Granger Mediation Analysis of Functional Magnetic Resonance Imaging Time Series

Granger Mediation Analysis of Functional Magnetic Resonance Imaging Time Series Yi Zhao and Xi Luo Department of Biostatistics Brown University June 8, 2017

Overview 1 Introduction 2 Model and Method 3 Simulation Study 4 An fmri Study 5 Discussion

Introduction

Motivation Credit: NSF Task-related functional MRI (fmri) fmri: measures brain activities task fmri: perform task under fmri scanner response conflict task GO trial: push the button STOP trial: withhold the pushing 1 / 29

Motivation Task-related functional MRI (fmri) fmri: measures brain activities Credit: NSF task fmri: perform task under fmri scanner response conflict task GO trial: push the button STOP trial: withhold the pushing Objective identify causal effects of task stimulus on brain activity infer brain connectivity (effective connectivity) 1 / 29

Functional MRI Data 10 5 10 6 uniformly spaced voxels 264 putative functional regions 1 1 Power et al., Neuron, 2011 2 / 29

Functional MRI Data 105 106 uniformly spaced voxels 264 putative functional regions1 time series or functional data 1 Power et al., Neuron, 2011 2 / 29

Response conflict task Brain regions of interest primary motor cortex (PMC): responsible for movement presupplementary motor area (presma): primary region for motor response prohibition Objective Quantify the causal effects stimulus presma, stimulus PMC presma PMC 2 2 Obeso et al., Brain Stimulation, 2013 3 / 29

Brain activities can be modeled as a linear superposition of task and spontaneous random fluctuations 3 task-related signal stimulus presma random fluctuation stimulus PMC 3 Cole et al., Neuron, 2014. 4 / 29

A E 1 presma (M) B Unmeasured confounding (U) E 2 Stimulus (Z) C PMC (R) M = ZA + E 1 R = ZC + MB + E 2 Task-related signal indirect effect (AB): stimulus presma PMC direct effect (C): stimulus PMC Random fluctuation spatiotemporal dependency Unmeasured confounding random fluctuation of other regions 4 ; system errors: head motion 5 4 Fox et al., Nature Neuroscience (2006), Mason et al., Science (2007) 5 Sobel and Lindquist, JASA, 2014 5 / 29

Data structure Randomized stimulus Participant 1 presma time series Participant 2 PMC time series. Causal effects for Participant 1 Participant N 6 / 29

Major challenges Unmeasured confounding Iterregional and temporal dependency Individual variation 7 / 29

Existing Methods Mediation Baron and Kenny (1996), Imai et al. (2010) Kenny et al. (2003): multilevel mediation Ten Have et al. (2007), Small (2011): instrumental variables Lindquist (2012): IV and functional mediation Robins et al. (2000), van der Laan and Petersen (2008), VanderWeele (2009): longitudinal mediation Granger causality (time series) Granger (1969) Granger causality of two time series time series y is said to Granger cause time series x if the current value of x can be predicted by the past values of x and y Goebel et al. (2003), Harrison et al. (2003), Valdes-Sosa (2004) vector autoregression (VAR) of fmri time series 8 / 29

Multilevel Granger mediation model Lower-level model fmri time series of one participant causal mediation + spatiotemporal dependency unmeasured confounding nonidentifiability issue Higher-level model individual variation identifiability and consistency 9 / 29

Model and Method

Lower-level model Z 1 Z 2 Z T 1 Z T M 1 ẽ 11 M 2 ẽ 12 M T 1 ẽ 1,T 1 M T ẽ 1T E 11 E 12 E 1,T 1 E 1T R 1 R 2 R T 1 R T E 21 E 22 E 2,T 1 E 2T ẽ 21 ẽ 22 ẽ 2,T 1 ẽ 2T Time 1 Time 2 Time T 1 Time T task-related signal: Z t M t, Z t R t, Z t M t R t spontaneous random fluctuations: E 1t M t, E 2t R t unmeasured confounding: U t 10 / 29

Lower-level model Z 1 Z 2 Z T 1 Z T M 1 ẽ 11 M 2 ẽ 12 M T 1 ẽ 1,T 1 M T ẽ 1T E 11 E 12 E 1,T 1 E 1T R 1 R 2 R T 1 R T E 21 E 22 E 2,T 1 E 2T ẽ 21 U 1 ẽ 22 U 2 ẽ 2,T 1 U T 1 ẽ 2T U T Time 1 Time 2 Time T 1 Time T task-related signal: Z t M t, Z t R t, Z t M t R t spontaneous random fluctuations: E 1t M t, E 2t R t unmeasured confounding: U t 10 / 29

