FMRI Group Analysis
Voxel-wise group analysis
Design matrix
Standard-space
brain atlas
subjects
Single-subject
effect size
Single-subject
statistics
effect
size
Single-subject
statistics
effect size
Single-subject
statistics
effect
size
statistics
Register
subjects into
a standard
space
Subject
groupings
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
Effect size subject-series
GLM
Group
effect size
statistics
subjects
Significant
voxels/clusters
Statistic Image
Contrast
Effect size
statistics
Thresholding
Multi-Level FMRI analysis
•
•
Mark
Steve
uses GLM at both lower and higher levels
typically need to infer across multiple subjects,
sometimes multiple groups and/or multiple sessions
Difference?
Group 1
Group 2
Karl
Will
Tom
Andrew
Josephine
Anna
Hanna Sebastian
Lydia Elisabeth
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 4
session 4
session 4
session 4
session 4
session 4
•
session 4
questions of interest involve comparisons at the
highest level
session 4
A simple example
Does the group activate on average?
Group
Mark
Steve
Karl
Will
Tom
Andrew
A simple example
Does the group activate on average?
Group
Mark
Steve
Karl
Will
0
Tom
Andrew
effect size
A simple example
Does the group activate on average?
Group
Mark
Yk = Xk
Steve
k
Karl
Will
Tom
Andrew
+ ⇥k
First-level GLM
on Mark’s 4D FMRI
data set
0
effect size
A simple example
Does the group activate on average?
Group
Mark
Yk = Xk
Mark’s
effect size
Steve
k
Karl
Will
Tom
Andrew
+ ⇥k
0
effect size
A simple example
Does the group activate on average?
Group
Mark
Yk = Xk
Steve
k
Karl
Will
Tom
Andrew
+ ⇥k
Mark’s
within-subject
variance
0
effect size
A simple example
Does the group activate on average?
Group
Mark
YK = XK
Steve
K
Karl
Will
Tom
Andrew
+ ⇥K
All first-level GLMs
on 6 FMRI data set
0
effect size
A simple example
Does the group activate on average?
Group
Mark
Steve
Karl
Will
Tom
Andrew
What group mean are we after? Is it:
1. The group mean for those exact 6 subjects?
Fixed-Effects (FE) Analysis
2. The group mean for the population from which
these 6 subjects were drawn?
Mixed-Effects (ME) analysis
Fixed-Effects Analysis
Do these exact 6 subjects activate on average?
Group
Mark
Steve
Karl
estimate group effect size as
straight-forward mean
across lower-level estimates
Will
Tom
Andrew
0
effect size
g
1
=
6
6
k
k=1
Fixed-Effects Analysis
Do these exact 6 subjects activate on average?
Group
Mark
Steve
YK = XK
K
⇧
⇧
⇧
Xg = ⇧
⇧
⇧
⇤
= Xg
1
1
1
1
1
1
⇥
⌃
⌃
⌃
⌃
⌃
⌃
⌅
K
Karl
Will
Tom
Andrew
+ ⇥K
g
0
Group
mean
effect size
g
1
=
6
6
k
k=1
Fixed-Effects Analysis
Do these exact 6 subjects activate on average?
Group
Mark
Steve
Karl
YK = XK
K = Xg
Will
K
Tom
+ ⇥K
g
Fixed Effects Analysis:
•
Consider only these 6 subjects
estimate the mean across these subject
only variance is within-subject variance
•
•
Andrew
A simple example
Does the group activate on average?
Group
Mark
Steve
Karl
Keith
Tom
Andrew
What group mean are we after? Is it:
1. The group mean for those exact 6 subjects?
Fixed-Effects (FE) Analysis
2. The group mean for the population from which
these 6 subjects were drawn?
Mixed-Effects (ME) analysis
Mixed-Effects Analysis
Does the population activate on average?
Group
Mark
YK = XK
Steve
K
Karl
Keith
•
Andrew
+ ⇥K
Consider the distribution over the
population from which our 6
subjects were sampled:
2
g
Tom
g
0
is the between-subject variance
g
effect size k
Mixed-Effects Analysis
Does the population activate on average?
Group
Mark
Steve
YK = XK
= Xg
K
⇧
⇧
⇧
Xg = ⇧
⇧
⇧
⇤
1
1
1
1
1
1
⇥
⌃
⌃
⌃
⌃
⌃
⌃
⌅
K
g
Karl
Keith
Tom
Andrew
+ ⇥K
+ ⇥g
g
0
Population
mean
betweensubject
variation
g
effect size k
Mixed-Effects Analysis
Does the population activate on average?
