Mixed effects and Group Modeling for fMRI data

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Mixed effects and Group Modeling

for fMRI data

Thomas Nichols, Ph.D.Department of Statistics &

Warwick Manufacturing GroupUniversity of Warwick

Zurich SPM CourseFebruary 16, 2012

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Outline

• Mixed effects motivation• Evaluating mixed effects methods• Two methods

– Summary statistic approach (HF) (SPM96,99,2,5,8)

– SPM8 Nonsphericity Modelling• Data exploration• Conclusions

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Overview




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Lexicon

Hierarchical Models• Mixed Effects Models• Random Effects (RFX) Models• Components of Variance

... all the same

... all alluding to multiple sources of variation(in contrast to fixed effects)

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Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Subj. 6

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Fixed vs.RandomEffects in fMRI

• Fixed Effects– Intra-subject

variation suggests all these subjects different from zero

• Random Effects– Intersubject

variation suggests population not very different from zero

Distribution of each subject’s estimated effect

Distribution of population effect

2FFX

2RFX

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Fixed Effects

• Only variation (over sessions) is measurement error

• True Response magnitude is fixed

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Random/Mixed Effects

• Two sources of variation– Measurement error– Response magnitude

• Response magnitude is random– Each subject/session has random magnitude–

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Random/Mixed Effects

• Two sources of variation– Measurement error– Response magnitude

• Response magnitude is random– Each subject/session has random magnitude– But note, population mean magnitude is fixed

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Fixed vs. Random

• Fixed isn’t “wrong,” just usually isn’t of interest

• Fixed Effects Inference– “I can see this effect in this cohort”

• Random Effects Inference– “If I were to sample a new cohort from the

population I would get the same result”

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Two Different Fixed Effects Approaches

• Grand GLM approach– Model all subjects at once

– Good: Mondo DF– Good: Can simplify modeling– Bad: Assumes common variance

over subjects at each voxel– Bad: Huge amount of data

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Two Different Fixed Effects Approaches

• Meta Analysis approach– Model each subject individually– Combine set of T statistics

• mean(T)n ~ N(0,1) • sum(-logP) ~ 2

n

– Good: Doesn’t assume common variance– Bad: Not implemented in software

Hard to interrogate statistic maps

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Overview




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Assessing RFX ModelsIssues to Consider

• Assumptions & Limitations– What must I assume?

• Independence?• “Nonsphericity”? (aka independence + homogeneous var.)

– When can I use it• Efficiency & Power

– How sensitive is it?• Validity & Robustness

– Can I trust the P-values?– Are the standard errors correct?– If assumptions off, things still OK?

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Overview




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Overview




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Holmes & Friston

• Unweighted summary statistic approach• 1- or 2-sample t test on contrast images

– Intrasubject variance images not used (c.f. FSL)• Proceedure

– Fit GLM for each subject i– Compute cbi, contrast estimate

– Analyze {cbi}i

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Holmes & Fristonmotivation...

p < 0.001 (uncorrected)

p < 0.05 (corrected)

SPM{t}

SPM{t}

1^

2^

3^

4^

5^

6^

^

• – c.f. 2 / nw

—

^

^

^

^

^

– c.f.

estimated mean activation imageFixed effects...

...powerful but wrong inference

n – subjects

w – error DF

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^

Holmes & FristonRandom Effects

1^

2^

3^

4^

5^

6^

^

^

^

^

^

• – c.f. 2/n = 2 /n + 2

/ nw^

– c.f.

level-one(within-subject)

variance 2^

an estimate of the mixed-effects

model variance 2

+ 2 / w

—

level-two(between-subject)

timecourses at [ 03, -78, 00 ] contrast images

p < 0.001 (uncorrected)

SPM{t}

(no voxels significant at p < 0.05 (corrected))

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Holmes & FristonAssumptions

• Distribution– Normality– Independent subjects

• Homogeneous Variance– Intrasubject variance homogeneous

2FFX same for all subjects

– Balanced designs

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Holmes & FristonLimitations• Limitations

– Only single image per subject

– If 2 or more conditions,Must run separate model for each contrast

• Limitation a strength!– No sphericity assumption

made on different conditions when each is fit with separate model

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Holmes & FristonEfficiency• If assumptions true

– Optimal, fully efficient

• If 2FFX differs between

subjects– Reduced efficiency– Here, optimal requires

down-weighting the 3 highly variable subjects

0

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Holmes & FristonValidity• If assumptions true

– Exact P-values

• If 2FFX differs btw subj.

