spm-like statistical analysis of fmri data -...
TRANSCRIPT
![Page 1: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/1.jpg)
SPM-Like Statistical Analysis of fMRI Data
Part II: Between-Subject Analysis
Alexis Roche
JIRFNI 2009, Marseille
![Page 2: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/2.jpg)
2
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 3: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/3.jpg)
3
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 4: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/4.jpg)
4
Group Analysis Goal
● Which functional networks are commonly activated in response to a given task?
![Page 5: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/5.jpg)
5
Processing pipeline
AnatomicalTemplate
Subject 1
Spatialsmoothing
Nonrigidmatching
Subject 2
Spatialsmoothing
Nonrigidmatching
… …
BOLDestimation
Contrast images
BOLDestimation
Group analysis input
![Page 6: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/6.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
6
Mass univariate one-sample analysis
Subject 1
…
H0: “mean effect
is zero”
Normalized contrast images
Observations in one voxel
Subject 2
Subject 3
…
![Page 7: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/7.jpg)
7
What kind of analysis?
● Random effect
✗ (H0) “the mean effect vanishes in the population of interest”
● Fixed effect
✗ (H0) “the mean effect vanishes in this particular cohort”
![Page 8: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/8.jpg)
8
FFX vs. RFX
Subj. 1
Subj. 2
Subj. 3
Subj. 4
Subj. 5
Subj. 60
Distribution of each subject’s BOLD estimate
Distribution of the mean estimate
σFFX
σRFX
FFX: very significant
RFX: bordeline
Source: T. Nichols
![Page 9: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/9.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
9
Inferences in fMRI
Type Fixed-effect(FFX)
Random-effect(RFX)
Single-subject Multi-subject Multi-subject
Goal Study functionalactivity in one
particular subject
Study functionalactivity in one
particular cohort
Study functionalactivity in apopulation
![Page 10: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/10.jpg)
10
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 11: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/11.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
11
One-sample t statistic
● Divide the sample mean by its standard error
=136.91 , std = 2/n=20.37
t= std
Source: www.statsci.org
Rainfalls in Sydney
![Page 12: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/12.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
12
One-sample parametric t-test
● t follows a Student distribution if the observations are normal with zero mean
● Hence reject the null hypothesis of zero mean if the observed t is too large
0 tobs
pvalue=P t≥t obs∣H 0
Student distribution with n-1 degrees of freedom
Reject H0
pvalue≤
Accept H0
yes
no
: accepted false positive rate
![Page 13: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/13.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
13
General parametric t-test
● The t-test is designed to test a contrasted effect in a general linear model
● Used in
✗ Within-subject analyses
✗ FFX analyses
✗ Other RFX analyses
● Generalization to multidimensional contrasts: F-test
H 0 c t=0, Y=X N 0, I
![Page 14: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/14.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
14
Shortcomings
● The parametric t-test assumes normal observations
● If normality doesn't hold,
✗ Biased specificity (false positive rate)
✗ Lack of sensitivity (true positive rate)
● This is a problem for small samples
✗ The Student distribution holds in the limit of large samples even if normality doesn't
![Page 15: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/15.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
15
Specificity bias
● Samples of size 10 drawn from a uniform law
Ground truthStudentPermutations
![Page 16: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/16.jpg)
Specificity bias in multiple testing
● Parametric multiple comparison correction usually uses the Euler Characteristic (EC) approximation
● Validity requirements
✗ Multivariate normal assumption
✗ High threshold
✗ Smooth data
● May produce severe bias!
FWER≈E u∣H 0
![Page 17: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/17.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
17
Lack of sensitivity
● The t statistic is not robust
✗ One single observation can cause t to collapse
t = Inf t = 0
● Outliers may occur more frequently than predicted by the normal distribution
![Page 18: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/18.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
18
Dealing with outliers
● Invalid approach
✗ Drop outliers and run the parametric t test
t = Inf ; P=0. Invalid because statistical independence is broken.
Sign test gives P=17%. Unequal proportions may be observed by chance!
● Valid approach
✗ Use a robust test statistic combined with a permutation test that works with all datapoints
![Page 19: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/19.jpg)
19
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 20: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/20.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
20
Beyond parametric tests
● Specificity bias calls for nonparametric calibration schemes
✗ Permutation tests
✗ Bootstrap tests
● Lack of sensitivity calls for robust test statistics
● Both approaches require relaxing normality
![Page 21: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/21.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
21
Sign permutation test
Observed t
P-value
Histogram of the simulated statistics
...
Original sample
t = 3.76
One observation permuted
t = 1.36
Two observations
permuted
t = 1.25
All observations
permuted
t =-3.76
![Page 22: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/22.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
22
Familywise error correction
Observed t
Corrected P-value
Histogram of the tmax statistics ...