Causal mediation with VAR(p) autoregressive error Regression models: task-induced brain activations Structural equation modeling for causal mediation M t = Z t A + E 1t R t = Z t C + M t B + E 2t Remaining residuals: resting-state random fluctuations 6 VAR(p) model E 1t = p j=1 ω 11 j E 1,t j + p j=1 ω 21 j E 2,t j + ɛ 1t E 2t = p j=1 ω 12 j E 1,t j + p j=1 ω 22 j E 2,t j + ɛ 2t ( ) ( ) ɛ1t σ 2 N (0, Σ), Σ = 1 δσ 1 δ 2 ɛ 2t δσ 1 σ 2 σ2 2 (ɛ 1t, ɛ 2t) Gaussian white noise, (ɛ 1t, ɛ 2t) 6 Fair et al., NeuroImage,2007 = (ɛ 1s, ɛ 2t) for s t 11 / 29

Z 1 Z 2 Z T 1 Z T M 1 ẽ 11 M 2 ẽ 12 M T 1 ẽ 1,T 1 M T ẽ 1T E 11 E 12 E 1,T 1 E 1T R 1 R 2 R T 1 R T E 21 E 22 E 2,T 1 E 2T ẽ 21 U 1 ẽ 22 U 2 ẽ 2,T 1 U T 1 ẽ 2T U T E 1t = (ω 11j E 1,t j + ω 21j E 2,t j) + ν 1U t + ẽ 1t = (ω 11j E 1,t j + ω 21j E 2,t j) + ɛ 1t j E 2t = (ω 12j E 1,t j + ω 22j E 2,t j) + ν 2U t + ẽ 2t = (ω 12j E 1,t j + ω 22j E 2,t j) + ɛ 2t j j j Assumption: additive unmeasured confounding ν 1 ν 2 0 ɛ 1t, ɛ 2t correlated 12 / 29

Assumptions Mediation model (A1) no interference the treatment assignment regime is the same (A2) models are correctly specified (A3) Z t is randomized with positive probabilities (A4) no unmeasured confounding effect on mediator-outcome relationship unmeasured confounding U t only affects current time mediator and outcome, and the effect is additive Granger causality (A5) {(E 1t, E 2t )} t stationary 7 7 Chang and Glover, NeuroImage, 2010. 13 / 29

Granger mediation analysis (GMA): causal mediation SEM + VAR(p) M t = Z t A + E 1t E 1t = p j=1, ω 11 j E 1,t j + p j=1 ω 21 j E 2,t j + ɛ 1t R t = Z t C + M t B + E 2t E 2t = p j=1 ω 12 j E 1,t j + p j=1 ω 22 j E 2,t j + ɛ 2t (E 1t, E 2t ) s not independent difficult to derive estimators in explicit form (ɛ 1t, ɛ 2t ) Gaussian white noise: (ɛ 1t, ɛ 2t ) = (ɛ 1s, ɛ 2s ) for t s reparametrize the model M t = Z t A + p j=1 φ 1 j Z t j + p j=1 ψ 11 j M t j + p j=1 ψ 21 j R t j + ɛ 1t R t = Z t C + M t B + p j=1 φ 2 j Z t j + p j=1 ψ 12 j M t j + p j=1 ψ 22 j R t j + ɛ 2t 14 / 29

GMA { M t = Z ta + E 1t R t = Z tc + M tb + E 2t { E 1t = j ω 11 j E 1,t j + j ω 21 j E 2,t j + ɛ 1t E 2t = j ω 12 j E 1,t j + j ω 22 j E 2,t j + ɛ 2t Reparametrized model M t = Z ta + j φ 1 j Z t j + j ψ 11 j M t j + j ψ 21 j R t j + ɛ 1t R t = Z tc + M tb + j φ 2 j Z t j + j ψ 12 j M t j + j ψ 22 j R t j + ɛ 2t Z t Z t+1 Z t Z t+1 ẽ 1t ẽ 1,t+1 M t ẽ 1t M t+1 ẽ 1,t+1 M t M t+1 E 1t E 1,t+1 R t R t+1 R t R t+1 E 2t E 2,t+1 ẽ 2t U t ẽ 2,t+1 U t+1 ẽ 2t U t ẽ 2,t+1 U t+1 Time t Time (t + 1) Time t Time (t + 1) 15 / 29

GMA (causal mediation SEM + VAR(p)) { { M t = Z ta + E 1t E 1t = j, ω11 j E 1,t j + j ω21 j E 2,t j + ɛ 1t R t = Z tc + M tb + E 2t E 2t = j ω12 j E 1,t j + j ω22 j E 2,t j + ɛ 2t Lemma Reparametrized model { M t = Z ta + p j=1 φ1 j Z t j + p j=1 ψ11 j M t j + p j=1 ψ21 j R t j + ɛ 1t R t = Z tc + M tb + p j=1 φ2 j Z t j + p j=1 ψ12 j M t j + p j=1 ψ22 j R t j + ɛ 2t Given (φ j, Ψ j ) and (A, B, C), Ω j can be uniquely determined. 16 / 29

