Mixed effects and Group Modeling for fMRI data Thomas Nichols, - - PowerPoint PPT Presentation

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Mixed effects and Group Modeling for fMRI data

Thomas Nichols, Ph.D. Department of Statistics & Warwick Manufacturing Group University of Warwick Zurich SPM Course February 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

• 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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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

Fixed vs. Random Effects 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

2

FFX

2

RFX

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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

• ver 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

• 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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Assessing RFX Models Issues 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

• 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

• 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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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 & Friston motivation...

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 image

Fixed effects... ...powerful but wrong inference

n – subjects w – error DF

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



^

Holmes & Friston Random 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 & Friston Assumptions

• Distribution

– Normality – Independent subjects

• Homogeneous Variance

– Intrasubject variance homogeneous

• 2

FFX same for all subjects

– Balanced designs

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Holmes & Friston Limitations

• 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 & Friston Efficiency

• If assumptions true

– Optimal, fully efficient

• If 2

FFX differs between

subjects

– Reduced efficiency – Here, optimal requires down-weighting the 3 highly variable subjects

 ˆ  ˆ

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Holmes & Friston Validity

• If assumptions true

– Exact P-values

• If 2

FFX 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

2

RFX

• 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 & Friston Robustness

(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

• 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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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 N Error 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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SPM8 Nonsphericity Modelling

• Assumptions & Limitations

– assumed to globally homogeneous – lk’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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SPM8 Nonsphericity Modelling

• 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

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

– Summary statistic approach (HF) (SPM96,99,2,5,8) – SPM8 Nonsphericity Modelling

• Data exploration
• Conclusions

Data: FIAC Data

• Acquisition

– 3 TE Bruker Magnet – For each subject: 2 (block design) sessions, 195 EPI images each – TR=2.5s, TE=35ms, 646430 volumes, 334mm vx.

• Experiment (Block Design only)

– Passive sentence listening – 22 Factorial Design

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

– Slice time correction, motion correction, sptl. norm. – 555 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 Deviation also useful

– Variance lowest in white 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 four RFX 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)