Mediation Analysis in Neuroimaging Studies

Transcription

1 Mediation Analysis in Neuroimaging Studies Yi Zhao Department of Biostatistics Johns Hopkins Bloomberg School of Public Health January 15, 2019

2 Overview Introduction Functional mediation analysis High-dimensional mediation analysis Multimodal neuroimaging data integration Discussion 2 / 35

3 Mediation analysis α Mediator (M β Treatment (X γ Outcome (Y Quantifies the intermediate effect of the mediator 3 / 35

4 Mediation analysis α Mediator (M β Treatment (X γ Outcome (Y Quantifies the intermediate effect of the mediator Helps clarify the underlying causal mechanism 3 / 35

5 Mediation analysis α Mediator (M β Treatment (X γ Outcome (Y Quantifies the intermediate effect of the mediator Helps clarify the underlying causal mechanism Popular approach: structural equation modeling (SEM M = Xα + ɛ 1 Y = Xγ + Mβ + ɛ 2 αβ: indirect (mediation effect γ: direct effect 3 / 35

6 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

7 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 Objective Resting-state fmri brain co-activation (functional connectivity impact on cognitive behaviors Credit: NSF 4 / 35

8 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 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 Credit: NSF 4 / 35

9 Challenges Large n with hierarchically nested data structure participants ( sessions tasks/trials population level inference Large p uniformly spaced voxels > 100 putative functional/anatomical regions high-dimensional problem Complex data output time series functional data 5 / 35

10 Challenges Large n with hierarchically nested data structure participants ( sessions tasks/trials population level inference Large p uniformly spaced voxels > 100 putative functional/anatomical regions high-dimensional problem Complex data output time series functional data 5 / 35

11 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., / 35

12 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., / 35

13 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., / 35

14 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

15 Functional mediation model For t [0, T ], Concurrent model Treatment X(t Mediator M(t Outcome Y (t M(t = X(tα(t + ɛ 1 (t Y (t = X(tγ(t + M(tβ(t + ɛ 2 (t Historical influence model M(t = X(sα(s, t ds + ɛ 1 (t Y (t = Ω 1 t Ω 2 t X(sγ(s, t ds + M(sβ(s, t ds + ɛ 2 (t Ω 3 t Ω k t = [(t δ k 0, t], δ k (0, + ], k = 1, 2, 3 if δ k [T, + ]: whole history 8 / 35

16 Concurrent model DE(t = E [ Y (t; {x(s, m(s} Ht Y (t; {x (s, m(s} Ht ] = ( x(t x (t γ(t IE(t = E [ Y (t; {x(s, m(s; {x(u} Hs } Ht Y (t; {x(s, m(s; {x (u} Hs } Ht ] = ( x(t x (t α(tβ(t Historical influence model [ ] DE(t = E Y (t; {x(s, m(s} Ht Y (t; {x (s, m(s} Ht ( = x(s x (s γ(s, t ds Ω 2 t [ ] IE(t = E Y (t; {x(s, m(s; {x(u} Hs } Ht Y (t; {x(s, m(s; {x (u} Hs } Ht ( = Ω 3 t Ω 1 s {x(s} Ht : history of variable x, H t = [0, t] (x(u x (uα(u, s du β(s, t ds M(t; {x(s} Ht : potential outcome of M at time t if X has the history {x(s} Ht Y (t; {x(s, m(s} Ht : potential outcome of Y at time t when the history X and M at level {x(s} Ht and {m(s} Ht 9 / 35

17 Historical influence model Direct effect (DE Indirect effect (IE (x(u x (uα(u, sβ(s, t DE(t = t t δ 0 (x(s x (sγ(s, t ds (x(s x (sγ(s, t t δ t s t 2δ t δ t s t δ u t IE(t = t s (x(u t δ 0 s δ 0 x (uα(u, sβ(s, t duds δ 1 = δ 2 = δ 3 = δ, δ small 10 / 35