M t = Z t A + p j=1 φ 1 j Z t j + p j=1 ψ 11 j M t j + p j=1 ψ 21 j R t j + ɛ 1t R t = Z t C + M t B + p j=1 φ 2 j Z t j + p j=1 ψ 12 j M t j + p j=1 ψ 22 j R t j + ɛ 2t Given δ = Cor(ɛ 1t, ɛ 2t ), estimate the rest by conditional likelihood Let θ 1 = {A, {φ 1j }, {ψ 11j }, {ψ 21j }}, θ 2 = {C, {φ 2j }, {ψ 12j }, {ψ 22j }}, ˆθ 1 = (X X) 1 X M, ˆσ 1 2 = M (I P X )M/(T p) ˆθ 2 = (X (I P M )X) 1 X (I P M )R + δˆσ 2/ˆσ 1(X X) 1 X M ˆB = (M M) 1 M(I X(X (I P M )X) 1 X (I P M ))R δˆσ 2/ˆσ 1 ˆσ 2 2 = R (I P MX P M )R/(T p)(1 δ 2 ) 17 / 29

M t = Z t A + p j=1 φ 1 j Z t j + p j=1 ψ 11 j M t j + p j=1 ψ 21 j R t j + ɛ 1t R t = Z t C + M t B + p j=1 φ 2 j Z t j + p j=1 ψ 12 j M t j + p j=1 ψ 22 j R t j + ɛ 2t Given δ = Cor(ɛ 1t, ɛ 2t ), estimate the rest by conditional likelihood Let θ 1 = {A, {φ 1j }, {ψ 11j }, {ψ 21j }}, θ 2 = {C, {φ 2j }, {ψ 12j }, {ψ 22j }}, ˆθ 1 = (X X) 1 X M, ˆσ 2 1 = M (I P X )M/(T p) ˆθ 2 = (X (I P M )X) 1 X (I P M )R + δˆσ 2/ˆσ 1(X X) 1 X M ˆB = (M M) 1 M(I X(X (I P M )X) 1 X (I P M ))R δˆσ 2/ˆσ 1 ˆσ 2 2 = R (I P MX P M )R/(T p)(1 δ 2 ) Theorem For fixed δ ( 1, 1), given the initial p observations, the conditional likelihood of the reparametrized model achieves the same maximum. 17 / 29

(δ, θ 2, B, σ 2 ) not identifiable under lower-level model time series of one participant fmri study with hierarchically nested data structure participants time series individual variation in the causal effects and overparametrization in the lower-level model 18 / 29

Lower-level model: for participant i M it = Z it A i + E i1t E i1t = j, ω i 11j E i1,t j + j ω i 21j E i2,t j + ɛ i1t R it = Z it C i + M it B i + E i2t E i2t = j ω i 12j E i1,t j + j ω i 22j E i2,t j + ɛ i2t Higher-level model: linear model for the coefficients A i A ɛ A i b i = B i = B + ɛ B i = b + η i C i C ɛ C i A, B, and C are the population-level causal effects of interest η i N (0, Λ) (δ i, θ i2, B i, σ 2i ) not identifiable Assumption (A6): δ i = δ across participants 19 / 29

l(υ) = N T N log P(R it, M it Z it, Z (p) i, M (p) t 1 i, R (p) t 1 i, θ i1, θ i2, B t 1 i, δ, σ 1i, σ 2i ) + log P(b i b, Λ) i=1 } t=1 {{ } l 1 : lower-level time series i=1 }{{} l 2 : causal coefficients Υ = {δ, b, Λ, (θ i1, θ i2, B i ), (σ 1i, σ 2i )} optimization problem max Υ:{Λ,(σ 1i,σ 2i )} S l(υ) S: constraint set for variance components 20 / 29

l(υ) = N T N log P(R it, M it Z it, Z (p) i, M (p) t 1 i, R (p) t 1 i, θ i1, θ i2, B t 1 i, δ, σ 1i, σ 2i ) + log P(b i b, Λ) i=1 } t=1 {{ } l 1 : lower-level time series i=1 }{{} l 2 : causal coefficients Block coordinate-descent optimize over l(υ) joint optimization of a large number of parameters fast block search for conditional convex function Two-stage optimize over l 1 first, and then over l 2 easy to compute b i s latent parameters, not actually observed 21 / 29

l(υ) = N T N log P(R it, M it Z it, Z (p) i, M (p) t 1 i, R (p) t 1 i, θ i1, θ i2, B t 1 i, δ, σ 1i, σ 2i ) + log P(b i b, Λ) i=1 } t=1 {{ } l 1 : lower-level time series i=1 }{{} l 2 : causal coefficients Block coordinate-descent Two-stage log likelihood (l 1 + l2) 34400 34350 34300 δ^ = 0.492 higher level log likelihood (l 2) 480 460 440 420 400 380 360 δ^ = 0.466 0.5 0.0 0.5 δ 0.5 0.0 0.5 δ 21 / 29

