general linear model and classical inference...−40 −20 0 20 40 60 80 parameter estimates general...

General linear model and classical inference

SPM for M/EEGMay 2019

Martin DietzCFIN, Aarhus University, DK

Introduction- Statistical parametric maps of MEG/EEG data- ERP example

General linear model (GLM)- Definition & design matrix- Parameter estimation

Classical inference- Contrasts and inference- Correlated regressors

Overview

Design matrix

Overview of SPM for M/EEG

Preprocessing

FilteringDown-samplingEpochingRe-referencingArtefact rejectionAveraging…

Image conversion

Design matrix

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Parameter estimates

General linear model

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mV

Evoked ResponsesA

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Sensor mapsB

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148 ms 128 ms

mV

SPMmip[4.25, 72, �100]

<<<

SPM{T 1

1}

negative

SPMresults:

./ee

g/no

rmal

/rfx

/rfx

_lef

tHe

ight

thr

esho

ld T

= 4

.024

701

{p<

0.00

1 (u

nc.)

}Ex

tent

thr

esho

ld k

= 1

000

voxe

ls

Desi

gn m

atri

x0.

51

1.5

2 4 6 8 10 12

cont

rast

(s)

2

00.

10.

20.

30.

40.

50.

60.

70.

80.

91

0500050005000St

atis

tics

: p

�val

ues

adju

sted

for

sea

rch

volu

mese

t�le

vel

pc

clus

ter�

leve

lpe

ak�l

evel

pFW

E�co

rrqFD

R�co

rrT

(Z�)

pun

corr

mm m

m ms

table shows 3 local maxima more than 8.0mm apart

Heig

ht t

hres

hold

: T

= 4.

02,

p =

0.00

1 (0

.875

)Ex

tent

thr

esho

ld:

k =

1000

vox

els,

p =

0.0

48 (

0.09

4)Ex

pect

ed v

oxel

s pe

r cl

uste

r, <

k> =

250

.388

Expe

cted

num

ber

of c

lust

ers,

<c>

= 0

.10

FWEp

: 7.

833,

FDR

p: I

nf,

FWEc

: 17

93,

FDRc

: In

f

Degr

ees

of f

reed

om =

[1.

0, 1

1.0]

FWHM

= 4

5.6

64.1

34.

0 mm

mm

ms;

21.5

23.

8 Vo

lume

: 83

8740

3 =

3671

64 v

oxel

s =

80.3

re

Voxe

l si

ze:

2.1

2.7

4.0

mm m

m ms

; (r

esel

0.02

30.

056

1793

0.01

10.

156

0.45

1

6.50

4.0

80.

000

17

�6

148

Statistical t-mapsC

SPMmip[59.5, 72, �100]

<<<

SPM{T 9}

negative

SPMresults:

./ee

g/no

rmal

/RFX

/rfx

_rig

htHe

ight

thr

esho

ld T

= 4

.296

806

{p<

0.00

1 (u

nc.)

}Ex

tent

thr

esho

ld k

= 5

00 v

oxel

s

Desi

gn m

atri

x0.

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rast

(s)

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00.

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91

0500050005000St

atis

tics

: p

�val

ues

adju

sted

for

sea

rch

volu

mese

t�le

vel

pc

clus

ter�

leve

lpFWE�corrqFDR�corrkE

puncorr

peak�l

evel

pFWE�corrqFDR�corrT

(Z�)

puncorr

mm mm ms


Height threshold: T = 4.30, p = 0.001 (0.902)

Extent threshold: k = 500 voxels, p = 0.125 (0.252)

Expected voxels per cluster, <k> = 221.208

Expected number of clusters, <c> = 0.29

FWEp: 9.681, FDRp: Inf, FWEc: 2317, FDRc: 2317

Degrees of freedom = [1.0, 9.0]

FWHM = 46.4 67.1 33.7 mm mm ms; 21.8 25.0 8.4

Volume: 8387403 = 367164 voxels = 75.9 resels

Voxel size: 2.1 2.7 4.0 mm mm ms; (resel = 45

0.0352

0.007

0.028

2317

0.003

0.046

0.200 9.85

4.61

0.000

15 �

3 128

0.098

0.221 8.50

4.35

0.000

�21 �3 108

0.101

0.205

903

0.046

0.685

0.836 5.14

3.43

0.000

�17 26 392

148 ms 128 ms

SPMmip

[4.2

5, 7

2, �

100]

<

< <

SPM{T11}

negative

SPMresults:./eeg/normal/rfx/rfx_leftHeight threshold T = 4.024701 {p<0.001 (unc.)}Extent threshold k = 1000 voxels

Design matrix0.5 1 1.5

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00

Statistics: p−values adjusted for search volumeset−levelp c

cluster−level peak−levelpFWE−corr

qFDR−corr

T (Z�) p

uncorrmm mm ms

table shows 3 local maxima more than 8.0mm apartHeight threshold: T = 4.02, p = 0.001 (0.875)Extent threshold: k = 1000 voxels, p = 0.048 (0.094)Expected voxels per cluster, <k> = 250.388Expected number of clusters, <c> = 0.10FWEp: 7.833, FDRp: Inf, FWEc: 1793, FDRc: Inf

