\(\def\loading{......LOADING......Please Wait} \def\RR{\bf R} \def\real{\mathbb{R}} \def\bold#1{\bf #1} \def\d{\mbox{Cord}} \def\hd{\widehat \mbox{Cord}} \DeclareMathOperator{\cov}{cov} \DeclareMathOperator{\var}{var} \DeclareMathOperator{\cor}{cor} \newcommand{\ac}[1]{\left\{#1\right\}} \DeclareMathOperator{\Ex}{\mathbb{E}} \DeclareMathOperator{\diag}{diag} \newcommand{\bm}[1]{\boldsymbol{#1}} \def\wait{......LOADING......Please Wait}\)

Granger Mediation Analysis for Multiple Time Series


Xi (Rossi) LUO


The University of Texas
Health Science Center
School of Public Health
Dept of Biostatistics
and Data Science


JSM, Philadelphia
August 6, 2020

Funding: NIH R01EB022911, P01AA019072, P20GM103645, P30AI042853; NSF/DMS (BD2K) 1557467

Slides viewable on web:
bit.ly/mediationjsm20

or

BigComplexData.com

Co-Author

Yi Zhao

Yi Zhao
Assistant Prof
Indiana Univ

fMRI Experiments

  • Task fMRI: performs tasks under brain scanning
  • Randomized stop/go task:
    • press button if "go";
    • withhold pressing if "stop"
  • Not resting-state: "do nothing" during scanning


Goal: infer brain activation and connectivity

fMRI data: blood-oxygen-level dependent (BOLD) signals from each cube/voxel (~millimeters), $10^5$ ~ $10^6$ voxels in total.

Multilevel fMRI Studies

Sub 1, Sess 1

Time 1

2

~200


Sub i, Sess j

Sub ~100, Sess ~4



Large, multilevel (subject, session, voxel) data
e.g. $1000 \times 4 \times 300 \times 10^6 \approx 1 $ trillion data points

Raw Data: Motor Region


Time (seconds)

Black: fMRI BOLD activity
Blue: stop onset times; Red: go onset times

$Z_t$: Stimulus onsets convoluted with Canonical HRF

$M_t$, $R_t$: fMRI time series from two brain regions

Review: Granger Causality/VAR

  • Given two (or more) time series $x_t$ and $y_t$ $$\begin{align*} x_t &= \sum_{j=1}^p \psi_{1j} x_{t-j} + \sum_{j=1}^p \phi_{1j} y_{t-j} + \epsilon_{1t} \\ y_t &= \sum_{j=1}^p \psi_{2j} y_{t-j} + \sum_{j=1}^p \phi_{2j} x_{t-j} + \epsilon_{2t} \end{align*}$$
  • Also called vector autoregressive models
  • $y$ Granger causes $x$ if $\phi_{1j} \ne 0$ Granger, 69
  • Models pair-wise connections not pathways

Granger Causality/VAR

  • Granger Causality (VAR) popular for fMRI
    • Over 10,000 google scholar results on "granger causality neuroimaging", as of May 29, 2019
  • Models multiple stationary time series
  • AR($p$) (small $p$) fits fMRI well Lingdquist, 08
  • Not for non-stationary/task fMRI
  • Cannot model stimulus effects

Conceptual Brain Model with Stimulus

Goal: quantify effects stimulipreSMAPMC regions Duann, Ide, Luo, Li (2009). J of Neurosci

Model: Mediation Analysis and SEM

$$\begin{align*}M &= Z a + \overbrace{U + \epsilon_1}^{E_1}\\ R &= Z c + M b + \underbrace{U g + \epsilon_2}_{E_2}, \quad \epsilon_1 \bot \epsilon_2\end{align*}$$
  • Indirect effect: $a \times b$; Direct effect: $c$
  • Correlated errors: $\delta = \cor(E_1, E_2) \ne 0$ if $U\ne 0$

