: Tightness