Mediation Analysis in Neuroimaging Studies Yi Zhao Department of Biostatistics Johns Hopkins Bloomberg School of Public Health January 15, 2019
Overview Introduction Functional mediation analysis High-dimensional mediation analysis Multimodal neuroimaging data integration Discussion 2 / 35
Mediation analysis Mediator ( M ) β α Treatment ( X ) Outcome ( Y ) γ • Quantifies the intermediate effect of the mediator 3 / 35
Mediation analysis Mediator ( M ) β α Treatment ( X ) Outcome ( Y ) γ • Quantifies the intermediate effect of the mediator • Helps clarify the underlying causal mechanism 3 / 35
Mediation analysis Mediator ( M ) β α Treatment ( X ) Outcome ( Y ) γ • Quantifies the intermediate effect of the mediator • Helps clarify the underlying causal mechanism • Popular approach: structural equation modeling (SEM) M = Xα + ǫ 1 Y = Xγ + Mβ + ǫ 2 • αβ : indirect (mediation) effect • γ : direct effect 3 / 35
Neuroimaging studies • Non-invasive techniques • e.g. structural/diffusion/functional MRI, PET, MEG/EEG • Functional MRI (fMRI) • brain activity: changes in brain hemodynamics • resting-state and task-based fMRI Credit: NSF 4 / 35
Neuroimaging studies • Non-invasive techniques • e.g. structural/diffusion/functional MRI, PET, MEG/EEG • Functional MRI (fMRI) • brain activity: changes in brain hemodynamics • resting-state and task-based fMRI Credit: NSF Objective • Resting-state fMRI • brain co-activation (functional connectivity) • impact on cognitive behaviors 4 / 35
Neuroimaging studies • Non-invasive techniques • e.g. structural/diffusion/functional MRI, PET, MEG/EEG • Functional MRI (fMRI) • brain activity: changes in brain hemodynamics • resting-state and task-based fMRI Credit: NSF Objective • Resting-state fMRI • brain co-activation (functional connectivity) • impact on cognitive behaviors • Task-based fMRI • causal effect of stimulus on brain activity • brain connectivity (effective connectivity) 4 / 35
Challenges • Large n with hierarchically nested data structure • participants ( → sessions) → tasks/trials • population level inference • Large p • 10 5 ∼ 10 6 uniformly spaced voxels • > 100 putative functional/anatomical regions • high-dimensional problem • Complex data output • time series • functional data 5 / 35
Challenges • Large n with hierarchically nested data structure • participants ( → sessions) → tasks/trials • population level inference • Large p • 10 5 ∼ 10 6 uniformly spaced voxels • > 100 putative functional/anatomical regions • high-dimensional problem • Complex data output • time series • functional data 5 / 35
Motivating example: response conflict task • Response conflict task • “GO” trial: button press • “STOP” trial: withhold pressing • Brain regions of interest • primary motor cortex (M1) : responsible for movement • presupplementary motor area (preSMA) : primary region for motor response prohibition • Objective : quantify causal effects • stimulus → preSMA, stimulus → M1 • preSMA → M1 (Obeso et al., 2013) 6 / 35
Motivating example: response conflict task • Response conflict task • “GO” trial: button press • “STOP” trial: withhold pressing • Brain regions of interest • primary motor cortex (M1) : responsible for movement • presupplementary motor area (preSMA) : primary region for motor response prohibition • Objective : quantify causal effects • stimulus → preSMA, stimulus → M1 • preSMA → M1 (Obeso et al., 2013) 6 / 35
Motivating example: response conflict task • Response conflict task • “GO” trial: button press • “STOP” trial: withhold pressing • Brain regions of interest • primary motor cortex (M1) : responsible for movement • presupplementary motor area (preSMA) : primary region for motor response prohibition • Objective : quantify causal effects • stimulus → preSMA, stimulus → M1 • preSMA → M1 (Obeso et al., 2013) 6 / 35
Mediation analysis Mediator M ( t ) Treatment X ( t ) Outcome Y ( t ) • Conflict response task: STOP/GO • Mediator region: preSMA, outcome region: M1 • Mediation model on functional measures • Dynamic causal effects 7 / 35
Functional mediation model Mediator M ( t ) For ∀ t ∈ [0 , T ] , • Concurrent model Treatment X ( t ) Outcome Y ( t ) M ( t ) = X ( t ) α ( t ) + ǫ 1 ( t ) Y ( t ) = X ( t ) γ ( t ) + M ( t ) β ( t ) + ǫ 2 ( t ) • Historical influence model � M ( t ) = X ( s ) α ( s, t ) d s + ǫ 1 ( t ) Ω 1 t � � Y ( t ) = X ( s ) γ ( s, t ) d s + M ( s ) β ( s, t ) d s + ǫ 2 ( t ) Ω 2 Ω 3 t t • Ω k t = [( t − δ k ) ∨ 0 , t ] , δ k ∈ (0 , + ∞ ] , k = 1 , 2 , 3 • if δ k ∈ [ T, + ∞ ] : whole history 8 / 35
