basics of fmri group analysis douglas n. greve. 2 fmri analysis overview higher level glm first...
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Basics of fMRI Group Analysis
Douglas N. Greve
2
fMRI Analysis Overview
Higher Level GLM
First Level GLM Analysis
First Level GLM Analysis
Subject 3
First Level GLM Analysis
Subject 4
First Level GLM Analysis
Subject 1
Subject 2
CX
CX
CX
CX
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
Raw Data
Raw Data
Raw Data
Raw Data
CX
3
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
4
Population vs Sample
Group Population(All members)Hundreds?Thousands?Billions?
Sample18 Subjects
• Do you want to draw inferences beyond your sample?
• Does sample represent entire population?• Random Draw?
5
Functional Anatomy/Brain Mapping
6
Visual/Auditory/Motor Activation Paradigm
15 sec ‘ON’, 15 sec ‘OFF’ • Flickering Checkerboard• Auditory Tone• Finger Tapping
7
Block Design: 15s Off, 15s On
8
Contrasts and Inference
p = 10-11, sig=-log10(p) =11 p = .10, sig=-log10(p) =1
OFFin N Var, Mean,,,
ONin N Var, Mean,,,
)2()1()1(
2
2
2
22
OFFOFFOFF
ONONON
OFFON
OFFOFFONON
OFFON
N
N
NNNN
t
OFFON Contrast
2
22
)2(
)1()1(st)Var(Contra
OFFON
OFFOFFONON
NN
NN
ONOFF
2ON2
OFF
Note: z, t, F monotonic with p
9
Matrix Model
y = X *
Task
base
Data fromone voxel
Design MatrixRegressors
= Vector ofRegressionCoefficients(“Betas”) Design Matrix
Obs
erva
tion
s
Contrast Matrix:C = [1 0]Contrast = C* = Task
10
Contrasts and the Full Model
ate)(multivariTest -F ˆˆˆF
e)(univariatTest - tˆ)(
ˆ
ˆ
ˆt
Cin Rows J
Estimate VarianceContrast ˆ)(1ˆˆ
Contrast ˆˆ
Variance Residual ˆˆ
ˆ
EstimatesParameter )(ˆ
),0(~ , ,
1JDOF,
21DOF
212
2
1
2
T
nTT
nTT
T
n
TT
n
CXXC
C
CXXCJ
C
DOF
nn
yXXX
NnnsynXy
11
Statistical Parametric Map (SPM)+3%
0%
-3%
ContrastAmplitude
CON, COPE, CES
ContrastAmplitudeVariance
(Error Bars)VARCOPE, CESVAR
Significance t-Map (p,z,F)(Thresholded
p<.01)sig=-log10(p)
“Massive Univariate Analysis” -- Analyze each voxel separately
12
Is Pattern Repeatable Across Subject?
Subject 1 Subject 2 Subject 3 Subject 4 Subject 5
13
Spatial Normalization
Subject 1
Subject 2
Subject 1
Subject 2
MNI305
Native Space MNI305 Space
Affine (12 DOF) Registration
14
Group Analysis
Does not have to be all positive!
15
GG
1
1
22
2
G
G
G
G
NDOF
N
t
G
G
“Random Effects (RFx)” Analysis
RFx
16
GG
“Random Effects (RFx)” Analysis
RFx
• Model Subjects as a Random Effect• Variance comes from a single source: variance across subjects
– 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
17
“Mixed Effects (MFx)” Analysis
MFx
RFx• Down-weight each subject based on variance.• Weighted Least Squares vs (“Ordinary” LS)
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“Mixed Effects (MFx)” Analysis
MFx
• 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 variance (easy to compute)
19
“Fixed Effects (FFx)” Analysis
FFx
RFx
i
G
i
G
DOFDOF
N
t
G
G
2
22
2
2i
20
“Fixed Effects (FFx)” Analysis
FFx
i
G
i
G
DOFDOF
N
t
G
G
2
22
2
• As if all subjects treated as a single subject (fixed effect)• Small error bars (with respect to RFx)• Large DOF• Same mean as RFx• Huge areas of activation• Not generalizable beyond sample.
21
Multi-Level Analysis
First Level (Time-Series)
GLM
DesignMatrix (X)
ContrastMatrix (C)
Contrast Size (Signed)
Contrast Variance
p/t/F/z
Raw Dataat a Voxel
Visualize
Higher Level
ROI Volume
Not recommended.Noisy.