Degrees of freedom = [1.0, 11.0]FWHM = 45.6 64.1 34.0 mm mm ms; 21.5 23.8 Volume: 8387403 = 367164 voxels = 80.3 reVoxel size: 2.1 2.7 4.0 mm mm ms; (resel

0.023 0.056 1793 0.011 0.156 0.451 6.50 4.08 0.000 17 −6 148

SPMmip

[4.2

5, 7

2, �

100]

<

< <

SPM{T11}

negative



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10

12

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2

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50

00



qFDR−corr

T (Z�) p

uncorrmm mm ms



0.023 0.056 1793 0.011 0.156 0.451 6.50 4.08 0.000 17 −6 148

Statistical Parametric Map

SPM{T138}

Contrast

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Inference andcorrection for multiple

comparisons

Statistical parametric maps

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mm

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mm

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time

frequency

3D M/EEG sourcereconstruction

2D time-frequency

2D+t scalp-time

1D time

time

SPMmip

[-10,

-92,

0]

<

< <

SPM{T11}

Positive

SPMresults:./au173801/projects/resultsHeight threshold T = 6.633283 {p<0.05 (FWE)}Extent threshold k = 2000 voxels

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Event-related potential (ERP) example

• Random presentation of ‘faces’ and ‘scrambled faces’

• 70 trials of each type

• 128 EEG channels

Is there a difference between the ERP of ‘faces’ and ‘scrambled faces’ ?

Question:

ERP example: ones channel

÷øöç

èæ +

-=

sf

2

sf

11nn

ts

µµ!

Compare size of effectto its standard error

(Signal-to-noise ratio)

Average

Trial-by-trial variability




Overview

Y = Xβ + ϵ Y = Xβ + ϵ

Data modelling

Faces Scrambled

EEG data

Y = Xβ + ϵx1β1<latexit sha1_base64="jJnS3ldCHcB8defuaF3EQ+LtCoE=">AAACg3icbVHdShtBFJ6s2mr6F+2lpQxNC5aWsJsK9kYU9KIXRWwxKiRpODs5iYPzs8ycrQlL3qAP0Nv6Vj5NO5tYaJIeOPDxfef/pJmSnuL4rhKtrK49eLi+UX30+MnTZ7XNrXNvcyewJayy7jIFj0oabJEkhZeZQ9Cpwov0+qjUL76j89KaMxpn2NUwNHIgBVCgvo16RTLppEhQgl6tHjfiqfFlkNyD+sHvF/pl7cfBaW+z0uv0rcg1GhIKvG8ncUbdAhxJoXBS7eQeMxDXMMR2gAY0+m4xHXvC3wSmzwfWBTfEp+y/GQVo78c6DZEa6MovaiX5P62d0+Bjt5AmywmNmDUa5IqT5eUNeF86FKTGAYBwMszKxRU4EBQuNddlWjtDMbdJMcqNFLaPC6yiETkIpEfSIE25VXEmw8r8BG/4V6vB/FVD2VLeOZZDSf795/AO83Y5uBpekiw+YBmcNxvJh0bzS1w/fM1mts622Su2wxK2xw7ZJ3bKWkwwx36yX+w2WoveRc1odxYaVe5znrM5i/b/AFkyyOs=</latexit>

x2β2<latexit sha1_base64="wtBhuZGF27XT7rl2JJwWQkm2iCs=">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</latexit>

Y = Xβ + ϵ

Y = Xβ + ϵ Y = Xβ + ϵ

‘Faces’ ‘Scrambled’ Residual error

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Y = Xβ + ϵ

Data

Y = Xβ + ϵY = Xβ + ϵY = Xβ + ϵ

b2

b1

Design matrix Parameters Error

X β

Y = Xβ + ϵ

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Y<latexit sha1_base64="pBKK5MpQWVGszNpbsJHrq+1IcwM=">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</latexit>


Y

N

1

+ e

N

1

=

N

p

b

1

p

X

The design matrix X embodies all knowledge about experimentally controlled factors and confounds

Y = Xβ + ϵ

ϵ ∼ N (0,σ2I)

GLM defined by


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Y = Xβ + ϵ

Matrix form

⎡

⎢⎢⎢⎢⎢⎢⎣

y1...yn...yN

⎤

⎥⎥⎥⎥⎥⎥⎦=

⎡

⎢⎢⎢⎢⎢⎢⎣

x11 . . . x1p . . . x1P...

. . ....

. . ....

xn1 . . . xnp . . . xnP...

. . ....