Group
Mark
Steve
Karl
YK = XK
K = Xg
•
Keith
Tom
Andrew
+ ⇥K
+ ⇥g
K
g
Mixed-Effects Analysis:
Consider the 6 subjects as samples from a wider population
estimate the mean across the population
between-subject variance accounts for random sampling
•
•
All-in-One Approach
Difference?
Group 1
Mark Steve
Karl
Will
Tom Andrew
Group 2
Josephine Anna Hanna Sebastian Lydia Elisabeth
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 1
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 2
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 3
session 4
session 4
session 4
session 4
session 4
session 4
•
session 4
Could use one (huge) GLM to infer group difference
•
•
•
difficult to ask sub-questions in isolation
computationally demanding
need to process again when new data is acquired
session 4
Summary Statistics Approach
In FEAT estimate levels one stage at a time
•
At each level:
•
•
•
Inputs are summary stats from levels
below (or FMRI data at the lowest
level)
Group
difference
Group
Outputs are summary stats or
statistic maps for inference
Need to ensure formal equivalence
between different approaches!
Subject
Session
FLAME
FMRIB’s Local Analysis of Mixed Effects
•
•
•
Fully Bayesian framework
•
•
•
use non-central t-distributions:
Input COPES,VARCOPES & DOFs
from lower-level
estimate COPES,VARCOPES &
DOFs at current level
pass these up
Infer at top level
Equivalent to All-in-One approach
Z-Stats
Group
difference
COPES
VARCOPES
DOFs
Group
COPES
VARCOPES
DOFs
Subject
COPES
VARCOPES
DOFs
Session
FLAME Inference
•
•
Default is:
•
FLAME1: fast approximation for all voxels (using
marginal variance MAP estimates)
Optional slower, slightly more accurate approach:
•
FLAME1+2:
•
FLAME1 for all voxels, FLAME2 for voxels close to
threshold
•
FLAME2: MCMC sampling technique
Choosing Inference Approach
1. Fixed Effects
Use for intermediate/top levels
2. Mixed Effects - OLS
Use at top level: quick and less accurate
3. Mixed Effects - FLAME 1
Use at top level: less quick but more accurate
4. Mixed Effects - FLAME 1+2
Use at top level: slow but even more accurate
FLAME vs. OLS
•
•
•
•
allow different within-level
variances (e.g. patients vs.
controls)
allow non-balanced designs
(e.g. containing behavioural
scores)
pat
ctl
0
effect size
...
allow un-equal group sizes
solve the ‘negative variance’
problem
<
Session
<
Subject
Group
FLAME vs. OLS
•
Two ways in which FLAME can give different Z-stats
compared to OLS:
FLAME
OLS
•
higher Z due to increased efficiency from using
lower-level variance heterogeneity
FLAME vs. OLS
•
Two ways in which FLAME can give different Z-stats
compared to OLS:
FLAME
OLS
•
Lower Z due to higher-level variance being
constrained to be positive (i.e. solve the implied
negative variance problem)
Multiple Group Variances
pat
ctl
•
can deal with multiple
variances
•
separate variance will be
0
estimated for each variance group (be aware of
#observations for each estimate, though!)
•
design matrices need to be ‘separable’, i.e. EVs only
have non-zero values for a single group
valid
invalid
group
effect size
Examples
Single Group Average
•
We have 8 subjects - all in one group - and want the
mean group average:
•
•
estimate mean
•
test significance of
mean > 0
estimate std-error
(FE or ME)
subject
Does the group activate on average?
0
effect size
>0?
Single Group Average
subject
Does the group activate on average?
0
effect size
Single Group Average
Does the group activate on average?
Unpaired Two-Group Difference
• We have two groups (e.g. 9 patients, 7 controls)
with different between-subject variance
•
•
estimate means
•
test significance of
difference in means
estimate std-errors
(FE or ME)
subject
Is there a significant group difference?
0
>0?
effect size
Unpaired Two-Group Difference
subject
Is there a significant group difference?
0
effect size
Unpaired Two-Group Difference
subject
Is there a significant group difference?
0
effect size
Unpaired Two-Group Difference
Is there a significant group difference?
Paired T-Test
8 subjects scanned under 2 conditions (A,B)
Is there a significant difference between conditions?
subject
•
0
effect size
Paired T-Test
8 subjects scanned under 2 conditions (A,B)
Is there a significant difference between conditions?
try non-paired t-test
subject
•
0
effect size
>0?
Paired T-Test
8 subjects scanned under 2 conditions (A,B)
Is there a significant difference between conditions?
data
0
subject
de-meaned data
subject
•
effect size
subject mean
accounts for large prop.
of the overall variance
0
effect size
Paired T-Test
8 subjects scanned under 2 conditions (A,B)
Is there a significant difference between conditions?
data
0
subject
de-meaned data
subject
•
effect size
subject mean
accounts for large prop.
of the overall variance
0
>0?
effect size
Paired T-Test
subject
Is there a significant difference between conditions?
