[Paper Review] Granger Mediation Analysis of Multiple Time Series with an Application to fMRI

Yi Zhao, Xi Luo|arXiv (Cornell University)|Sep 15, 2017

Functional Brain Connectivity Studies2 references3 citations

TL;DR

This paper introduces Granger Mediation Analysis (GMA), a novel framework for causal mediation analysis in multiple time series, integrating vector autoregressive models with causal mediation modeling to handle temporal dependence and feedback effects. It enables valid estimation of direct and indirect effects in fMRI data, reducing bias and improving power compared to existing methods, particularly in capturing dynamic feedback between brain regions.

ABSTRACT

It becomes increasingly popular to perform mediation analysis for complex data from sophisticated experimental studies. In this paper, we present Granger Mediation Analysis (GMA), a new framework for causal mediation analysis of multiple time series. This framework is motivated by a functional magnetic resonance imaging (fMRI) experiment where we are interested in estimating the mediation effects between a randomized stimulus time series and brain activity time series from two brain regions. The stable unit treatment assumption for causal mediation analysis is thus unrealistic for this type of time series data. To address this challenge, our framework integrates two types of models: causal mediation analysis across the variables and vector autoregressive models across the temporal observations. We further extend this framework to handle multilevel data to address individual variability and correlated errors between the mediator and the outcome variables. These models not only provide valid causal mediation for time series data but also model the causal dynamics across time. We show that the modeling parameters in our models are identifiable, and we develop computationally efficient methods to maximize the likelihood-based optimization criteria. Simulation studies show that our method reduces the estimation bias and improve statistical power, compared to existing approaches. On a real fMRI data set, our approach not only infers the causal effects of brain pathways but accurately captures the feedback effect of the outcome region on the mediator region.

Motivation & Objective

• To address the limitations of traditional mediation analysis in time series data, where the Stable Unit Treatment Value Assumption (SUTVA) is violated due to temporal dependence.
• To develop a unified framework that integrates causal mediation analysis with vector autoregressive (VAR) models to model dynamic, time-dependent relationships across multiple time series.
• To extend the framework to multilevel data to account for individual variability and correlated errors between mediator and outcome variables.
• To ensure model identifiability and develop computationally efficient likelihood-based optimization methods for parameter estimation.
• To evaluate the method’s performance through simulations and real fMRI data, demonstrating improved bias reduction and statistical power.

Proposed method

• Proposes a hybrid model combining structural equation modeling for mediation with multivariate vector autoregressive (VAR) models to capture temporal dynamics in treatment, mediator, and outcome time series.
• Models the joint distribution of time series using a multivariate normal likelihood with a structured mean and covariance matrix that accounts for cross-variable and temporal dependencies.
• Incorporates a hierarchical structure to model individual-level variation and correlated errors between mediator and outcome, enabling multilevel mediation analysis.
• Derives explicit, closed-form estimators for model parameters (e.g., β, γ, δ) via likelihood maximization, with projection onto the parameter space to ensure constraints are respected.
• Uses a two-stage optimization: first, conditional on δ, the likelihood is convex and separable, enabling efficient computation of variance components and regression coefficients.
• Employs bootstrap resampling (200 samples) to estimate standard errors and construct 95% confidence intervals for key parameters like the transition matrix Ω.

Experimental results

Research questions

• RQ1 Can a mediation framework be developed that properly accounts for temporal dependence in time series data, particularly in fMRI studies?
• RQ2 How can feedback effects from the outcome region back to the mediator region be modeled and estimated in a causal mediation framework?
• RQ3 Does the proposed GMA framework reduce estimation bias and improve statistical power compared to existing methods like MACC-h and KKB?
• RQ4 To what extent does the inclusion of multilevel structure improve estimation accuracy in the presence of individual variability and correlated errors?
• RQ5 Is the model robust to model mis-specification in the error structure, such as higher-order Markov errors?

Key findings

• GMA-h significantly reduces bias in estimating the indirect effect (B) and direct effect (C) compared to MACC-h and KKB, especially as the true feedback parameter δ increases.
• GMA-h achieves lower mean squared error (MSE) in estimating both B and C, with MSE converging to zero as sample size (N) and time series length (Ti) increase.
• In the real fMRI application, GMA-h successfully captured the feedback effect of the M1 region on the preSMA region, which was not detectable with standard Granger causality or mediation models.
• The estimated transition matrix Ω from GMA-h showed strong agreement with Granger causality for off-diagonal elements, but GMA-h provided more accurate estimates of diagonal elements, indicating better modeling of within-series dynamics.
• Under a MAR(3) error model, the third-order coefficients (Ω3) were close to zero, suggesting that a MAR(2) model is sufficient for this dataset, validating model parsimony.
• The likelihood function is conditionally convex, ensuring stable and efficient optimization, and the parameter estimators are identifiable under the proposed model structure.