18 Historical influence model Direct effect (DE Indirect effect (IE (x(u x (uα(u, sβ(s, t DE(t = t 0 (x(s x (sγ(s, t ds (x(s x (sγ(s, t 0 t s t s t u IE(t = t s (x(u 0 0 x (uα(u, sβ(s, t duds δ 1 = δ 2 = δ 3 = δ, δ [T, + ] 10 / 35

19 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 ds / 35

20 Concurrent model M(t = X(tα(t + ɛ 1 (t Y (t = X(tγ(t + M(tβ(t + ɛ 2 (t Historical influence model M(t = X(sα(s, t ds + ɛ 1 (t Y (t = Ω 1 t Ω 2 t X(sγ(s, t ds + M(sβ(s, t ds + ɛ 2 (t Ω 3 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

21 Model selection mean squared error: θ i observed M i or Y i MSE(ˆθ = 1 N N i=1 0 T (ˆθ i (t θ i (t 2 dt Concurrent Historical Historical ( X ( M δ = 2 δ = 4 δ = 6 δ = 10 δ = 20 δ = 30 δ = M δ = δ = δ = Y δ = δ = δ = δ = / 35

22 Mediator: presma-post (MNI: ( 4, 8, 60 STOP trial: δ MX = 20, δ Y X = 6, δ Y M = 4 14 / 35

23 Challenges Large n with hierarchically nested data structure participants ( sessions tasks/trials population level inference Large p uniformly spaced voxels > 100 putative functional/anatomical regions high-dimensional problem Complex data output time series functional data 15 / 35

24 Single modality Mediator p (M p Treatment (X a 1 a p a 2. Mediator 2 (M 2 b 2 Mediator 1 (M 1 b 1 c b p Outcome (Y 16 / 35

25 Single modality Mediator p (M p a p d 1p Mediator 2 (M 2. d 2p b p Treatment (X a 1 a 2 d 12 Mediator 1 (M 1 c b 1 b 2 Outcome (Y 16 / 35

26 Single modality Mediator p (M p a p d 1p Mediator 2 (M 2. d 2p b p Treatment (X a 1 a 2 d 12 Mediator 1 (M 1 c b 1 b 2 Outcome (Y Objective Identify significant brain regions (mediators Estimate mediation effects 16 / 35

27 Single modality Mediator p (M p a p d 1p Mediator 2 (M 2. d 2p b p Treatment (X a 1 a 2 d 12 Mediator 1 (M 1 c b 1 b 2 Outcome (Y Challenges Ordering of the mediators unknown Large number of mediators (> number of observations 16 / 35

28 Zhao and Luo (2016 Full model Reduced model ɛ 1p η 1p M p M p a p M 2 d 1p. ɛ 12 d 2p b p α p M 2. η 12 β p X a 1 a 2 d 12 ɛ M 11 1 c b 1 b 2 Y ɛ 2 X α 1 α 2 M 1 γ η 11 β 1 β 2 Y η 2 M 1 = Xa 1 + ɛ 11 M 2 = Xa 2 + M 1 d 12 + ɛ 12. M p = Xa p + M 1 d 1p + + M p 1 d p 1,p + ɛ 1p Y = Xc + M 1 b M pb p + ɛ 2 M 1 = Xα 1 + η 11 M 2 = Xα 2 + η 12. M p = Xα K + η 1p Y = Xγ + M 1 β M pβ p + η 2 17 / 35

29 Zhao and Luo (2016 Full model Reduced model ɛ 1p η 1p M p M p a p M 2 d 1p. ɛ 12 d 2p b p α p M 2. η 12 β p X a 1 a 2 d 12 ɛ M 11 1 c b 1 b 2 Y ɛ 2 X α 1 α 2 M 1 γ η 11 β 1 β 2 Y η 2 M 1 = Xa 1 + ɛ 11 M 2 = Xa 2 + M 1 d 12 + ɛ 12. M p = Xa p + M 1 d 1p + + M p 1 d p 1,p + ɛ 1p Y = Xc + M 1 b M pb p + ɛ 2 M 1 = Xα 1 + η 11 M 2 = Xα 2 + η 12. M p = Xα K + η 1p Y = Xγ + M 1 β M pβ p + η 2 17 / 35

