MIXED EFFECTS MODELS FOR TIME SERIES

Outline MIXED EFFECTS MODELS FOR TIME SERIES Cristina Gorrostieta Hakmook Kang Hernando Ombao Brown University Biostatistics 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

ANALYSIS OF BRAIN SIGNALS Electrophysiologic data: multi-channel EEG, local field potentials Hemodynamic data: fmri time series at several ROIs

ANALYSIS OF BRAIN SIGNALS Goals of our research Characterize dependence in a brain network Temporal: Y 1 (t) [Y 1 (t 1), Y 2 (t 1),...] Spectral: interactions between oscillatory activities at Y 1, Y 2 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 for bipolar than healthy

ANALYSIS OF BRAIN SIGNALS Desirable Components of Model for Connectivity Fixed Effects Condition-specific or group-specific connectivity Test for differences across conditions or groups Random Effects In multiple subjects: subject-specific random deviation from the group effect In multiple trials: trial-specific random deviation from the mean condition effect Model variation in connectivity between subjects/participants in a group; and between trials within a subject

MIXED EFFECTS MODELS A Simple Regression Model Data for one subject: {(x t, Y t ), t = 1,...,T} A simple model Y t = f(x t )+ǫ t, ǫ t (0,σ 2 ) f(x t ) = β 0 +β 1 x t parametric f(x t ) = β k ψ k (x t ) non-parametric k β s are fixed effects Estimation and inference via maximum likelihood

MIXED EFFECTS MODELS Multi-subject data: {(xt n, Y t n ); t = 1,...,T; n = 1,...,N} Fixed effects only model Y n t = β 0 +β 1 x n t +ǫ n t Y n t Mixed effects model = k β k ψ(x n t )+ǫn t Y n t = β n 0 +βn 1 x n t +ǫ n t Y n t = k β n k ψ(x n t )+ǫ n t Decompose β n k = β k + b n k ; where bn k (0,τ 2 ) Eβk n = β k fixed effect (average across all subjects) bk n subject-specific random deviation from the group effect

ILLUSTRATION: ESTIMATING THE HEMODYNAMIC RESPONSE IN FMRI Goal is to estimate the brain hemodynamic response function 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 is slow Experiment: 30 fmri trials; each trial 20 seconds; TR = 1 second

ILLUSTRATION: ESTIMATING THE HEMODYNAMIC RESPONSE IN FMRI fmri time series for several trials

ILLUSTRATION: ESTIMATING THE HEMODYNAMIC RESPONSE IN FMRI {φ k (t)} B-splines basis

ILLUSTRATION: ESTIMATING THE HEMODYNAMIC RESPONSE IN FMRI Time series for trial n HRF for trial n Y n (t) = f n (t)+ǫ n (t) f n (t) = f(t)+δ n (t) f(t) = k β k φ k (t) δ n (t) = k b n k φ k(t) f(t) - over-all olfaction HRF δ n trial-specific deviation from the over-all HRF Implemented in R (nlmix) and SAS (proc mixed)

ILLUSTRATION: ESTIMATING THE HEMODYNAMIC RESPONSE IN FMRI Trial-specific HRF estimate

ILLUSTRATION: ESTIMATING THE HEMODYNAMIC RESPONSE IN FMRI Over-all HRF estimate

ESTIMATION Model: Y n = X n β + Z n b n +η n Between-subjects Random Effect: b n N(0, D) Within-subject error: η n N(0, V) Unconditional distribution Y n N(X n β, Z n DZ n + V) Conditional distribution (subject n) Y n b n N(X n β + Z n b n, V) The subject-specific mean function E[Y n b n ] = X n β + Z n b n

