our entry in the functional imaging analysis contest

Our entry in the Functional Imaging

Analysis contest

Jonathan Taylor

Stanford

Keith Worsley

McGill

What is functional Magnetic Resonance Imaging (fMRI) data?

Time series of ~200 “frames”, 3D images of brain “activity”, taken every ~2.5s (~8min)

Meanwhile, subject receives stimulus or external task (e.g on/off every 10s)

Several (~4) time series (“runs”) per session Several (~2) sessions per subject Several (~15) subjects Statistics problem: find the regions of the brain

activated by the stimulus or task

Why a Functional Imaging Analysis Contest (FIAC)?

Competing packages produce slightly different results, which is “correct”?

Simulated data? Real data, compare analyses “Contest” session at 2005 Human Brain Map

conference 9 entrants Results in a special issue of Human Brain

Mapping in May, 2006

The main participants

SPM (Statistical Parametric Mapping, 1993), University College, London, “SAS”, (MATLAB)

AFNI (1995), NIH, more display and manipulation, not much stats (C)

FSL (2000), Oxford, the “upstart” (C) …. FMRISTAT (2001), McGill, stats only (MATLAB) BRAINSTAT (2005), Stanford/McGill, Python

version of FMRISTAT

0 50 100 150 200 250 300 350-1

0

1

2Alternating hot and warm stimuli separated by rest (9 seconds each).

hot

warm

hot

warm

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Hemodynamic response function: difference of two gamma densities

0 50 100 150 200 250 300 350-1

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2Responses = stimuli * HRF, sampled every 3 seconds

Time, seconds

Effect of stimulus on brain response

Stimulus is delayed and dispersed by ~6s

Modeled by convolving the stimulus with the “hemodynamic response function”

0

500

1000First scan of fMRI data

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T statistic for hot - warm effect

0 100 200 300

870880890 hot

restwarm

Highly significant effect, T=6.59

0 100 200 300

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820hotrestwarm

No significant effect, T=-0.74

0 100 200 300

790800810

Drift

Time, seconds

fMRI data, pain experiment, one slice

T = (hot – warm effect) / S.d. ~ t110 if no effect

How fMRI differs from other repeated measures data

Many reps (~200 time points) Few subjects (~15) Df within subjects is high, so not worth

pooling sd across subjects Df between subjects low, so use spatial

smoothing to boost df Data sets are huge ~4GB, not easy to use R

directly

FMRISTAT / BRAINSTATstatistical analysis strategy

Analyse each voxel separately Borrow strength from neighbours when needed

Break up analysis into stages 1st level: analyse each time series separately 2nd level: combine 1st level results over runs 3rd level: combine 2nd level results over subjects

Cut corners: do a reasonable analysis in a reasonable time (or else no one will use it!)

MATLAB / Python

1st level: Linear model with AR(p) errors

Data Yt = fMRI data at time t

xt = (responses,1, t, t2, t3, … )’ to allow for drift

Model Yt = xt’β + εt

εt = a1εt-1 + … + apεt-p + σFηt, ηt ~ N(0,1) i.i.d.

Fit in 2 stages: 1st pass: fit by least squares, find residuals, estimate AR

parameters a1 … ap

2nd pass: whiten data, re-fit by least squares

Higher levels:Mixed effects model

Data Ei = effect (contrast in β) from previous level

Si = sd of effect from previous level

zi = (1, treatment, group, gender, …)’

Model Ei = zi’γ + Siεi

F + σRεiR (Si high df, so assumed fixed)

εiF ~ N(0,1) i.i.d. fixed effects error

εiR ~ N(0,1) i.i.d. random effects error

Fit by ReML Use EM for stability, 10 iterations

Where we use spatial information

1st level: smooth AR parameters to lower variability and increase “df”

Higher levels: smooth Random / Fixed effects sd ratio to lower variability and increase “df”

Final level: use random field theory to correct for multiple comparisons

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0

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1st level: Autocorrelation

AR(1) model: εt = a1 εt-1 + σFηt

Fit the linear model using least squares εt = Yt – Yt

â1 = Correlation (εt , εt-1)

Estimating errort’s changes their correlation structure slightly, so â1 is slightly biased:

Raw autocorrelation Smoothed 12.4mm Bias corrected â1

~ -0.05 ~ 0~ -0.05 ~ 0

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FWHMacor

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FWHMacor

How much smoothing?