30 Zhao and Luo (2016 Full model Reduced model ɛ 1p η 1p M p M p a p M 2 d 1p. ɛ 12 d 2p b p α p M 2. η 12 β p X a 1 a 2 d 12 ɛ M 11 1 c b 1 b 2 Y ɛ 2 X α 1 α 2 M 1 γ η 11 β 1 β 2 Y η 2 M 1 = Xa 1 + ɛ 11 M 2 = Xa 2 + M 1 d 12 + ɛ 12. M p = Xa p + M 1 d 1p + + M p 1 d p 1,p + ɛ 1p Y = Xc + M 1 b M pb p + ɛ 2 M 1 = Xα 1 + η 11 M 2 = Xα 2 + η 12. M p = Xα K + η 1p Y = Xγ + M 1 β M pβ p + η 2 17 / 35

31 adjacency matrix of mediators: Full Model 0 d 12 d 13 d 1p 0 d 23 d 2p. = dp 1,p 0 p p X a1 a2 Mp d2p. d1p ap ɛ12 M2 d12 b2 ɛ11 M1 b1 c ɛ1p bp Y ɛ2 influence matrix: (I p 1 Reduced Model γ = c, β 1 = b 1,..., β p = b p ( α 1 α p = (a 1 a p (I p 1 ( η 11 η 1p = (ɛ 11 ɛ 1p (I p 1 X α1 α2 αp M2 η11 M1 γ Mp. η12 β2 β1 η1p βp Y η2 18 / 35

32 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

33 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

34 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

35 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

36 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

37 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

38 Example with 3 mediators Full Model ɛ M 13 3 Reduced Model M 3 η 13 X d d a 3 ɛ M 12 2 b 3 a 2 d 12 b 2 ɛ M 11 a 1 1 b 1 c α 3 η 12 M 2 β 3 α 2 ɛ β 2 2 η 11 α 1 M 1 β 1 Y X γ 0 d 12 d 13 1 d 12 d 13 + d 12d 23 = 0 0 d 23, (I 3 1 = 0 1 d (α 1 α 2 α 3 = (a 1 a 2 a 3 (I 3 1 = (a 1 a 1 d 12 + a 2 a 1 (d 13 + d 12 d 23 + a 2 d 23 + a 3 Y η 2 19 / 35

39 (η 11 η 1p = (ɛ ɛ 1p (I p 1 } {{ } } 11 {{ } reduced model full model Sequential ignorability of mediators (sequentially conditionally independent Cov [vec(ɛ 1 ] = Ω I n = diag{ω 2 1,..., ω 2 p} I n Cov [vec(η 1 ] = ( I p 1 Ω ( Ip 1 In = Σ 1 I n Σ 1 diagonal matrix = 0 (mediators causally independent ( I p 1 Ω ( Ip 1 LDL decomposition of Σ1 Σ 1 positive-definite, decomposition unique p > n, ˆΣ 1 not of full rank decomposition not unique require knowledge of ordering of the mediators 20 / 35

40 Full Model Reduced Model ɛ1p η1p Mp Mp ap d1p. d2p αp. M2 ɛ12 bp M2 η12 βp a2 d12 b2 α2 β2 X a1 M1 c ɛ11 b1 Y ɛ2 X α1 M1 γ η11 β1 Y η2 α j : total effect of X on M j (e.g., α 2 = a 1d 12 + a 2 α j β j : M j mediation effect as the last mediator in pathway to Y dependency in M j s correlation in η 1j s (e.g., η 12 = η 11d 12 + ɛ / 35

41 Pathway Lasso min 1 p ( 2 l+λ α j β j + φ j (αj 2 + βj 2 + γ +ω j=1 } {{ } P 1 :pathway lasso penalty p ( α j + β j j=1 }{{} P 2 :lasso penalty loss function l = tr [ W 1(M Xα (M Xα ] + w 2(Y Xγ Mβ (Y Xγ Mβ λ 0, φ j > 1/2, ω 0 tuning parameters λ = 0: lasso penalty ω = 0: pathway lasso 22 / 35

