Our entry in the Functional Imaging Analysis contest
description
Transcript of 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
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2Alternating hot and warm stimuli separated by rest (9 seconds each).
hot
warm
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warm
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Hemodynamic response function: difference of two gamma densities
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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”
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T statistic for hot - warm effect
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restwarm
Highly significant effect, T=6.59
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No significant effect, T=-0.74
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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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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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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
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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
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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
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Mixed effects sd, ~100 df
Random sd / fixed sd
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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
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dfrandom = 3, dffixed = 4 110 = 440, FWHMdata = 8mm:
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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
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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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High FWHM: use Random Field Theory
Low FWHM: use Bonferroni
Final level: Multiple comparisons correction
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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
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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
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Magnitude sd (relative to error)
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Delay sd (seconds)
Event
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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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Subj Mixed effects
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Magnitude (%BOLD), diff - same sentence, event experiment
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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Subj Mixed effects
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Magnitude (%BOLD), diff - same sentence, block experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 5904
Random /fixed effects sdsmoothed 7.103mm
FWHM (mm)
P=0.05 threshold for local maxima is +/- 5.67
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Subj Mixed effects
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Delay shift (secs), diff - same sentence, event experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed 10.6778mm
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P=0.05 threshold for local maxima is +/- 4.31
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Subj Mixed effects
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Delay shift (secs), diff - same sentence, block experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed 8.8952mm
FWHM (mm)
P=0.05 threshold for local maxima is +/- 4.3
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Subj Mixed effects
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Magnitude (%BOLD), diff - same sentence, event experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 6214
Random /fixed effects sdsmoothed 7.0105mm
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P=0.05 threshold for local maxima is +/- 5.68
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Subj Mixed effects
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Magnitude (%BOLD), diff - same sentence, block experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: min fMRI > 5904
Random /fixed effects sdsmoothed 7.103mm
FWHM (mm)
P=0.05 threshold for local maxima is +/- 5.67
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Delay shift (secs), diff - same sentence, event experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed 10.6778mm
FWHM (mm)
P=0.05 threshold for local maxima is +/- 4.31
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Subj Mixed effects
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Delay shift (secs), diff - same sentence, block experiment
Slice range is -74<x<70mm, -46<y<4mm, z=-2mm; Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed 8.8952mm
FWHM (mm)
P=0.05 threshold for local maxima is +/- 4.3
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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
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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.
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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
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• 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.
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1
2
0
0.5
1
1.5
2
-4
-2
0
2
4
0
200
1
200
3
200
4
200
6
200
7
100
8
200
9
200
10
200
11
200
12
200
13
200
14
200
15
200 100
Subject id, event experiment Mixed effects
Ef
Sd
T
df
Delay shift (secs), stimulus average
Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed
13.3482mm
FWHM (mm)
P=0.05 threshold for peaks is +/- 3.8943
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
209
1
209
3
214
4
210
6
211
7
210
8
210
9
207
10
212
11
214
12
214
13
210
14
210
15
216 100
Subject id, block experiment Mixed effects
Ef
Sd
T
df
Delay shift (secs), stimulus average
Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed
13.5901mm
FWHM (mm)
P=0.05 threshold for peaks is +/- 3.983
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
273
1
271
3
276
4
281
6
274
7
136
8
265
9
293
10
264
11
268
12
265
13
264
14
296
15
270 100
Subject id, event experiment Mixed effects
Ef
Sd
T
df
Delay shift (secs), diff - same speaker
Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed
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
Subject id, block experiment Mixed effects
Ef
Sd
T
df
Delay shift (secs), diff - same speaker
Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed
14.3951mm
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
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
Subject id, event experiment Mixed effects
Ef
Sd
T
df
Delay shift (secs), interaction
Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed
16.9013mm
FWHM (mm)
P=0.05 threshold for peaks is +/- 3.8306
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
Subject id, block experiment Mixed effects
Ef
Sd
T
df
Delay shift (secs), interaction
Contour is: magnitude, stimulus average, T statistic > 5
Random /fixed effects sdsmoothed
14.4178mm
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
STAT_SUMMARY example: single run, hot-warm
Detected by DLM,but not by BON or RFT
Detected by BON andDLM but not by RFT