ESTIMATION Marginal Model Y n = X n β +ǫ n ǫ n = Z n b n +η n N(0,Σ(θ)) ǫ n Y n N(X n β,σ(θ)) Estimates β and θ obtained by REML Random effects estimated by empirical Bayes Prior b n N(0, D) Likelihood Y n N(X n β, Z n DZ n +σ 2 I) Posterior Prior Likelihood 2 Plug in estimates D, β, σ b n = [Z nz n + D 1 σ 2 ] 1 Z n [ Y n X n β ]

MIXED-EFFECTS MODELS IN TIME SERIES Experiment N = 15 right-handed college students Experiment: subjects see visual targets and must move joystick Two Conditions Free choice - subject freely chooses any target Instructed - subject must choose the specified target Regions of interest (7 areas that show highest differential activation) PFC, SMA, etc.

MIXED-EFFECTS MODELS IN TIME SERIES

MIXED-EFFECTS MODELS IN TIME SERIES Total of R ROIs Y n r (t) the fmri time series at the ROI r for subject n Entire network: Y n (t) = [Y n 1 (t),..., Y n R (t)]. General additive model Y n (t) = F n (t) + E n (t) The components F n (t) - mean (deterministic) component E n (t) - stochastic component

MIXED-EFFECTS MODELS IN TIME SERIES The mean component F n (t) Decomposition F n (t) = D n (t)+m n (t)+β n 1 X 1(t)+...+β n C X C(t), The mean component includes systematic changes in the BOLD signal that is due to Scanner drift Physiological signals of non-interest (e.g., cardiac and respiratory) Experimental conditions

MIXED-EFFECTS MODELS IN TIME SERIES Methods for estimating the mean component Worsley and Friston, 1995; Nichols and Holmes, 2002. Implemented in Statistical Parametric Mapping (SPM) FMRIB Statistical Laboratory (FSL) Analysis of Functional NeuroImages (AFNI)

MIXED-EFFECTS MODELS IN TIME SERIES The stochastic component E n (t) E n (t) captures between-roi connectivity Cov[Y n (t + h), Y n (t)] = Cov[F n (t + h)+e n (t + h), F n (t)+e n (t)] = Cov[E n (t + h), E n (t)] E n (t) cannot be observed directly; we use the residuals: E n (t) = Y n (t) F n (t) R n (t) = Y n (t) F n (t)

MIXED-EFFECTS MODELS IN TIME SERIES Vector Auto-Regressive Model E(t) = [E 1 (t), E 2 (t)] VAR(1) Model E(t) = Φ 1 E(t 1)+η(t) Φ 1 lag-1 connectivity matrix; componentsφ kl VAR(1) Equations E 1 (t) = φ 11 E 1 (t 1)+φ 12 E 2 (t 1)+η 1 (t) E 2 (t) = φ 21 E 1 (t 1)+φ 22 E 2 (t 1)+η 2 (t)

MIXED-EFFECTS MODELS IN TIME SERIES Conditional MLE Model E(t) = Φ 1 E(t 1)+η(t), t = 1,...,T Define 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(Φ 1 X(t 1),Σ) Conditional likelihood function L C [Φ 1 E(1)] = f(e(2) E(1))... f(e(t) E(T 1))) Conditional approach leads to a closed form for the estimator of Φ 1 Full likelihood - no closed form

MIXED-EFFECTS MODELS IN TIME SERIES The ME-VAR(1) Model [ ] E (n) (t) = Φ 1,k W 1 (t)+φ 2,k W 2 (t)+b (n) 1 E (n) (t 1)+e (n) (t) W (t) is the indicator function When condition 1 is active then W 1 (t) = 1 and W 2 (t) = 0 When condition 2 is active then W 1 (t) = 0 and W 2 (t) = 1 b (n) 1 models between-subject variation in connectivity b (n) 1 (0,σ 2 1 )

MIXED-EFFECTS MODELS IN TIME SERIES Subject-specific connectivity matrix (condition on b (n) 1 ) When W 1 (t) = 1, the connectivity matrix for subject n is Φ 1,1 + b (n) 1 When W 2 (t) = 1, the connectivity matrix for subject n is Φ 1,2 + b (n) 1 The model can be utilized to test for Lagged dependence between each pair of ROIs H 0 : Φ 1,1 = 0,Φ 1,2 = 0 Granger causality in each experimental condition Testing for differences between conditions H 0 : 1 = Φ 1,1 Φ 1,2 = 0

CONNECTIVITY ANALYSIS OF FMRI DATA Step 0. Selection of pre-defined ROIs. Step 1. Estimate the mean function at each ROI F n (t). Step 2. Fit the ME-VAR model to the residuals R n (t).

