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Outline
MIXED EFFECTS MODELS FOR TIME SERIES
Cristina Gorrostieta Hakmook Kang Hernando Ombao
Brown UniversityBiostatistics Section
February 16, 2011
Outline
OUTLINE OF TALK
1 SCIENTIFIC MOTIVATION
2 BACKGROUND ON MIXED EFFECTS MODELS
3 MIXED EFFECTS VAR
4 SPATIO-SPECTRAL MIXED EFFECTS MODELS
Outline
OUTLINE OF TALK
1 SCIENTIFIC MOTIVATION
2 BACKGROUND ON MIXED EFFECTS MODELS
3 MIXED EFFECTS VAR
4 SPATIO-SPECTRAL MIXED EFFECTS MODELS
Outline
OUTLINE OF TALK
1 SCIENTIFIC MOTIVATION
2 BACKGROUND ON MIXED EFFECTS MODELS
3 MIXED EFFECTS VAR
4 SPATIO-SPECTRAL MIXED EFFECTS MODELS
Outline
OUTLINE OF TALK
1 SCIENTIFIC MOTIVATION
2 BACKGROUND ON MIXED EFFECTS MODELS
3 MIXED EFFECTS VAR
4 SPATIO-SPECTRAL MIXED EFFECTS MODELS
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio -Spectral Mixed Effects
ANALYSIS OF BRAIN SIGNALS
Electrophysiologic data: multi-channel EEG, local fieldpotentialsHemodynamic data: fMRI time series at several ROIs
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio -Spectral Mixed Effects
ANALYSIS OF BRAIN SIGNALS
Goals of our research
Characterize dependence in a brain network
Temporal: Y1(t) ∼ [Y1(t − 1),Y2(t − 1), . . .]′
Spectral: interactions between oscillatory activities at Y1,Y2
Develop estimation and inference methods for connectivity
Investigate potential for connectivity as a biomarker
Predicting behavior
Motor intent (left vs. right movement)State of learning
Differentiating patient groups (bipolar vs. healthy children)
Connectivity between left DLPFC ⇆ right STG is greater forbipolar than healthy
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio -Spectral Mixed Effects
ANALYSIS OF BRAIN SIGNALS
Desirable Components of Model for Connectivity
Fixed EffectsCondition-specific or group-specific connectivityTest for differences across conditions or groups
Random EffectsIn multiple subjects: subject-specific random deviation fromthe group effectIn multiple trials: trial-specific random deviation from themean condition effectModel variation in connectivity betweensubjects/participants in a group; and between trials within asubject
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED EFFECTS MODELS
A Simple Regression Model
Data for one subject: {(xt ,Yt), t = 1, . . . ,T}
A simple model
Yt = f (xt) + ǫt , ǫt ∼ (0, σ2)
f (xt) = β0 + β1xt parametric
f (xt) =∑
k
βkψk (xt) non-parametric
β’s are fixed effects
Estimation and inference via maximum likelihood
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED EFFECTS MODELS
Multi-subject data: {(xnt ,Y
nt ); t = 1, . . . ,T ;n = 1, . . . ,N}
Fixed effects only model
Y nt = β0 + β1xn
t + ǫnt
Y nt =
∑
k
βkψ(xnt ) + ǫnt
Mixed effects model
Y nt = βn
0 + βn1 xn
t + ǫnt
Y nt =
∑
k
βnkψ(x
nt ) + ǫnt
Decompose βnk = βk + bn
k ; where bnk ∼ (0, τ2)
Eβnk = βk – fixed effect (average across all subjects)
bnk – subject-specific random deviation from the group effect
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ILLUSTRATION: ESTIMATING THE HEMODYNAMIC
RESPONSE IN FMRI
Goal is to estimate the brain hemodynamic responsefunction in olfaction stimuli
Standard HRF are estimated from visual stimuli
In visual stimuli rise and decay are fast
In olfaction, rise is slow, peak is persistent and decay isslow
Experiment: 30 fMRI trials; each trial 20 seconds; TR = 1second
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ILLUSTRATION: ESTIMATING THE HEMODYNAMIC
RESPONSE IN FMRI
fMRI time series for several trials
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ILLUSTRATION: ESTIMATING THE HEMODYNAMIC
RESPONSE IN FMRI
{φk (t)} B-splines basis
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ILLUSTRATION: ESTIMATING THE HEMODYNAMIC
RESPONSE IN FMRI
Time series for trial n
Y n(t) = f n(t) + ǫn(t)
HRF for trial n
f n(t) = f (t) + δn(t)
f (t) =∑
k
βkφk (t)
δn(t) =∑
k
bnkφk (t)
f (t) - over-all olfaction HRFδn trial-specific deviation from the over-all HRFImplemented in R (nlmix) and SAS (proc mixed)
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ILLUSTRATION: ESTIMATING THE HEMODYNAMIC
RESPONSE IN FMRI
Trial-specific HRF estimate
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ILLUSTRATION: ESTIMATING THE HEMODYNAMIC