Mediation Analysis in fMRI

  • Mediation analysis (usually assuming $U=0$)Baron&anp;Kenny, 86; Sobel, 82; Holland 88; Preacher&Hayes 08; Imai et al, 10; VanderWeele, 15;...
  • Parametric Wager et al, 09 and functional Lindquist, 12 mediation, under (approx.) independent errors
    • Stimulus $\rightarrow$ brain $\rightarrow$ user reported ratings, one brain mediator
    • Assuming $U=0$ between ratings and brain
  • Multiple mediator and multiple pathways
    • Dimension reduction by arXiv1511.09354Chen, Crainiceanu, Ogburn, Caffo, Wager, Lindquist, 15
    • Pathway Lasso penalization Zhao, Luo, 16
  • This talk: integrating Granger causality and mediation analysis

Model & Method

Our Mediation Model

$$\begin{align*}M_{t} = Z_{t} a + E_{1t},\quad R_t = Z_t c + M_t b + E_{2t}\end{align*}$$
  • Temporal VAR errors $$\begin{align*} E_{1t}&=& \sum_{j=1}^{p}\left(\omega_{11_{j}}E_{1,t-j}+\omega_{21_{j}}E_{2,t-j}\right)+\epsilon_{1t} \\ E_{2t}&=&\sum_{j=1}^{p}\left(\omega_{12_{j}}E_{1,t-j}+\omega_{22_{j}}E_{2,t-j}\right)+\epsilon_{2t} \end{align*}$$
  • Spatial errors: $\epsilon_{1t}, \epsilon_{2t}$ $$ \begin{pmatrix} \epsilon_{1t} \\ \epsilon_{2t} \end{pmatrix}\sim\mathcal{N}\left(\boldsymbol{\mathrm{0}},\boldsymbol{\Sigma}\right), \quad \boldsymbol{\Sigma}=\begin{pmatrix} \sigma_{1}^{2} & \delta\sigma_{1}\sigma_{2} \\ \delta\sigma_{1}\sigma_{2} & \sigma_{2}^{2} \end{pmatrix} $$

Equivalent Form

$$\begin{align*} \scriptsize M_{t}& \scriptsize =Z_{t}A+\sum_{j=1}^{p}\left(\phi_{1j}Z_{t-j}+\psi_{11_{j}}M_{t-j}+\psi_{21_{j}}R_{t-j}\right)+\epsilon_{1t} \\ \scriptsize R_{t}& \scriptsize =Z_{t}C+M_{t}B+\sum_{j=1}^{p}\left(\phi_{2j}Z_{t-j}+\psi_{12_{j}}M_{t-j}+\psi_{22_{j}}R_{t-j}\right)+\epsilon_{2t} \end{align*} $$
  • Nonzero $\phi$'s and $\psi$'s denote the temporal influence from stimulus to mediator/outcome and etc
  • $A$, $B$, $C$ are causal following a similar proof in Sobel, Lindquist, 04

Defining Indirect/Direct Effects

  • The (natural) indirect/direct effects for time series can be defined similarly for time series
  • $$ \mathrm{AIE}(z_{i},z_{i}^{\prime}) =(z_{i}-z_{i}^{\prime})AB, \quad \mathrm{ADE}=(z_{i}-z_{i}^{\prime})C. $$
  • We established causality using potential outcomes.
  • Ours also model time series and multilevel dependence

Causal Conditions

  • The treatment randomization regime is the same across time and participants.
  • Models are correctly specified, and no treatment-mediator interaction.
  • At each time point $t$, the observed outcome is one realization of the potential outcome with observed treatment assignment $\mathbf{\bar S}_{t}$, where $\mathbf{ \bar S}_{t}=( \mathbf{S}_{1},\dots,\mathbf{S}_{t})$.
  • The treatment assignment is random across time.
  • The causal effects are time-invariant.
  • The time-invariant covariance matrix is not affected by the treatment assignments.

