Basics of fMRI Group Analysis
Douglas N. Greve
fMRI Analysis Overview
Subject 1
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
Subject 2
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
C
First Level
GLM Analysis
X
C
Higher Level GLM
Subject 3
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
X
C
C
Subject 4
Preprocessing
MC, STC, B0
Smoothing
Normalization
Raw Data
First Level
GLM Analysis
X
C
2
Overview
• Population vs Sample
• First-Level (Time-Series) Analysis Review
• Types of Group Analysis
– Random Effects, Mixed Effects, Fixed Effects
• Multi-Level General Linear Model (GLM)
• Examples (One Group, Two Groups, Covariates)
• Longitudinal
3
Population vs Sample
Group Population
(All members)
Hundreds?
Thousands?
Billions?
Sample
18 Subjects
• Do you want to draw
inferences beyond your
sample?
• Does sample represent entire
population?
• Random Draw?
4
Functional Anatomy/Brain Mapping
5
Visual/Auditory/Motor Activation Paradigm
15 sec ‘ON’, 15 sec ‘OFF’
• Flickering Checkerboard
• Auditory Tone
• Finger Tapping
6
Block Design: 15s Off, 15s On
7
Contrasts and Inference

OFF
2
OFF
t
ON
2
 ON
 ON   OFF
2
2
( N ON  1) ON
 ( N OFF  1) OFF
( N ON  N OFF  2) 2
Contrast ON  OFF
2
2
( NON  1) ON
 ( NOFF  1) OFF
Var(Contrast) 
( NON  NOFF  2) 2
2
 ON ,  ON
, N ON  Mean,Var, N in ON
2
 OFF ,  OFF
, N OFF  Mean,Var, N in OFF
p = 10-11, sig=-log10(p) =11
Note: z, t, F monotonic with p
p = .10, sig=-log10(p) =1
8
Matrix Model
y
=
X
*

Observations
Task
base
=
Data from
one voxel
Vector of
Regression
Coefficients
(“Betas”)
Design Matrix
Regressors
Design Matrix
Contrast Matrix:
C = [1 0]
Contrast = C* = Task
9
Contrasts and the Full Model
y  X  n, y  s  n, n ~ N (0,  n2 )
ˆ  ( X T X ) 1 X T y P aramet erEst imat es
T
ˆ
n
nˆ
2
ˆ n 
DOF
ˆ  Cˆ
Residual Variance
Cont rast


1
ˆ
ˆ      C ( X T X ) 1 C T ˆ n2 Cont rastVariance Est imat e
J
J  Rows in C
ˆ
Cˆ
t DOF 

t - T est (univariate)
ˆ 
C ( X T X ) 1 C T ˆ n2
2

ˆ 1ˆ
FDOF, J  ˆ T 


F - T est (mult ivariat e)
10
Statistical Parametric Map (SPM)
+3%
0%
-3%
Contrast
Amplitude
CON, COPE, CES
“Massive Univariate Analysis”
-- Analyze each voxel separately
Contrast
Amplitude
Variance
(Error Bars)
VARCOPE, CESVAR
Significance
t-Map (p,z,F)
(Thresholded
p<.01)
sig=-log10(p)
11
Is Pattern Repeatable Across Subject?
Subject 1
Subject 2
Subject 3
Subject 4
Subject 5
12
Spatial Normalization
Native Space
MNI305 Space
Subject 1
Subject 1
MNI305
Subject 2
Subject 2
Affine (12 DOF) Registration
13
Group Analysis
Does not have to
be all positive!
14
“Random Effects (RFx)” Analysis
t
G
 2
G
G
G
 2 
G
 G2
NG 1
DOF  N G  1
RFx
15
“Random Effects (RFx)” Analysis
• Model Subjects as a Random Effect
• Variance comes from a single source:
variance across subjects
G
G
– Mean at the population mean
– Variance of the population variance
• Does not take first-level noise into account
(assumes 0)
• “Ordinary” Least Squares (OLS)
• Usually less activation than individuals
• Sometimes more
RFx
16
“Mixed Effects (MFx)” Analysis
MFx
RFx
• Down-weight each subject based on variance.
• Weighted Least Squares vs (“Ordinary” LS)
17
“Mixed Effects (MFx)” Analysis
• Down-weight each subject based on variance.
• Weighted Least Squares vs (“Ordinary” LS)
• Protects against unequal variances across group or
groups (“heteroskedasticity”)
• May increase or decrease significance with respect to
simple Random Effects
• More complicated to compute
• “Pseudo-MFx” – simply weight by first-level
MFx
variance (easy to compute)
18
“Fixed Effects (FFx)” Analysis
t
G
FFx
 2
G
 2 
G