RESPONSE IN FMRI
Over-all HRF estimate
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ESTIMATION
Model: Yn = Xnβ + Znbn + ηn
Between-subjects Random Effect: bn ∼ N(0,D)
Within-subject error: ηn ∼ N(0,V )
Unconditional distribution
Yn ∼ N(Xnβ,ZnDZ′
n + V )
Conditional distribution (subject n)
Yn | bn ∼ N(Xnβ + Znbn,V )
The subject-specific mean function
E [Yn | bn] = Xnβ + Znbn
Outline of Talk Scientific Motivation Background on Mixed Effects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
ESTIMATION
Marginal Model Yn = Xnβ + ǫn
ǫn = Znbn + ηn
ǫn ∼ N(0,Σ(θ))
Yn ∼ N(Xnβ,Σ(θ))
Estimates β̂ and θ̂ obtained by REML
Random effects estimated by empirical Bayes
Prior bn ∼ N(0,D)Likelihood Yn ∼ N(Xnβ,ZnDZ
′
n + σ2I)Posterior ∝ Prior × LikelihoodPlug in estimates D̂, β̂, σ̂2
b̂n =[Z′
nZn + D̂−1σ̂2]−1
Z′
n
[Yn − Xnβ̂
]
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
Experiment
N = 15 right-handed college students
Experiment: subjects see visual targets and must movejoystick
Two Conditions
Free choice - subject freely chooses any targetInstructed - subject must choose the specified target
Regions of interest (7 areas that show highest differentialactivation)
PFC, SMA, etc.
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
PM
PFC
PMv
SMA
PMd
IPS
SPL
0.22%0
L R
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
Total of R ROIs
Y nr (t) the fMRI time series at the ROI r for subject n
Entire network: Yn(t) = [Y n1 (t), . . . ,Y
nR(t)]
′.
General additive model
Yn(t) = Fn(t) + En(t)
The componentsFn(t) - mean (deterministic) componentEn(t) - stochastic component
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
The mean component F n(t)
Decomposition
Fn(t) = Dn(t) + Mn(t) + βn1 ⊗ X1(t) + . . .+ βn
C ⊗ XC(t),
The mean component includes systematic changes in theBOLD signal that is due to
Scanner driftPhysiological signals of non-interest (e.g., cardiac andrespiratory)Experimental conditions
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
Methods for estimating the mean component
Worsley and Friston, 1995;Nichols and Holmes, 2002.
Implemented inStatistical Parametric Mapping (SPM)FMRIB Statistical Laboratory (FSL)Analysis of Functional NeuroImages (AFNI)
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
The stochastic component E n(t)
En(t) captures between-ROI connectivity
Cov[Yn(t + h),Yn(t)] = Cov[Fn(t + h) + En(t + h),Fn(t) + En(t)]
= Cov[En(t + h),En(t)]
En(t) cannot be observed directly; we use the residuals:
En(t) = Yn(t)− Fn(t)
Rn(t) = Yn(t)− F̂n(t)
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
Vector Auto-Regressive Model
E(t) = [E1(t),E2(t)]′
VAR(1) Model
E(t) = Φ1E(t − 1) + η(t)
Φ1 lag-1 connectivity matrix; components φkℓ
VAR(1) Equations
E1(t) = φ11E1(t − 1) + φ12E2(t − 1) + η1(t)
E2(t) = φ21E1(t − 1) + φ22E2(t − 1) + η2(t)
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
Conditional MLE
Model E(t) = Φ1E(t − 1) + η(t), t = 1, . . . ,TDefine E(t − 1) = {E(t − 1),E(t − 2), . . .}Suppose η(t) iid N(0,Σ)Condition on past data E(t − 1)
E(t) | E(t − 1) ∼ N(Φ1X(t − 1),Σ)
Conditional likelihood function
LC [Φ1|E(1)] = f (E(2)|E(1))× . . . × f (E(T ) | E(T − 1)))
Conditional approach leads to a closed form for theestimator of Φ1
Full likelihood - no closed form
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
The ME-VAR(1) Model
E(n)(t) =[Φ1,kW1(t) + Φ2,kW2(t) + b(n)
1
]E(n)(t − 1) + e(n)(t)
W�(t) is the indicator functionWhen condition 1 is active then W1(t) = 1 and W2(t) = 0When condition 2 is active then W1(t) = 0 and W2(t) = 1
b(n)1 models between-subject variation in connectivity
b(n)1 ∼ (0, σ2
1)
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
MIXED-EFFECTS MODELS IN TIME SERIES
Subject-specific connectivity matrix (condition on b(n)1 )
When W1(t) = 1, the connectivity matrix for subject n is
Φ1,1 + b(n)1
When W2(t) = 1, the connectivity matrix for subject n is
Φ1,2 + b(n)1
The model can be utilized to test for
Lagged dependence between each pair of ROIs
H0 : Φ1,1 = 0,Φ1,2 = 0
Granger causality in each experimental conditionTesting for differences between conditions
H0 : ∆1 = Φ1,1 −Φ1,2 = 0
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONNECTIVITY ANALYSIS OF FMRI DATA
Step 0. Selection of pre-defined ROIs.
