Basics of fMRI Group Analysis Douglas N. Greve. 2 fMRI Analysis Overview Higher Level GLM First...

Basics of fMRI Group Analysis

Douglas N. Greve

2

fMRI Analysis Overview

Higher Level GLM

First Level GLM Analysis


Subject 3


Subject 4


Subject 1

Subject 2

CX

CX

CX

CX

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

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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

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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

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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

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Is Pattern Repeatable Across Subject?

Subject 1 Subject 2 Subject 3 Subject 4 Subject 5

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Spatial Normalization

Subject 1

Subject 2

Subject 1

Subject 2

MNI305

Native Space MNI305 Space

Affine (12 DOF) Registration

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Group Analysis

Does not have to be all positive!

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GG

1

1

22

2

G

G

G

G

NDOF

N

t

G

G

“Random Effects (RFx)” Analysis

RFx

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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

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“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)

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“Fixed Effects (FFx)” Analysis

FFx

RFx

i

G

i

G

DOFDOF

N

t

G

G

2

22

2

2i

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“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.

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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)

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Two Groups GLM Analysis

=

11100

G1

G2

y = X *

Data fromone voxel

Obs

erva

tion

s(L

ow-L

evel

Con

tras

ts)

00011

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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

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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

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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


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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


One regressor for Age.

Different Offset Same Slope (DOSS)

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=

11100

G1

G2

Age

00011

2133641747


One regressor for Age indicates that groups have same slope – makes difference between group means/intercepts independent of age.

Different Offset Same Slope (DOSS)

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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

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=

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

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• Slope with respect to Age differs between groups• Interaction between Group and Age• Intercept different as well

Group/Covariate Interaction

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=

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)


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1. Does Slope differ between groups?Is there an interaction between group and age? C = [0 0 1 -1], Contrast = C* = Age1- Age1


=

11100

G1

G2

Age1

Age2

00011

213364 0 0

0 0 01747

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=

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?

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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

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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

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LongitudinalVisit 1

Visit 2

Subject 1 Subject 2 Subject 3 Subject 4 Subject 5

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Longitudinal

Did something change between visits?• Drug or Behavioral Intervention?• Training?• Disease Progression?• Aging?• Injury?• Scanner Upgrade?

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Longitudinal

Paired DifferencesBetween Subjects

Subject 1, Visit 1

Subject 1, Visit 2

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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

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fMRI Analysis Overview

Higher Level GLM



Subject 3


Subject 4


Subject 1

Subject 2

CX

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Raw Data

Raw Data

Raw Data

Raw Data

CX

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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

Basics of fMRI Group Analysis Douglas N. Greve. 2 fMRI Analysis Overview Higher Level GLM First...

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