2
i
2

 i
N G 2
DOF   DOFi
RFx
19
“Fixed Effects (FFx)” Analysis
•
•
•
•
•
•
As if all subjects treated as a single subject (fixed effect)
Small error bars (with respect to RFx)
Large DOF

t G
Same mean as RFx
 2
Huge areas of activation
2

 2   i2
Not generalizable beyond sample.
N G 
G
G
DOF   DOFi
FFx
20
Multi-Level Analysis
Raw Data
at a Voxel
First Level
(Time-Series)
GLM
Contrast Size (Signed)
Higher Level
Contrast Variance
p/t/F/z
Contrast
Matrix (C)
ROI Volume
Not recommended.
Noisy.
Visualize
Design
Matrix (X)
21
Multi-Level Analysis
Subject 1
First
Level
Contrast Size 1
C
Subject 2
First
Level
Subject 3
Contrast Size 2
Higher Level GLM
C
First
Level
Contrast Size 3
C
Subject 4
First
Level
Contrast Size 4
C
22
Higher Level GLM Analysis
(Low-Level Contrasts)
Observations
y
=
=
X * 
1
1
1
1
1
Design Matrix
(Regressors)
Data from
one voxel
One-Sample Group Mean (OSGM)
G
G
G
Vector of
Regression
Coefficients
(“Betas”)
Contrast Matrix:
C = [1]
Contrast = C* = G
23
Two Groups GLM Analysis
(Low-Level Contrasts)
Observations
y
=
X * 
=
1
1
1
0
0
0
0
0
1
1
G1
G2
Data from
one voxel
24
Contrasts: Two Groups GLM Analysis
1. Does Group 1 by itself differ from 0?
C = [1 0], Contrast = C* = G1
2. Does Group 2 by itself differ from 0?
C = [0 1], Contrast = C* = G2
=
1
1
1
0
0
0
0
0
1
1
G1
G2
3. Does Group 1 differ from Group 2?
C = [1 -1], Contrast = C* = G1- G2
4. Does either Group 1 or Group 2 differ from 0?
C has two rows: F-test (vs t-test)
1 0 Concatenation of contrasts #1 and #2
C=
01
25
One Group, One Covariate (Age)
y
=
X * 
(Low-Level Contrasts)
Observations
Contrast
=
1
1
1
1
1
21
33
64
17
47
G
Age
Intercept: G
Slope: Age
Age
Data from
one voxel
26
Contrasts: One Group, One Covariate
1. Does Group offset/intercept differ from 0?
Does Group mean differ from 0 regressing
out age?
C = [1 0], Contrast = C* = G
(Treat age as nuisance)
Contrast
2. Does Slope differ from 0?
C = [0 1], Contrast = C* = Age
=
1
1
1
1
1
21
33
64
17
47
G
Age
Intercept: G
Slope: Age
Age
27
Two Groups, One Covariate
• Somewhat more complicated design
• Slopes may differ between the groups
• What are you interested in?
• Differences between intercepts? Ie, treat covariate
as a nuisance?
• Differences between slopes? Ie, an interaction
between group and covariate?
28
Two Groups, One (Nuisance) Covariate
Is there a difference between the
group means?
Synthetic Data
29
Two Groups, One (Nuisance) Covariate
Raw Data
Effect of Age
Means After Age
“Regressed Out”
(Intercept, Age=0)
• No difference between groups
• Groups are not well matched for age
• No group effect after accounting for age
• Age is a “nuisance” variable (but important!)
• Slope with respect to Age is same across groups
30
Two Groups, One (Nuisance) Covariate
(Low-Level Contrasts)
Observations
y
=
X * 
=
1
1
1
0
0
0
0
0
1
1
21
33
64
17
47
G1
G2
Age
One regressor for Age.
Data from
one voxel
Different Offset Same Slope (DOSS)
31
Two Groups, One (Nuisance) Covariate
=
1
1
1
0
0
0
0
0
1
1
Different Offset Same Slope (DOSS)
21
33
64
17
47
G1
G2
Age
One regressor for Age indicates that
groups have same slope – makes
difference between group
means/intercepts independent of age.
32
Contrasts: Two Groups + Covariate
1. Does Group 1 mean differ from 0 (after
regressing out effect of age)?
C = [1 0 0], Contrast = C* = G1
2. Does Group 2 mean differ from 0
(after regressing out effect of age)?
C = [0 1 0], Contrast = C* = G2