Step 1. Estimate the mean function at each ROI Fn(t).
Step 2. Fit the ME-VAR model to the residuals Rn(t).
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONNECTIVITY ANALYSIS OF FMRI DATA
Fitting the ME-VAR Model
Optimal lag over was p∗ = 2 using the BayesianInformation Criterion (Pmax = 8)
In physical time: TR = 2 seconds; optimal delay/lag is 4seconds.
Computations were carried out in SAS proc mixed (we useexisting machinery!)
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONNECTIVITY ANALYSIS OF FMRI DATA
SMA
IPS
SPL
PFC
PMd PMd
PMv
L R
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONNECTIVITY ANALYSIS OF FMRI DATA
SMA
IPS
SPL
PFC
PMd PMd
PMv
L R
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONNECTIVITY ANALYSIS OF FMRI DATA
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONCLUSION
Statistical vs Clinical significance
Connectivity parameter of 0.018 can be statisticallysignificant. Is it clinically meaningful?
How do we measure clinical significance?
Impact of increase (or decrease) in connectivity measureon some clinical outcome
Should we be testing for H0 : θ = 0 or H0 : |θ| > η?
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
CONCLUSION
Granger causality – not physiological causality
Connectivity in the hemodynamic activity – not necessarilyneuronal — temporal ordering could be switched becauseof the poor temporal resolution in fMRI
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS
Spatio-temporal models
In fMRI, spatial correlation adds computationalcomplications!
Current approaches
Bowman (2005, 2007, 2008); Worsley (1999); Valdes-Sosaet al. (2004)Spatial - parametric; “functional" distanceBayesian hierarchial4D-Wavelet packetsPartial least squares
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS
Spatio-Spectral Model
With H. Kang (PhD student) [John Van Ryzin Award 2011]
Goals:
Estimate ROI-specific activationUse all local and global information
Approach
Spectral domain: Fourier coefficients are approxuncorrelatedSpatio-spectral covariance matrix is block diagonal -simplified!Multi-scale correlation - local (within ROI) and global(between ROIs)
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS
ROI1
ROI3ROI2
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS
Spatio-temporal Model
Ycv (t) =P∑
p=1
[βpc + bp
cv ]Xp(t) + dc(t) + ǫcv (t)
βpc is the ROI-specific fixed effect due to stimulus p;
bpcv is a zero-mean voxel-specific random deviation
local spatial covariance between voxels within an ROICov(bp
cv , bpc′v ′) = δ(c − c′)ψb(‖ v − v ′ ‖)
dc(t) is a zero-mean ROI-specific signalglobal spatial covariationCov(dc(t), dc′ (t)) = ψd (c, c′)
ǫcv (t) is the within-voxel temporal noise
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS
Problem: Spatio-temporal covariance structure iscomplicated!
A solution: Fourier transform
X(t) stationary time series with covariance matrix Σd(ωk ) =
∑t X(t)exp(−i2πωk t) Fourier coefficient
I(ωk ) = |d(ωk )|2
I(ω1), . . . , I(ωM) are approx uncorrelated
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS
Ycv (t) time series at voxel v in ROI c
Ycv (ω) corresponding Fourier coefficient
Spatio-Spectral Model
Ycv (ωk ) =P∑
p=1
[βpc + bp
cv ]Xp(ωk ) + dc(ωk ) + ǫcv (ωk )
βpc is the ROI-specific fixed effect due to stimulus p
bpcv is a zero-mean voxel-specific random effect
dc(t) is a zero-mean ROI-specific random effect
Outline of Talk Scientific Motivation Background on Mixed Ef fects Models Mixed Effects VAR Spatio-Spectral Mixed Effects
SPATIO-SPECTRAL MIXED EFFECTS MODELS