Estimation: Conditional Likelihood

  • The full likelihood for our model is too complex
  • Given the initial $p$ time points, the conditional likelihood is $$ \begin{align*} & \tiny \ell\left(\boldsymbol{\Theta},\delta~|~\mathbf{Z},\mathcal{I}_{p}\right) = \sum_{t=p+1}^{T}\log f\left((M_{t},R_{t})~|~\mathbf{X}_{t}\right) \\ & \tiny = -\frac{T-p}{2}\log\sigma_{1}^{2}\sigma_{2}^{2}(1-\delta^{2})-\frac{1}{2\sigma_{1}^{2}}\|\mathbf{M}-\mathbf{X}\boldsymbol{\theta}_{1}\|_{2}^{2} \\ & \tiny -\frac{1}{2\sigma_{2}^{2}(1-\delta^{2})}\|(\mathbf{R}-\mathbf{M}B-\mathbf{X}\boldsymbol{\theta}_{2})-\kappa(\mathbf{M}-\mathbf{X}\boldsymbol{\theta}_{1})\|_{2}^{2} \end{align*} $$

Full Model for Multilevel Data

Multilevel Data: Two-level Likelihood

  • Second level model, for each subject $i$ $$(A_i,B_i,C_i) = (A,B,C) + (\eta^A_i, \eta^B_i, \eta^C_i)$$ where errors $\eta$ are normally distributed
  • The two level likelihood is conditional convex
  • Two-stage fitting: plug-in estimates from the first level
  • Block coordinate fitting: jointly optimize first level likelihood + second level likelihood
Theorem: Assume assumptions (A1)-(A6) are satisfied. Assume $\mathbb{E}(Z_{i_{t}}^{2})=q\lt \infty$, for $i=1,\dots,N$. Let $T=\min_{i}T_{i}$.
1. If $\boldsymbol{\Lambda}$ is known, then the two-stage estimator $\hat{\delta}$ maximizes the profile likelihood of model asymptotically, and $\hat{\delta}$ is $\sqrt{NT}$-consistent.
2. If $\boldsymbol{\Lambda}$ is unknown, then the profile likelihood of model has a unique maximizer $\hat{\delta}$ asymptotically, and $\hat{\delta}$ is $\sqrt{NT}$-consistent, provided that $1/\varpi=\bar{\kappa}^{2}/\varrho^{2}=\mathcal{O}_{p}(1/\sqrt{NT})$, $\kappa_{i}=\sigma_{i_{2}}/\sigma_{i_{1}}$, $\bar{\kappa}=(1/N)\sum\kappa_{i}$, and $\varrho^{2}=(1/N)\sum(\kappa_{i}-\bar{\kappa})^{2}$.
Using the two-stage estimator $\hat{\delta}$, the CMLE of our model is consistent, as well as the estimator for $\mathbf{b}=(A,B,C)$.

Theory: Summary

  • Under regularity conditions, $N$ subs, $T$ time points
  • Our $\hat \delta$ is $\sqrt{NT}$-consistent
    • This relaxes the unmeasured confounding assumption in mediation analysis
  • Our $(\hat{A},\hat{B}, \hat{C})$ is also consistent

Simulations & Real Data

Comparison

  • Our methods: GMA-h and GMA-ts
  • Previous methods: BK Baron & Kenny, MACC Zhao and Luo, KKB Kenny et al
  • Other methods do not model the temporal correlations or time series like ours

Simulations

Low bias for $AB$

Low bias for temporal cor

Gray dash lines are the truth

GMA performs the best, and recovers the temporal correlations

Real Data Experiment

  • Public data: OpenFMRI ds30
  • Stop-go experiment: withhold (STOP) from pressing buttons
  • Expect "STOP" stimuli to deactivate brain region M1
  • Goal: quantify the role of region preSMA

Result

Result

  • STOP deactivates M1 directly ($C$) and indirectly ($AB$)
  • preSMA mediates a good portion of the total effect
    • Help resolve the debates among neuroscientists
  • Other methods under-estiamte the effects
  • Novel feedback findings: M1 → preSMA after lag 1 and 2 (not shown)

Discussion

  • Mediation analysis for multiple time series
  • Method: Granger causality + mediation
    • Optimizing complex likelihood
  • Theory: identifiability, consistency
    • Causal assumptions based on potential outcomes
  • Result: low bias and improved accuracy
  • Extension: functional mediation
  • Paper in Biometrics 2019 at 10.1111/biom.13056
  • CRAN pkg: gma and references within

Thank you!


Comments? Questions?




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