• Concurrent model E � Y ( t ; { x ( s ) , m ( s ) } H t ) − Y ( t ; { x ′ ( s ) , m ( s ) } H t ) � DE( t ) = � x ( t ) − x ′ ( t ) � = γ ( t ) E � Y ( t ; { x ( s ) , m ( s ; { x ( u ) } H s ) } H t ) − Y ( t ; { x ( s ) , m ( s ; { x ′ ( u ) } H s ) } H t ) � IE( t ) = � x ( t ) − x ′ ( t ) � = α ( t ) β ( t ) • Historical influence model E � Y ( t ; { x ( s ) , m ( s ) } H t ) − Y ( t ; { x ′ ( s ) , m ( s ) } H t ) � DE( t ) = � � x ( s ) − x ′ ( s ) � = γ ( s, t ) d s Ω 2 t E � Y ( t ; { x ( s ) , m ( s ; { x ( u ) } H s ) } H t ) − Y ( t ; { x ( s ) , m ( s ; { x ′ ( u ) } H s ) } H t ) � IE( t ) = � �� � ( x ( u ) − x ′ ( u )) α ( u, s ) d u = β ( s, t ) d s Ω 3 Ω 1 s t • { x ( s ) } H t : history of variable x , H t = [0 , t ] • M ( t ; { x ( s ) } H t ) : potential outcome of M at time t if X has the history { x ( s ) } H t • Y ( t ; { x ( s ) , m ( s ) } H t ) : potential outcome of Y at time t when the history X and M at level { x ( s ) } H t and { m ( s ) } H t 9 / 35
• Historical influence model Direct effect (DE) Indirect effect (IE) ( x ( u ) − x ′ ( u )) α ( u, s ) β ( s, t ) � t ( x ( s ) − x ′ ( s )) γ ( s, t ) DE( t ) = t − δ ∨ 0 ( x ( s ) − x ′ ( s )) γ ( s, t ) d s t − δ s t t − 2 δ s t t − δ t − δ t � t � s IE( t ) = s − δ ∨ 0 ( x ( u ) − x ′ ( u )) α ( u, s ) β ( s, t ) d u d s u t − δ ∨ 0 • δ 1 = δ 2 = δ 3 = δ , δ small 10 / 35
• Historical influence model Direct effect (DE) Indirect effect (IE) ( x ( u ) − x ′ ( u )) α ( u, s ) β ( s, t ) � t ( x ( s ) − x ′ ( s )) γ ( s, t ) DE( t ) = 0 ( x ( s ) − x ′ ( s )) γ ( s, t ) d s s t 0 s t t � t � s IE( t ) = 0 ( x ( u ) − x ′ ( u )) α ( u, s ) β ( s, t ) d u d s u 0 • δ 1 = δ 2 = δ 3 = δ , δ ∈ [ T, + ∞ ] 10 / 35
Response conflict task fMRI study 1 • N = 121 right-handed healthy participants • randomized STOP/GO trials: 90 GO trials and 32 STOP trials • mediator region: preSMA-post (MNI: (-4,-8,60)) • outcome region: M1 (MNI: (-41,-20,62)) • TR = 2 s, 184 time points • X ( t ) : convolution of event onsets and canonical HRF • M ( t ) and Y ( t ) : BOLD signals after motion correction 1 OpenfMRI ds000030 11 / 35
• Concurrent model M ( t ) = X ( t ) α ( t ) + ǫ 1 ( t ) Y ( t ) = X ( t ) γ ( t ) + M ( t ) β ( t ) + ǫ 2 ( t ) • Historical influence model � M ( t ) = X ( s ) α ( s, t ) d s + ǫ 1 ( t ) Ω 1 t � � Y ( t ) = X ( s ) γ ( s, t ) d s + M ( s ) β ( s, t ) d s + ǫ 2 ( t ) Ω 2 Ω 3 t t • Ω k t = [( t − δ k ) ∨ 0 , t ] , δ k ∈ (0 , + ∞ ] , k = 1 , 2 , 3 • if δ k ∈ [ T, + ∞ ] : whole history • δ = 2 , 4 , 6 , 10 , 20 , 30 , ∞ (seconds) 12 / 35
Model selection • mean squared error: θ i observed M i or Y i � T N � θ ) = 1 θ i ( t ) − θ i ( t )) 2 d t MSE(ˆ (ˆ N 0 i =1 Historical Historical ( ∼ X ) Concurrent ( ∼ M ) δ = 2 δ = 4 δ = 6 δ = 10 δ = 20 δ = 30 δ = ∞ 353.460 352.645 352.244 351.988 351.652 351.179 351.272 357.396 M δ = 2 212.331 212.308 211.960 212.333 212.378 212.130 212.343 δ = 4 211.324 211.227 211.062 211.064 211.124 211.070 211.572 δ = 6 211.883 211.663 211.541 211.546 211.592 211.575 212.110 Y 220.203 δ = 10 214.277 214.035 213.909 213.953 213.989 213.971 214.510 δ = 20 218.383 218.098 217.878 217.928 218.312 218.247 218.765 δ = 30 221.183 220.915 220.666 220.685 221.041 221.266 221.727 δ = ∞ 295.291 294.938 294.904 294.695 294.820 294.742 301.385 13 / 35
Mediator: preSMA-post (MNI: ( − 4 , − 8 , 60) ) • STOP trial: δ MX = 20 , δ Y X = 6 , δ Y M = 4 14 / 35
Challenges • Large n with hierarchically nested data structure • participants ( → sessions) → tasks/trials • population level inference • Large p • 10 5 ∼ 10 6 uniformly spaced voxels • > 100 putative functional/anatomical regions • high-dimensional problem • Complex data output • time series • functional data 15 / 35
Single modality Mediator p ( M p ) . . . a p Mediator 2 ( M 2 ) b p a 2 b 2 a 1 b 1 Mediator 1 ( M 1 ) Treatment ( X ) Outcome ( Y ) c 16 / 35
Single modality Mediator p ( M p ) d 2 p . . . d 1 p a p Mediator 2 ( M 2 ) b p a 2 b 2 d 12 a 1 b 1 Mediator 1 ( M 1 ) Treatment ( X ) Outcome ( Y ) c 16 / 35
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