42 Multimodal neuroimaging (Liu et al / 35

43 Structural and functional imaging Exposure Outcome Hypothesis: structural functional Objective: integrating DTI and fmri through mediation analysis 24 / 35

44 γ p2 ω p1p 2 M 2p2 M 1p1 θ p1 ω 2p2 π p2 X α p1 φ 1p1 α 1 α 2 ω p12... ω p11 φ 2p1 δ M 12 φ 12 ω 1p2 θ 2 ω ψ 2p2 ψ 1p2 θ 1 M 22 π 2 Y M 11 ω 12 ω 21 γ 2 ψ 12 π 1 γ 1 ω 11 M 21 Two blocks of mediators: {M 11,..., M 1p1 } and {M 21,..., M 2p2 } Within block, ordering unknown 25 / 35

45 Full model Reduced model β = α(i Φ 1, ζ = γ(i Ψ 1, Λ = Ω(I Ψ 1 26 / 35 γp2 ωp1p2 M2p2 ζp2 λp1p2 M2p2 M1p1 θp1 ω2p2 πp2 M1p1 θp1 λ2p2 πp2 X αp1 ωp12. φ1p1.. ωp11 δ φ2p1 α2 M12 α1 φ12... ψ2p2 ω1p2 ψ1p2 θ2 ω22 θ1 M22 π2 Y X βp1 β1 β2 λp12... λp11 δ M12 θ2 λ1p2 λ22... θ1 M22 π2 Y M11 ω12 ω21 γ2 ψ12 π1 M11 λ12 λ21 ζ2 π1 γ1 Full model ω11 M21 ( M 1 M 2 Y = Reduced model ( M 1 M 2 Y = α γ δ (X M 1 M 2 Φ Ω θ (ɛ + η ξ β ζ δ (X M 1 M 2 Λ θ (ε + ϑ ξ Ψ ζ1 π π λ11 M21

46 Full model Reduced model γp2 ωp1p2 M2p2 ζp2 λp1p2 M2p2 M1p1 θp1 ω2p2 πp2 M1p1 θp1 λ2p2 πp2 X αp1 ωp12. φ1p1.. ωp11 δ φ2p1 α2 M12 α1 φ12... ψ2p2 ω1p2 ψ1p2 θ2 ω22 θ1 M22 π2 Y X βp1 β1 β2 λp12... λp11 δ M12 θ2 λ1p2 λ22... θ1 M22 π2 Y M11 ω12 ω21 γ2 ψ12 π1 M11 λ12 λ21 ζ2 π1 γ1 ω11 M21 ζ1 λ11 M21 β jθ j: indirect effect of M 1j not through either M 1s s (for s > j or M 2 s ζ k π k : indirect effect of M 2k not through either M 1 s or M 2t s (for t > k β jλ jk π k : indirect effect through M 1j and M 2k but not through either M 1s s (for s > j or (M 1t for t > k dependency in M 1j s correlation in ε 1j s dependency in M 2k s correlation in ϑ 2k s 27 / 35

47 min tr [ W 1 (M 1 Xβ (M 1 Xβ ] + tr [ W 2 (M 2 Xζ M 1 Λ (M 2 Xζ M 1 Λ ] + w (Y Xδ M 1 θ M 2 π (Y Xδ M 1 θ M 2 π such that p 1 β j θ j t 1, j=1 p 2 ζ k π k t 2, k=1 p 1 p 2 β j λ jk π k t 3, j=1 k=1 δ t 4 p 1 j=1 p 2 k=1 p 1 p 2 j=1 k=1 δ t 4 ( βj θ j + ν 1 (β 2 j + θ2 j r 1, ( ζk π k + ν 2 (ζ 2 k + π2 k r 2, λ jk r 3, ν 1, ν 2 1/2 r 1 t 1, r 2 t 2, r 3 t 3 ν 1 ν 2 /r 1 r 2 28 / 35

