The University of Texas Health Science Center at Houston
School of Public Health
Dept of Biostatistics and Data Science
IMSI, Univ of Chicago, USA August 28, 2024
Funding: NIH R01MH126970, R01EB022911
fMRI Experiments
Task fMRI: performs tasks under brain scanning
Randomized stop/go task:
press button if "go";
withhold pressing if "stop"
Resting-state fMRI: "do nothing" during scanning
Challenging to infer causality from noisy, indirect fMRI measures
Judea Pearl wrote: in The Causal Foundations of Structural Equation Modeling
...causal effects in observational studies can only be substantiated from a combination of data and untested theoretical assumptions, not from the
data alone.
Multilevel fMRI Studies
Subject 1, Session 1
Time 1
2
…
~T
⋮
Subject i, Session j
…
⋮
Sub N, Sess K
…
Subject x Session x Time, millions of data points
Brain Mediation Model
Goal: quantify the direct and indirect effects
Scalar Mediation using SEM
$$\small \begin{align*}M &= Z a + \overbrace{U + \epsilon_1}^{E_1}\qquad R = Z c + M b + \underbrace{U g + \epsilon_2}_{E_2}, \quad \epsilon_1 \bot
\epsilon_2\end{align*}$$
Indirect effect: $a \times b$; Direct effect: $c$
Untestable assumption: $\delta=cor(E_1, E_2) =0 $, no confounders Baron&Kenny, 86; Sobel, 82; Holland 88; Preacher&Hayes 08; Imai et al, 10; VanderWeele, 15; Lindquist, 12 ...
We first prove two different models generate same single-trial BOLD activations if only observing $Z$, $M$, and $R$
without measuring $U$
Multilevel Mediation Zhao and Luo, 2023
We optimize the joint likelihood of all multilevel mediation data, allowing $\delta \ne 0$
We prove a unique solution and consistent estimation for $\delta$.
From data alone, we estimate the causal effects, without the untestable assumption of no unmeasured confounding.
Large fMRI data is helpful for resolving causality
Model Identifiability and Bias
Unique $\delta$ for ML
Effects doubled
Causal effects in fMRI actually larger after removing unmeasured confounding
Granger Mediation Zhao and Luo, 2019
We build multilevel, time series mediation model, based on Granger causality or VAR
A spatial-temporal model for brain pathways
We specify additional causal assumptions
We prove asymptotic convergence rates
Granger Mediation Model
→ Time
Simulations Match Theory
Low bias for $AB$
Low bias for temporal cor
Gray dash lines are the truth
GMA performs the best, and recovers the temporal correlations
Stronger indirect pathways while other methods under-estimate the effects/ratios
Help resolve the debates among neuroscientists
Novel feedback findings: M1 → preSMA after lag 1 and 2 (not shown)
Theorem. Under the assumptions and the above models, $\mathrm{TE}(t;\bar{z}_{t},\bar{z}_{t}^{*})$ is parametrically identified:
$$
\begin{align}
\scriptsize
\int_{\Omega_{t}^{2}}(z_{s}-z_{s}^{*})\gamma(s,t)~\mathrm{d}s+\int_{\Omega_{t}^{1}}(z_{s}-z_{s}^{*})\int_{\Omega_{t}^{3}}\alpha(s,u)\beta(u,t)~\mathrm{d}u~\mathrm{d}s.
\end{align}
$$
and the direct and indirect components are also parametrically identified.
Simulation Comparison
Challenging for scalar mediation methods to capture functional causal effects
Brain mediation effects are dynamic depending on stimulus patterns and history
High Dim Mediation Zhao and Luo, 2022
Full Model
Reduced Model
$$\scriptsize \sum_{k=1}^K \| M_k - Z A_k \|_2^2 + \| R - Z C - \sum_k M_k B_k \|_2^2 + \mbox{Pen}(A, B)$$