Simulation Study

Simulation Study N = 50 subjects, T i Poisson(100), Z it Bern(0.5) Each time point is a randomized trial A = 0.5, B = 1, C = 0.5 Λ = 0.5I 3, σ i1 = 1, σ i2 = 2 (E 1t, E 2t ) stationary: p = 1, Ω transition matrix, (cov (E 0 ) = Ξ) ( ) ( ) E1t = Ω E1,t 1 + ɛ t, ɛ t N (0, Σ), Ξ = Ω ΞΩ + Σ E 2t E 2,t 1 Methods Granger mediation using block coordinate descent (GMA-h); Granger mediation with two-stage approach (GMA-ts); CMA-h (and CMA-ts) by Zhao and Luo (2014) (R macc package); multilevel SEM (KKB) 8 ; Baron and Kenny (BK) method. 8 Kenny et al., Psychological methods, 2003. 22 / 29

δ estimate ˆδ δ δ^ 0.5 0.0 0.5 GMA h GMA ts CMA h CMA ts True δ^ δ 0.04 0.02 0.00 0.02 GMA h GMA ts CMA h CMA ts 0.5 0.0 0.5 δ 0.5 0.0 0.5 δ 23 / 29

Direct effect Indirect effect 0.5 0.0 0.5 1.0 C^ GMA h GMA ts CMA h KKB (δ = 0) BK (δ = 0) True (C = 0.5) AB^ p 1.0 0.5 0.0 GMA h GMA ts CMA h KKB (δ = 0) BK (δ = 0) True (AB = 0.5) 0.5 0.0 0.5 0.5 0.0 0.5 δ δ 24 / 29

Ω 2 1 0 1 ω^ 11 ω^ 21 ω^ 12 ω^ 22 True 0.5 0.0 0.5 δ GMA-h estimates 25 / 29

Asymptotic property ˆδ ) MSE (ˆδ δ^ 0.44 0.46 0.48 0.50 50 100 200 500 1000 2000 5000 N GMA h GMA ts True (δ = 0.5) MSE(δ^) 0.000 0.002 0.004 0.006 0.008 0.010 50 100 200 500 1000 2000 5000 N GMA h GMA ts N: number of participants; T i = N: number of time points 26 / 29

An fmri Study

fmri study N = 96 with T i = 295 time points randomized STOP/GO trials outcome (R): time series of the primary motor cortex (PMC) mediator (M): time series of the presupplementary motor area (presma) Methods GMA-h, GMA-ts CMA-h KKB BK participant-level bootstrap repeated 500 times 27 / 29

Causal effects Transition matrix Ω 1.0 0.5 0.0 0.5 1.0 GMA h GMA ts CMA h KKB (δ = 0) BK (δ = 0) Ω^ 0.4 0.2 0.0 0.2 0.4 ω^ 11 = 0.401 ω^ 12 = 0.020 ω^ 21 = 0.231 ω^ 22 = 0.344 Direct Indirect (product) δ Indirect (difference) GMA-h 0.095 (-0.012, 0.209) GMA-ts 0.065 (-0.041, 0.178) CMA-h 0.073 (-0.045, 0.201) 0 100 200 300 400 500 ω 21 0: feedback effect of PMC (R) on presma (M) p = 1, estimate of Ω 1 under VAR(1) p = 2, Ω 2 0 28 / 29

Discussion

Discussion 9 Propose a two-level mediation model for fmri time series. remove unmeasured confounding capture both interregional and temporal dependencies address individual variation issue Jointly optimize a large number of parameters Improve estimation accuracy and consistency Identify feedback effect of (outcome) PMC on (mediator) presma Future work theory about δ identifiability and consistency (convergence rate) covariates more general form of the spatio/temporal correlation structure spatial variation in the haemodynamic response function (HRF) relax the concurrent influence assumption of unmeasured confounding 9 2017 JSM Mental Health Section one of two highest rate papers 29 / 29

Acknowledgements Advisor Xi (Rossi) Luo, PhD, Department of Biostatistics, Brown University Dissertation Committee Reader Joseph Hogan, ScD, Department of Biostatistics, Brown University Jerome Sanes, PhD, Department of Neuroscience, Brown University Eli Upfal, PhD, Department of Computer Science, Brown University Yen-Tsung Huang, MD & ScD, Departments of Epidemiology and Biostatistics, Brown University Brian Caffo, PhD, Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health

Thank you!