Original sample
tmax = 4.95
t map
... tmax = 4.63
t mapOne subject permuted
…... tmax = 4.17
t mapTwo subjects permuted
... tmax = 2.93
t mapAll subjects permuted
![Page 23: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/23.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
23
Sign permutation test
● Conservative (exact in a conditional sense)
● Works under mild assumptions
✗ One-sided tests require distribution symmetry (more general than normality)
● Manages multiple comparisons
✗ Alternative to random field theory
Distancewww.madic.org/download/
SnPMwww.sph.umich.edu/ni-stat/SnPM/
![Page 24: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/24.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
24
Other permutation tests
● Sign permutations are suited for one-sample RFX
● Two-sample RFX
✗ Shuffle group labels
● Within-subject or FFX
✗ Shuffle experimental conditions
CamBAwww-bmu.psychiatry.cam.ac.uk/software/
SnPMwww.sph.umich.edu/ni-stat/SnPM/
![Page 25: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/25.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
25
Bootstrap test
● Similar to the permutation test, except it uses another resampling method
Subtract the mean
Draws with replacement
…
![Page 26: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/26.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
26
Bootstrap test
● Approximate (not necessarily conservative)
● Asymptotically exact under weaker assumptions than the permutation test
● Manages multiple comparisons
![Page 27: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/27.jpg)
27
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 28: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/28.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
28
Robust test statistics
● The t statistic has optimal detection power for normally distributed data
● Other statistics may perform better in distributions that tend to produce outliers
✗ Median
✗ Sign statistic
✗ Wilcoxon signed rank
✗ Empirical likelihood ratio
✗ ...
![Page 29: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/29.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
29
ROC curves● Detection power in a normal distribution (sampling
10 subjects)
Student statisticWilcoxon statistic
![Page 30: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/30.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
30
ROC curves● Detection power in a Laplace distribution (sampling
10 subjects)
Student statisticWilcoxon statistic
![Page 31: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/31.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
31
ROC curves● Detection power in a heavy-tailed distribution
(sampling 10 subjects)
Student statisticWilcoxon statistic
![Page 32: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/32.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
32
Median
● Highly robust location estimator
✗ Resistant to contamination by (n-1)/2 outliers
median = 0.27 median = 0.27
![Page 33: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/33.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
33
Sign statistic
● Number of positive observations
● Similar to the median
● Permutation distribution data-independent (Binomial)
t s=Card {i ,i0}
Binomial law for 10 subjects
![Page 34: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/34.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
34
Wilcoxon signed rank
tw=∑irank ∣i∣ signi
10 subjects
● More sensitive than the sign statistic in moderate-tailed distributions
● Permutation distribution is also data-independent
![Page 35: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/35.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
35
Iterated reweighted least squares
● Algorithm sketch
C =∑i wii−2
Weight each observation according to its “closeness” to the current mean estimate
Update the estimate by minimizing the weighted least squares
Start with a robust mean estimate (e.g. the median)
● From there, compute a robust t statistic by analogy with ordinary least squares
● Not evaluated yet within permutation tests
![Page 36: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/36.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
36
Speech activation in three-month-olds
Cluster level:
Pcorr. = 0.34
Cluster level:
Pcorr. = 0.08
Cluster level:
Pcorr. = 0.02
✔ 10 subjects✔ Single threshold tests at P=0.01 (uncorrected)
Parametric t test (SPM) Permutation t test Wilcoxon test
Courtesy of Ghislaine Dehaene
Broca’s area
![Page 37: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/37.jpg)
37
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 38: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/38.jpg)
Mixed effects
● Observed effects have two distinct variability sources
✗ Within-subject (errors in first-level analysis)
✗ Between-subject (intrinsic variability of BOLD responses)
● Both sources can produce outliers
✗ Artifactual outliers
✗ Behavioral outliers
● Mixed-effect statistics are robust to artifactual outliers
![Page 39: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/39.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
39
Two-level linear model
● Accounts for mixed effects
First level
βG
β1
βi
βn
Y1
Yi
Yn
Unknown populationmean effect
Unknown individualeffects Observed fMRI data
Second level
i=X GG Y i=X i ii
● Equivalent to a generalized linear model with heteroscedastic noise
![Page 40: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/40.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
40
Two-level linear model
● Full mixed-effect approach
✗ Estimate βG based on the raw scans
✗ Implemented in FSL, SPM, ...
● Simplification: sequential approach
✗ Run first-level analyses, then estimate βG based on non-exhaustive summary statistics
✗ Implemented in fMRIstat/NiPy, Distance, ...