CONNECTIVITY ANALYSIS OF FMRI DATA Fitting the ME-VAR Model Optimal lag over was p = 2 using the Bayesian Information Criterion (Pmax = 8) In physical time: TR = 2 seconds; optimal delay/lag is 4 seconds. Computations were carried out in SAS proc mixed (we use existing machinery!)

CONNECTIVITY ANALYSIS OF FMRI DATA

CONCLUSION Statistical vs Clinical significance Connectivity parameter of 0.018 can be statistically significant. Is it clinically meaningful? How do we measure clinical significance? Impact of increase (or decrease) in connectivity measure on some clinical outcome Should we be testing for H 0 : θ = 0 or H 0 : θ > η?

CONCLUSION Granger causality not physiological causality Connectivity in the hemodynamic activity not necessarily neuronal temporal ordering could be switched because of the poor temporal resolution in fmri

SPATIO-SPECTRAL MIXED EFFECTS MODELS Spatio-temporal models In fmri, spatial correlation adds computational complications! Current approaches Bowman (2005, 2007, 2008); Worsley (1999); Valdes-Sosa et al. (2004) Spatial - parametric; functional" distance Bayesian hierarchial 4D-Wavelet packets Partial least squares

SPATIO-SPECTRAL MIXED EFFECTS MODELS Spatio-Spectral Model With H. Kang (PhD student) [John Van Ryzin Award 2011] Goals: Estimate ROI-specific activation Use all local and global information Approach Spectral domain: Fourier coefficients are approx uncorrelated Spatio-spectral covariance matrix is block diagonal - simplified! Multi-scale correlation - local (within ROI) and global (between ROIs)

SPATIO-SPECTRAL MIXED EFFECTS MODELS ROI1 ROI2 ROI3

SPATIO-SPECTRAL MIXED EFFECTS MODELS Spatio-temporal Model Y cv (t) = P [βc p + bcv]x p p (t)+d c (t)+ǫ cv (t) p=1 β p c is the ROI-specific fixed effect due to stimulus p; b p cv is a zero-mean voxel-specific random deviation local spatial covariance between voxels within an ROI Cov(b p cv, b p c v ) = δ(c c )ψ b ( v v ) d c (t) is a zero-mean ROI-specific signal global spatial covariation Cov(d c (t), d c (t)) = ψ d (c, c ) ǫ cv (t) is the within-voxel temporal noise

SPATIO-SPECTRAL MIXED EFFECTS MODELS Problem: Spatio-temporal covariance structure is complicated! A solution: Fourier transform X(t) stationary time series with covariance matrix Σ d(ω k ) = t X(t) exp( i2πω kt) Fourier coefficient I(ω k ) = d(ω k ) 2 I(ω 1 ),...,I(ω M ) are approx uncorrelated

SPATIO-SPECTRAL MIXED EFFECTS MODELS Y cv (t) time series at voxel v in ROI c Y cv (ω) corresponding Fourier coefficient Spatio-Spectral Model P Y cv (ω k ) = [βc p + bcv]x p p (ω k )+d c (ω k )+ǫ cv (ω k ) p=1 β p c is the ROI-specific fixed effect due to stimulus p b p cv is a zero-mean voxel-specific random effect d c (t) is a zero-mean ROI-specific random effect

SPATIO-SPECTRAL MIXED EFFECTS MODELS