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Higher Level GLM
First Level
C
Contrast Size 1
Subject 1
First Level
C
Contrast Size 2
Subject 2
First Level
C
Contrast Size 3
Subject 3
First Level
C
Contrast Size 4
Subject 4
Multi-Level Analysis
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GG
Higher Level GLM Analysis
=
11111
G
y = X *
Data fromone voxel
Design Matrix(Regressors)
Vector ofRegressionCoefficients(“Betas”)
Obs
erva
tion
s(L
ow-L
evel
Con
tras
ts)
Contrast Matrix:C = [1]Contrast = C* = G
One-Sample Group Mean (OSGM)
24
Two Groups GLM Analysis
=
11100
G1
G2
y = X *
Data fromone voxel
Obs
erva
tion
s(L
ow-L
evel
Con
tras
ts)
00011
25
Contrasts: Two Groups GLM Analysis
1. Does Group 1 by itself differ from 0?C = [1 0], Contrast = C* = G1
= 11100
G1
G2
00011
2. Does Group 2 by itself differ from 0?C = [0 1], Contrast = C* = 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) Concatenation of contrasts #1 and #21 0
0 1C =
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One Group, One Covariate (Age)
=
11111
G
Age
y = X *
Data fromone voxel
Obs
erva
tion
s(L
ow-L
evel
Con
tras
ts)
2133641747
Intercept: G
Slope: Age
Contrast
Age
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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)
= 11111
G
Age
2133641747
2. Does Slope differ from 0?C = [0 1], Contrast = C* = Age
Intercept: G
Slope: Age
Contrast
Age
28
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?
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Two Groups, One (Nuisance) Covariate
Is there a difference between the group means?
Synthetic Data
30
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
Two Groups, One (Nuisance) Covariate
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=
11100
G1
G2
Age
y = X *
Data fromone voxel
Obs
erva
tion
s(L
ow-L
evel
Con
tras
ts)
00011
2133641747
Two Groups, One (Nuisance) Covariate
One regressor for Age.
Different Offset Same Slope (DOSS)
32
=
11100
G1
G2
Age
00011
2133641747
Two Groups, One (Nuisance) Covariate
One regressor for Age indicates that groups have same slope – makes difference between group means/intercepts independent of age.
Different Offset Same Slope (DOSS)
33
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
3. Does Group 1 mean differ from Group 2 mean(after regressing out effect of age)?C = [1 -1 0], Contrast = C* = G1- G2
=
11100
G1
G2
Age
00011
2133641747
34
=
11100
G1
G2
Age
00011
2133641747
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
Contrasts: Two Groups + Covariate
35
• Slope with respect to Age differs between groups• Interaction between Group and Age• Intercept different as well
Group/Covariate Interaction
36
=
11100
G1
G2
Age1
Age2
y = X *
Data fromone voxel
Obs
erva
tion
s(L
ow-L
evel
Con
tras
ts)
00011
213364 0 0
0 0 01747
Group-by-Age Interaction
Different Offset Different Slope (DODS)
Group/Covariate Interaction
37
1. Does Slope differ between groups?Is there an interaction between group and age? C = [0 0 1 -1], Contrast = C* = Age1- Age1
Group/Covariate Interaction
=
11100
G1
G2
Age1
Age2
00011
213364 0 0
0 0 01747
38
Group/Covariate Interaction
=
11100
G1
G2
Age1
Age2
00011
213364 0 0
0 0 01747
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
Very tricky!This tests for difference at Age=0What about Age = 12?What about Age = 20?
39
Group/Covariate InteractionIf 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 slopes3. If there is no difference, re-analyze with single
regressor model (DOSS)4. If there is a difference, proceed with caution
40
Interaction between Condition and GroupExample:• Two First-Level Conditions: Angry and Neutral Faces• Two Groups: Healthy and Schizophrenia
Desired Contrast = (Neutral-Angry)Sch - (Neutral-Angry)Healthy
Two-level approach1. Create First Level Contrast (Neutral-Angry)2. Second Level:
• Create Design with Two Groups• Test for a Group Difference
41
LongitudinalVisit 1
Visit 2
Subject 1 Subject 2 Subject 3 Subject 4 Subject 5
42
Longitudinal
Did something change between visits?• Drug or Behavioral Intervention?• Training?• Disease Progression?• Aging?• Injury?• Scanner Upgrade?
43
Longitudinal
Paired DifferencesBetween Subjects
Subject 1, Visit 1
Subject 1, Visit 2
44
Longitudinal Paired Analysis
=
11111
V
y = X *
Paired Diffsfrom one voxel
Design Matrix(Regressors)
Obs
erva
tion
s(V
1-V
2 D
iffe
renc
es in
Low
-Lev
el C
ontr
asts
)
Contrast Matrix:C = [1]Contrast = C* = V
One-Sample Group Mean (OSGM): Paired t-Test
45
fMRI Analysis Overview
Higher Level GLM
First Level GLM Analysis
First Level GLM Analysis
Subject 3
First Level GLM Analysis
Subject 4
First Level GLM Analysis
Subject 1
Subject 2
CX
CX
CX
CX
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
PreprocessingMC, STC, B0
SmoothingNormalization
Raw Data
Raw Data
Raw Data
Raw Data
CX
46
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
47
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