![Page 41: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/41.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
41
Mixed-effect statistics
● Exploit the first-level variances of the observed effects
● The higher the variance, the less reliable a subject
Subj 1
First-level summary statistics
Subj 2
…
i
Test statistic
std i
![Page 42: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/42.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
42
Expectation-Maximization algorithm
● Estimate the population mean βG and variance σ
G
from the (effect+variance) observations
E-step
Estimate the true effects and their posterior variances,
M-step
Update the population mean and variance estimates,
● Then, form a mixed-effect version of the t statistic
● Many variants exist
i=G
2
G2 i
2i
i2
G2 i
2 G , i2=
G2 i
2
G2 i
2 G=1n∑i
i , G2 =
1n∑i
[i2G−i
2]
![Page 43: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/43.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
43
FIAC’05 dataset✔ 15 subjects✔ Sentence x Speaker interaction✔ Single threshold tests at P=0.01 (uncorrected)
Parametric t test (SPM) Permutation t test Permutation MFX test
Right Superior Temporal Sulcus
Cluster level:
Pcorr. = 0.13
Cluster level:
Pcorr. = 0.09
Cluster level:
Pcorr. = 0.02
![Page 44: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/44.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
44
Two-sample testing
● Mixed-effect t statistic derived similarly to the one-sample case
● Permutation test by label switching
![Page 45: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/45.jpg)
JIRFNI 2009, Marseille26/05/0906/10/07
45
Localizer dataset✔ Two-sample test: females > males (10 subjects in each group)✔ Mental calculus task✔ Single threshold tests at P=0.01 (uncorrected)
Parametric t test (SPM) Permutation t test Permutation MFX test
Left Intra-Parietal Sulcus
Cluster level:
Pcorr. = 0.01
Cluster level:
Pcorr. = 0.13
Cluster level:
Pcorr. = 0.09
![Page 46: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/46.jpg)
46
Outline
1. Types of inferences in fMRI
2. Parametric tests
3. Nonparametric tests
4. Robust statistics
5. Mixed-effect models
6. Sphericity correction
![Page 47: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/47.jpg)
47
Linear models for group
● Spherical GLM: linear prediction + i.i.d. errors
Y=X , ~N 0,2 In
Xβ
![Page 48: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/48.jpg)
48
The issue of sphericity
“Since the data obtained in many areas of psychological inquiry are not likely to conform to these requirements … researchers using the conventional procedure will erroneously claim treatment effects when none are present, thus filling their literatures with false positive claims.” – Keselman et al. 2001
![Page 49: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/49.jpg)
49
Examples of non-spherical models
● Two-sample analysis under unequal variance
● Mixed effects
● Repeated measures ANOVA
![Page 50: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/50.jpg)
50
Two-sample model1st level:
2nd level:
Controls Blinds
XV
Source: W. Penny
![Page 51: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/51.jpg)
51
One-sample mixed-effect model
Subject 1
Subject 2
...
...
...
...0
σ1
σ
σ2V=2 I ndiag 1 ,
2 ... ,n2
Population
V
1st level:
2nd level:
![Page 52: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/52.jpg)
JIRFNI 2009, Marseille26/05/09
Repeated Measures ANOVA
Contrasted effects
Effect covariances
BaselineDrug 1 Drug 2
Subjects
Placebo
Drug1/Drug2 Drug1/Placebo Drug1/Baseline
Drug2/Placebo Drug2/Baseline
Placebo/Baseline
![Page 53: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/53.jpg)
JIRFNI 2009, Marseille26/05/09
Non-spherical linear model
Y= X Subject 1Subject 2
Subject nSubject 1Subject 2
Subject nSubject 1Subject 2
Subject nSubject 1Subject 2
Subject n
Drug 1
Drug 2
Placebo
Baseline
Variance matrix
Design matrix
![Page 54: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/54.jpg)
JIRFNI 2009, Marseille26/05/09
Non-sphericity consequences
● Specificity issue
✗ t (F) statistics deviate from expected Student (Fisher) distributions
● Sensitivity issue
✗ t (F) statistics have sub-optimal detection power
![Page 55: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/55.jpg)
JIRFNI 2009, Marseille26/05/09
Non-sphericity correction (1)
● Estimate β by ordinary least squares
● Estimate V by sample variance
● Use Satterthwaite's degrees of freedom approximation
DOF≈ traceQ V 2
trace Q V Q V Q=In−X X t X −1 X t
![Page 56: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/56.jpg)
JIRFNI 2009, Marseille26/05/09
Non-sphericity correction (2)
● Estimate hyperparameters of V by restricted maximum likelihood (ReML)
● Compute usual statistics from V-1/2Y and V-1/2X
V−1 /2Y=V−1 /2 X , ~N 0, InFiltered data Filtered predictors
![Page 57: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/57.jpg)
JIRFNI 2009, Marseille26/05/09
Global/local correction
● SPM performs global sphericity correction
✗ Implicit assumption that the variance structure is constant across voxels
✗ Repeated measures ANOVA are then inconsistent with one-sample tests
![Page 58: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/58.jpg)
JIRFNI 2009, Marseille26/05/09
Utilisation dans BrainVisa
![Page 59: SPM-Like Statistical Analysis of fMRI Data - Freeirmfmrs.free.fr/Uploads/JIRFNI_2009/JIRFNI_Marseille_Roche_between_subject.pdf · SPM-Like Statistical Analysis of fMRI Data Part](https://reader030.vdocuments.us/reader030/viewer/2022040117/5e067beba71375112878b807/html5/thumbnails/59.jpg)
JIRFNI 2009, Marseille26/05/09
Utilisation dans BrainVisa