– Standard errors not OK• Est. of 2

RFX may be biased

– DF not OK• Here, 3 Ss dominate• DF < 5 = 6-1

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2RFX

• In practice, Validity & Efficiency are excellent– For one sample case, HF almost impossible to break

• 2-sample & correlation might give trouble– Dramatic imbalance or heteroscedasticity

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Holmes & FristonRobustness

(outlier severity)

Mumford & Nichols. Simple group fMRI modeling and inference. Neuroimage, 47(4):1469--1475, 2009.

False Positive Rate Power Relative to Optimal

(outlier severity)

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Overview




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SPM8 Nonsphericity Modelling

• 1 effect per subject– Uses Holmes & Friston approach

• >1 effect per subject– Can’t use HF; must use SPM8 Nonsphericity

Modelling– Variance basis function approach used...

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y = X + N 1 N p p 1 N 1

N

NError covariance

SPM8 Notation: iid case

• 12 subjects,4 conditions – Use F-test to find

differences btw conditions• Standard Assumptions

– Identical distn– Independence– “Sphericity”... but here

not realistic!

X

Cor(ε) = λ I

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y = X + N 1 N p p 1 N 1

N

N

Error covariance

Errors can now have different variances and there can be correlations

Allows for ‘nonsphericity’

Multiple Variance Components

• 12 subjects, 4 conditions • Measurements btw

subjects uncorrelated• Measurements w/in

subjects correlated

Cor(ε) =Σk λkQk

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Non-Sphericity Modeling• Errors are not

independent and not identical

Qk’s:

Error Covariance

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Non-Sphericity Modeling

• Errors are independent but not identical– Eg. Two Sample T

Two basis elements

Error Covariance

Qk’s:

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• Assumptions & Limitations– assumed to globally

homogeneous– k’s only estimated from voxels with large F– Most realistically, Cor() spatially heterogeneous

– Intrasubject variance assumed homogeneous

Cor(ε) =Σk λkQk

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• Efficiency & Power– If assumptions true, fully efficient

• Validity & Robustness– P-values could be wrong (over or under) if

local Cor() very different from globally assumed

– Stronger assumptions than Holmes & Friston

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Overview




Data: FIAC Data• Acquisition

– 3 TE Bruker Magnet– For each subject:

2 (block design) sessions, 195 EPI images each– TR=2.5s, TE=35ms, 646430 volumes, 334mm vx.

• Experiment (Block Design only)– Passive sentence listening– 22 Factorial Design

• Sentence Effect: Same sentence repeated vs different• Speaker Effect: Same speaker vs. different

• Analysis– Slice time correction, motion correction, sptl. norm.– 555 mm FWHM Gaussian smoothing– Box-car convolved w/ canonical HRF– Drift fit with DCT, 1/128Hz

Look at the Data!

• With small n, really can do it!

• Start with anatomical– Alignment OK?

• Yup– Any horrible

anatomical anomalies?

• Nope

Look at the Data!

• Mean & Standard Deviationalso useful– Variance

lowest inwhite matter

– Highest around ventricles

Look at the Data!

• Then the functionals– Set same

intensity window for all [-10 10]

– Last 6 subjects good

– Some variability in occipital cortex

Feel the Void!

• Compare functional with anatomical to assess extent of signal voids

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Conclusions

• Random Effects crucial for pop. inference

• When question reduces to one contrast– HF summary statistic approach

• When question requires multiple contrasts– Repeated measures modelling

• Look at the data!

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References for fourRFX Approaches in fMRI

• Holmes & Friston (HF)– Summary Statistic approach (contrasts only)– Holmes & Friston (HBM 1998). Generalisability, Random Effects & Population Inference. NI, 7(4

(2/3)):S754, 1999.

• Holmes et al. (SnPM)– Permutation inference on summary statistics– Nichols & Holmes (2001). Nonparametric Permutation Tests for Functional Neuroimaging: A Primer

with Examples. HBM, 15;1-25.– Holmes, Blair, Watson & Ford (1996). Nonparametric Analysis of Statistic Images from Functional

Mapping Experiments. JCBFM, 16:7-22.

• Friston et al. (SPM8 Nonsphericity Modelling)– Empirical Bayesian approach– Friston et al. Classical and Bayesian inference in neuroimaging: theory. NI 16(2):465-483, 2002 – Friston et al. Classical and Bayesian inference in neuroimaging: variance component estimation in

fMRI. NI: 16(2):484-512, 2002.

• Beckmann et al. & Woolrich et al. (FSL3)– Summary Statistics (contrast estimates and variance)– Beckmann, Jenkinson & Smith. General Multilevel linear modeling for group analysis in fMRI. NI

20(2):1052-1063 (2003)– Woolrich, Behrens et al. Multilevel linear modeling for fMRI group analysis using Bayesian inference.

NI 21:1732-1747 (2004)

Mixed effects and Group Modeling for fMRI data

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