1
1
= 1
0
0
0
0
0
1
1
21
33
64
17
47
G1
G2
Age
3. Does Group 1 mean differ from Group 2 mean
(after regressing out effect of age)?
C = [1 -1 0], Contrast = C* = G1- G2
33
Contrasts: Two Groups + Covariate
4. Does Slope differ from 0 (after
regressing out the effect of group)?
Does not have to be a “nuisance”!
C = [0 0 1], Contrast = C* = Age
1
1
= 1
0
0
0
0
0
1
1
21
33
64
17
47
G1
G2
Age
34
Group/Covariate Interaction
• Slope with respect to Age differs between groups
• Interaction between Group and Age
• Intercept different as well
35
Group/Covariate Interaction
(Low-Level Contrasts)
Observations
y
Data from
one voxel
Different Offset Different Slope (DODS)
=
X * 
=
1
1
1
0
0
0
0
0
1
1
21
33
64
0
0
0
0
0
17
47
G1
G2
Age1
Age2
Group-by-Age Interaction
36
Group/Covariate Interaction
1. Does Slope differ between groups?
Is there an interaction between group and age?
C = [0 0 1 -1], Contrast = C* = Age1- Age1
1
1
1
= 0
0
0
0
0
1
1
21 0
33 0
64 0
0 17
0 47
G1
G2
Age1
Age2
37
Group/Covariate Interaction
Does this contrast make sense?
2. Does Group 1 mean differ from Group 2
mean (after regressing out effect of age)?
C = [1 -1 0 0], Contrast = C* = G1- G2
1
1
1
= 0
0
0
0
0
1
1
21 0
33 0
64 0
0 17
0 47
G1
G2
Age1
Age2
Very tricky!
This tests for difference at Age=0
What about Age = 12?
What about Age = 20?
38
Group/Covariate Interaction
If you are interested in the difference between the
means but you are concerned there could be a
difference (interaction) in the slopes:
1. Analyze with interaction model (DODS)
2. Test for a difference in slopes
3. If there is no difference, re-analyze with single
regressor model (DOSS)
4. If there is a difference, proceed with caution
39
Interaction between Condition and Group
Example:
• Two First-Level Conditions: Angry and Neutral Faces
• Two Groups: Healthy and Schizophrenia
Desired Contrast = (Neutral-Angry)Sch - (Neutral-Angry)Healthy
Two-level approach
1. Create First Level Contrast (Neutral-Angry)
2. Second Level:
• Create Design with Two Groups
• Test for a Group Difference
40
Longitudinal
Visit 1
Visit 2
Subject 1
Subject 2
Subject 3
Subject 4
Subject 5
41
Longitudinal
Did something change between visits?
• Drug or Behavioral Intervention?
• Training?
• Disease Progression?
• Aging?
• Injury?
• Scanner Upgrade?
42
Longitudinal
Subject 1, Visit 1
Subject 1, Visit 2
Paired Differences
Between Subjects
43
Longitudinal Paired Analysis
(V1-V2 Differences in
Low-Level Contrasts)
Observations
y
Paired Diffs
from one voxel
=
X * 
=
1
1
1
1
1
Design Matrix
(Regressors)
One-Sample Group Mean (OSGM): Paired t-Test
DV
Contrast Matrix:
C = [1]
Contrast = C* = DV
44
fMRI Analysis Overview
Subject 1
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
Subject 2
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
C
First Level
GLM Analysis
X
C
Higher Level GLM
Subject 3
Raw Data
Preprocessing
MC, STC, B0
Smoothing
Normalization
First Level
GLM Analysis
X
X
C
C
Subject 4
Preprocessing
MC, STC, B0
Smoothing
Normalization
Raw Data
First Level
GLM Analysis
X
C
45
Summary
• Higher Level uses Lower Level Results
– Contrast and Variance of Contrast
• Variance Models
– Random Effects
– Mixed Effects – protects against heteroskedasticity
– Fixed Effects – cannot generalize beyond sample
• Groups and Covariates (Intercepts and Slopes)
• Covariate/Group Interactions
• Longitudinal – Paired Differences
46
47