48 A multimodal neuroimaging study N = 30 primary progressive aphasia (PPA patients semantic (7: fluent speech, impaired word comprehension nonfluent (9: difficulty producing grammatical sentences and/or motor speech impairment (apraxia of speech logopenic (14: word-finding difficulties and disproportionately impaired sentence repetition X = 1 if semantic, X = 0 otherwise Outcome (Y : word naming accuracy total effect: (p-value= Imaging modalities: DTI (M 1 and resting-state fmri (M 2 Baseline covariates: age, sex, year of onset, language severity Study interest: brain structural and functional pathways on word naming accuracy comparing semantic vs. non-semantic 29 / 35

49 DTI: FA value of p1 = 12 fiber tracks UNC (uncinate fasciculus ILF (inferior longitudinal fasciculus IFO (inferior fronto-occipital fasciculus SLF (superior longitudinal fasciculus: FP (fronto-parietal, FT (fronto-temporal, PT (parietal-temporal (Dick and Tremblay, 2012 both left and right fmri: functional connectivity of 19 regions (p2 = language areas IFG_opercularis IFG_orbitalis IFG_triangularis SMG FuG STG STG_pole MTG MTG_pole ITG 6 DMN MFG_DPFC AG PCC 30 / 35

50 Single-modality result DTI: 4.01% effect mediated fmri: 25.02% effect mediated ITG_L MFG_DPFC_R MTG_L AG_R MTG_L MFG_DPFC_R X Y MTG_L MTG_L_pole SMG_L MFG_DPFC_R X SMG_L MTG_L Y IFG_triangularis_R SMG_L IFG_triangularis_L STG_L_pole UNC_L IFG_orbitalis_R PCC_R IFG_opercularis_R SMG_L IFG_opercularis_L STG_L_pole 31 / 35

51 Single-modality result DTI: 4.01% effect mediated fmri: 25.02% effect mediated 31 / 35

52 Two-modality result X DTI fmri Y : 4.19% effect mediated MTG_L_pole MFG_DPFC_R MTG_L MTG_L_pole ILF_L STG_L_pole AG_R STG_L PCC_L STG_L AG_L STG_L MTG_L_pole STG_L MTG_L X UNC_L SMG_L PCC_R Y SMG_L AG_R SMG_L MTG_L SMG_L STG_L IFG_triangularis_L AG_R SLF_PT_L IFG_opercularis_L MTG_L_pole IFG_opercularis_L STG_L IFG_opercularis_L IFG_orbitalis_R 32 / 35

53 Two-modality result X DTI fmri Y : 4.19% effect mediated 32 / 35

54 Two-modality result X DTI fmri Y : 4.19% effect mediated (Petrides, / 35

55 Discussion Mediation analysis in neuroimaging applications Functional mediation analysis dynamic effective connectivity limitations and future directions unmeasured confounding, sensitivity analysis covariates: scalar and functional dense/sparse functional data High-dimensional mediation analysis ordering of the mediators simultaneous mediator selection and mediation effect estimation multimodal/multiview data integration: imaging and omics R packages: macc, gma, cfma, spcma (github 33 / 35

56 Acknowledgements Brown University Xi (Rossi Luo, PhD Department of Biostatistics Joseph Hogan, ScD Department of Biostatistics Yen-Tsung Huang, MD, ScD Departments of Epidemiology and Biostatistics Johns Hopkins University Brian Caffo, PhD Department of Biostatistics Martin Lindquist, PhD Department of Biostatistics Jerome Sanes, PhD Department of Neuroscience Eli Upfal, PhD Department of Computer Science UC Berkeley Kyrana Tsapkini, PhD Department of Neurology Lexin Li, PhD Department of Biostatistics and Epidemiology R01 DC by the National Institutes of Health (National Institute of Deafness and Communication Disorders 34 / 35

57 Thank you! 35 / 35