Hot stimulus Hot-warm stimulus

Target = 100 df

Residual df = 110

Target = 100 df

Residual df = 110

FWHM = 10.3mm FWHM = 12.4mm

dfacor = dfresidual(2 + 1) 1 1 2 acor(contrast of data)2

dfeff dfresidual dfacor

FWHMacor2 3/2

FWHMdata2

= +

• Variability in acor lowers df• Df depends on contrast • Smoothing acor brings df back up:

Contrast of data, acor = 0.79Contrast of data, acor = 0.61

FWHMdata = 8.79

dfeff dfeff

Higher order AR model? Try AR(3):

… has little effect on the T statistics:

AR(1) seemsto be adequate

a1

a2

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AR(1), df=100 AR(2), df=99

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AR(3), df=98No correlation

biases T up ~12% → more false positives

Run 1 Run 2 Run 3 Run 4

Effect, E i

Sd, S

i

T stat, E i / S i

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1 2nd level

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2nd level: 4 runs, 3 df for random effects sd

… and T>15.96 for P<0.05 (corrected):

… very noisy sd:

… so no response is detected …

• Basic idea: increase df by spatial smoothing (local pooling) of the sd.

• Can’t smooth the random effects sd directly, - too much anatomical structure.

• Instead,

random effects sd

fixed effects sd

which removes the anatomical structure before smoothing.

Solution: Spatial smoothing of the sd ratio

sd = smooth fixed effects sd )

Random effects sd, 3 df Fixed effects sd, 440 df

0

0.05

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Mixed effects sd, ~100 df

Random sd / fixed sd

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1.5Smoothed sd ratio

randomeffect, sdratio ~1.3

divide multiply

^ Average Si

dfratio = dfrandom(2 + 1)1 1 1

dfeff dfratio dffixed

How much smoothing?

FWHMratio2 3/2

FWHMdata2

= +

dfrandom = 3, dffixed = 4 110 = 440, FWHMdata = 8mm:

0 20 40 Infinity0

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FWHMratio

dfeff

random effectsanalysis, dfeff = 3

fixed effects analysis, dfeff = 440

Target = 100 df FWHM = 19mm

Run 1 Run 2 Run 3 Run 4

Effect, E i

Sd, S

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Final result: 19mm smoothing, 100 df

… less noisy sd:

… and T>4.93 for P<0.05 (corrected):

… and now we can detect a response!

In between: use Discrete Local Maxima (DLM)

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Bonferroni

Random Field Theory

Discrete Local Maxima (DLM)

High FWHM: use Random Field Theory

Low FWHM: use Bonferroni

Final level: Multiple comparisons correction

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Gaussian T, 20 df T, 10 dfP

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ue

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True

Bonferroni Random Field Theory

Discrete Local Maxima

DLM can ½ P-value when FWHM ~3 voxels

In between: use Discrete Local Maxima (DLM)

High FWHM: use Random Field Theory

Low FWHM: use Bonferroni

FIAC paradigm 16 subjects 4 runs per subject

2 runs: event design 2 runs: block design

4 conditions per run Same sentence, same speaker Same sentence, different speaker Different sentence, same speaker Different sentence, different speaker

3T, 191 frames, TR=2.5s

Events

Blocks

Response

0 50 100 150 200 250 300 350 400 450 500-0.2

0

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0 50 100 150 200 250 300 350 400 450 500-0.2

0

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Seconds

Beginning of block/run

1st snt in blockS snt, S spk, B1S snt, S spk, B2S snt, D spk, B1S snt, D spk, B2D snt, S spk, B1D snt, S spk, B2D snt, D spk, B1D snt, D spk, B2 Constant Linear Quadratic Cubic Spline Whole brain avg

Design matrix for block expt

B1, B2 are basis functions for magnitude and delay:

Motion and slice time correction (using FSL) 5 conditions

Smoothing of temporal autocorrelation to control the effective df

1st level analysis

3 contrasts Beginning of block/run

Same sent, same speak

Same sent, diff speak

Diff sent, same speak

Diff sent, diff speak

Sentence 0 -0.5 -0.5 0.5 0.5

Speaker 0 -0.5 0.5 -0.5 0.5

Interaction 0 1 -1 -1 1

0

0.5

1

1.5

2

Diff sente Diff speak Interac

Magnitude sd (relative to error)

Event

Block

00.20.40.60.8

11.21.41.6

Diff sente Diff speak Interac

Delay sd (seconds)

Event

Block

Sd of contrasts (lower is better) for a single run, assuming additivity of responses • For the magnitudes, event and block have similar efficiency

• For the delays, event is much better.

Efficiency

2nd level analysis Analyse events and blocks separately Register contrasts to Talairach (using FSL)

Bad registration on 2 subjects - dropped Combine 2 runs using fixed FX

Combine remaining 14 subjects using random FX 3 contrasts × event/block × magnitude/delay = 12

Threshold using best of Bonferroni, random field theory, and discrete local maxima (new!)