0
effect size
Paired T-Test
subject
Is there a significant difference between conditions?
0
effect size
Paired T-Test
Is there a significant difference between conditions?
EV1models the A-B
paired difference; EVs
2-9 are confounds
which model out each
subject’s mean
Paired T-Test
Is there a significant difference between conditions?
Multi-Session & Multi-Subject
• 5 subjects each have three sessions.
Does the group activate on average?
•
Use three levels: in the second level we
model the within-subject repeated measure
Multi-Session & Multi-Subject
• 5 subjects each have three sessions.
Does the group activate on average?
•
Use three levels: in the third level we model
the between-subjects variance
Multi-Session & Multi-Subject
•
•
•
5 subjects each have three sessions.
Does the group activate on average?
Use three levels:
•
in the second level we model the within subject
repeated measure typically using fixed effects(!)
as #sessions are small
•
in the third level we model the between subjects
variance using fixed or mixed effects
Reducing variance
subject
subject
Does the group activate on average?
0
effect size
>0?
mean effect size large
relative to std error
0
effect size
>0?
mean effect size small
relative to std error
Reducing variance
subject
subject
Does the group activate on average?
0
effect size
>0?
mean effect size large
relative to std error
0
effect size
>0?
mean effect size large
relative to std error
Single Group Average & Covariates
•
We have 7 subjects - all in one group. We also have
additional measurements (e.g. age; disability score;
behavioural measures like reaction times):
•
use covariates to
‘explain’ variation
•
•
estimate mean
subject
Does the group activate on average?
estimate std-error
(FE or ME)
0
effect size
Single Group Average & Covariates
•
We have 7 subjects - all in one group. We also have
additional measurements (e.g. age; disability score;
behavioural measures like reaction times):
•
use covariates to
‘explain’ variation
•
•
estimate mean
subject
Does the group activate on average?
estimate std-error
(FE or ME)
0
effect size
slow
RT
fast
Single Group Average & Covariates
•
We have 7 subjects - all in one group. We also have
additional measurements (e.g. age; disability score;
behavioural measures like reaction times):
•
use covariates to
‘explain’ variation
•
•
estimate mean
subject
Does the group activate on average?
estimate std-error
(FE or ME)
0
effect size
slow
RT
fast
Single Group Average & Covariates
•
We have 7 subjects - all in one group. We also have
additional measurements (e.g. age; disability score;
behavioural measures like reaction times):
•
use covariates to
‘explain’ variation
•
•
estimate mean
subject
Does the group activate on average?
estimate std-error
(FE or ME)
0
effect size
slow
RT
fast
Single Group Average & Covariates
•
We have 7 subjects - all in one group. We also have
additional measurements (e.g. age; disability score;
behavioural measures like reaction times):
•
use covariates to
‘explain’ variation
•
•
estimate mean
subject
Does the group activate on average?
estimate std-error
(FE or ME)
0
effect size
slow
RT
fast
Single Group Average & Covariates
Does the group activate on average?
•
•
use covariates to ‘explain’ variation
need to de-mean additional covariates!
FEAT Group Analysis
•
Run FEAT on raw FMRI data to get first-level .feat
directories, each one with several (consistent) COPEs
•
•
low-res copeN/varcopeN
.feat/stats
when higher-level FEAT is run, highres copeN/
varcopeN
.feat/reg_standard
FEAT Group Analysis
•
Run second-level FEAT to get one .gfeat directory
•
Inputs can be lowerlevel .feat dirs or
lower-level COPEs
•
the second-level GLM analysis is run separately
for each first-level COPE
•
each lower-level COPE generates its own .feat
directory inside the .gfeat dir
That’s all folks
Appendix:
Group F-tests
•
3 groups of subjects
Is any of the groups activating on average?
ANOVA: 1-factor 4-levels
•
8 subjects, 1 factor at 4 levels
Is there any effect?
•
•
EV1 fits cond. D, EV2 fits cond A relative to D etc.
F-test shows any difference between levels
ANOVA: 2-factor 2-levels
•
8 subjects, 2 factor at 2 levels. FE Anova: 3 F-tests give
standard results for factor A, B and interaction
•
If both factors are random effects then Fa=fstat1/fstat3,
Fb=fstat2/fstat3ME
•
ME: if fixed fact. is A, Fa=fstat1/fstat3
ANOVA: 3-factor 2-levels
•
•
16 subjects, 3 factor at 2 levels.
•
For random/mixed effects need different Fs.
Fixed-Effects ANOVA:
Understanding FEAT dirs
•
First-level analysis:
Understanding FEAT dirs
•
Second-level analysis:
That’s all folks