  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.