3rd level analysis

Part of slice z = -2 mm

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Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 6214

Random /fixed effects sdsmoothed 7.0105mm

FWHM (mm)

P=0.05 threshold for local maxima is +/- 5.68

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Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5


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Events: 0.14±0.04s; Blocks: 1.19±0.23s Both significant, P<0.05 (corrected) (!?!) Answer: take a look at blocks:

Events vs blocks for delaysin different – same sentence

Different sentence(sustained interest)

Same sentence (lose interest)

Best fitting block

Greatermagnitude

Greater delay

SPM BRAINSTAT

Magnitude increase for Sentence, Event Sentence, Block Sentence, Combined Speaker, Combined at (-54,-14,-2)

Magnitude decrease for

Sentence, Block Sentence, Combined

at (-54,-54,40)

Delay increase forSentence, Eventat (58,-18,2)inside the region where all conditions are activated

Conclusions

Greater %BOLD response for different – same sentences (1.08±0.16%) different – same speaker (0.47±0.0.8%)

Greater latency for different – same sentences (0.148±0.035 secs)

z=-12 z=2 z=5

3

1,4

21

3 3 31

3

The main effects of sentence repetition (in red) and of speaker repetition (in blue). 1: Meriaux et al, Madic; 2: Goebel et al, Brain voyager; 3: Beckman et al, FSL; 4: Dehaene-Lambertz et al, SPM2.

Brainstat:combinedblock andevent, threshold at T>5.67, P<0.05.

-5 0 5 10 15 20 25-0.4

-0.2

0

0.2

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0.6

t (seconds)

Estimating the delay of the response

• Delay or latency to the peak of the HRF is approximated by a linear combination of two optimally chosen basis functions:

HRF(t + shift) ~ basis1(t) w1(shift) + basis2(t) w2(shift)

• Convolve bases with the stimulus, then add to the linear model

basis1 basis2HRF

shift

delay

-5 0 5-3

-2

-1

0

1

2

3

shift (seconds)

• Fit linear model, estimate w1 and w2

• Equate w2 / w1 to estimates, then solve for shift (Hensen et al., 2002)

• To reduce bias when the magnitude is small, use

shift / (1 + 1/T2)

where T = w1 / Sd(w1) is the T statistic for the magnitude

• Shrinks shift to 0 where there is little evidence for a response.

w1

w2

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Contour is: average anatomy > 2000

Random /fixed effects sdsmoothed

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P=0.05 threshold for peaks is +/- 5.1687

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16.9641mm

FWHM (mm)

Peaks not significant at P=0.05

0.5

1

1.5

0

5

10

15

20

y (mm)

x (m

m)

-60-40-200

-50

0

50

0

5

10

15

20

-2

-1

0

1

2

0

0.5

1

1.5

2

-4

-2

0

2

4

0

201

1

202

3

200

4

206

6

201

7

201

8

200

9

200

10

204

11

204

12

206

13

201

14

205

15

204 100


Ef

Sd

T

df

Delay shift (secs), diff - same speaker



14.3951mm

FWHM (mm)


0.5

1

1.5

0

5

10

15

20

y (mm)

x (m

m)

-60-40-200

-50

0

50

0

5

10

15

20

-2

-1

0

1

2

0

0.5

1

1.5

2

-4

-2

0

2

4

0

278

1

278

3

279

4

280

6

264

7

126

8

277

9

287

10

264

11

272

12

260

13

277

14

264

15

264 100


Ef

Sd

T

df

Delay shift (secs), interaction



16.9013mm

FWHM (mm)


0.5

1

1.5

0

5

10

15

20

y (mm)

x (m

m)

-60-40-200

-50

0

50

0

5

10

15

20

-2

-1

0

1

2

0

0.5

1

1.5

2

-4

-2

0

2

4

0

204

1

200

3

207

4

200

6

204

7

205

8

202

9

203

10

202

11

204

12

206

13

201

14

201

15

200 100


Ef

Sd

T

df

Delay shift (secs), interaction



14.4178mm

FWHM (mm)


0.5

1

1.5

0

5

10

15

20

y (mm)

x (m

m)

-60-40-200

-50

0

50

0

5

10

15

20

STAT_SUMMARY example: single run, hot-warm

Detected by DLM,but not by BON or RFT

Detected by BON andDLM but not by RFT

our entry in the functional imaging analysis contest

Documents

level results

dffinal level

runs3rd level

correlation t

reasonable time

matlab python1st level

real data

simulated data