. . ....

xN1 . . . xNp . . . xNP

⎤

⎥⎥⎥⎥⎥⎥⎦

⎡

⎢⎢⎢⎢⎢⎢⎣

β1...βp...βP

⎤

⎥⎥⎥⎥⎥⎥⎦+

⎡

⎢⎢⎢⎢⎢⎢⎣

ϵ1...ϵn...ϵN

⎤

⎥⎥⎥⎥⎥⎥⎦

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Linear regression


Y = Xβ + ϵSpecial cases of the GLM

• t-test

• F-test

• Analysis of variance (ANOVA)

• Analysis of covariance (AnCova)

• multiple regression

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Y = Xβ + ϵY = Xβ + ϵ

X βY = Xβ + ϵY = Xβ + ϵ

Voxel-wise general linear model

2D pixel, 3D space-time or 3D voxel

2D or 3D imageperi-stimulus time (ms)

mV

Evoked ResponsesA

−100 −50 0 50 100 150 200 250 300 350 400−1.5

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0.5

1

1.5

2

peri-stimulus time (ms)−100 −50 0 50 100 150 200 250 300 350 400

−1.5

−1

−0.5

0

0.5

1

1.5

2

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Sensor mapsB

−0.4

−0.2

0

0.2

0.4

0.6

0.8

−0.4

−0.2

0

0.2

0.4

0.6

0.8

148 ms 128 ms

mV

SPMmip[4.25, 72, �100]

<<<

SPM{T 1

1}

negative

SPMresults:

./ee

g/no

rmal

/rfx

/rfx

_lef

tHe

ight

thr

esho

ld T

= 4

.024

701

{p<

0.00

1 (u

nc.)

}Ex

tent

thr

esho

ld k

= 1

000

voxe

ls

Desi

gn m

atri

x0.

51

1.5

2 4 6 8 10 12

cont

rast

(s)

20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10500050005000Statistics: p

�val

ues

adju

sted

for

sea

rch

volu

mese

t�le

vel

pc

clus

ter�

leve

lpe

ak�l

evel

pFW

E�co

rrqFD

R�co

rrT

(Z�)

pun

corr

mm m

m ms


Heig

ht t

hres

hold

: T

= 4.

02,

p =

0.00

1 (0

.875

)Ex

tent

thr

esho

ld:

k =

1000

vox

els,

p =

0.0

48 (

0.09

4)Ex

pect

ed v

oxel

s pe

r cl

uste

r, <

k> =

250

.388

Expe

cted

num

ber

of c

lust

ers,

<c>

= 0

.10

FWEp

: 7.

833,

FDR

p: I

nf,

FWEc

: 17

93,

FDRc

: In

f

Degr

ees

of f

reed

om =

[1.

0, 1

1.0]

FWHM

= 4

5.6

64.1

34.

0 mm

mm

ms;

21.5

23.

8 Vo

lume

: 83

8740

3 =

3671

64 v

oxel

s =

80.3

re

Voxe

l si

ze:

2.1

2.7

4.0

mm m

m ms

; (r

esel

0.023

0.056

1793

0.011

0.156

0.451 6.50

4.08

0.000

17 �

6 148

Statistical t-mapsCSPMmip

[59.5, 72, �100]<<

<

SPM{

T 9}

nega

tive

SPMr

esul

ts:

./ee

g/no

rmal

/RFX

/rfx

_rig

htHe

ight

thr

esho

ld T

= 4

.296

806

{p<

0.00

1 (u

nc.)

}Ex

tent

thr

esho

ld k

= 5

00 v

oxel

s

Desi

gn m

atri

x0.

51

1.5

1 2 3 4 5 6 7 8 9 10

cont

rast

(s)

2

00.

10.

20.

30.

40.

50.

60.

70.

80.

91

0500050005000St

atis

tics

: p

�val

ues

adju

sted

for

sea

rch

volu

mese

t�le

vel

pc

clus

ter�

leve

lpFWE�

corrqFD

R�co

rrkE

puncorr

peak�l

evel

pFW

E�co

rrqFD

R�corrT

(Z�)

pun

corr

mm m

m ms


Heig

ht t

hres

hold

: T

= 4.

30,

p =

0.00

1 (0

.902

)Ex

tent

thr

esho

ld:

k =

500

voxe

ls,

p =

0.12

5 (0

.252

)Ex

pect

ed v

oxel

s pe

r cl

uste

r, <

k> =

221

.208

Expe

cted

num

ber

of c

lust

ers,

<c>

= 0

.29

FWEp

: 9.

681,

FDR

p: I

nf,

FWEc

: 23

17,

FDRc

: 23

17

Degr

ees

of f

reed

om =

[1.

0, 9

.0]

FWHM

= 4

6.4

67.1

33.

7 mm

mm

ms;

21.8

25.

0 8.

4 Vo

lume

: 83

8740

3 =

3671

64 v

oxel

s =

75.9

res

els

Voxe

l si

ze:

2.1

2.7

4.0

mm m

m ms

; (r

esel

= 4

5

0.03

52

0.00

70.

028

2317

0.00

30.

046

0.20

0

9.85

4.6

10.

000

15

�3

128

0.09

80.

221 8.

50 4

.35

0.00

0�2

1 �3

108

0.10

10.

205

903

0.04

60.

685

0.83

6

5.14

3.4

30.

000

�17

26

392

148 ms 128 ms

mm

mm

mm

mm

time

mm

time

frequency

3D M/EEG sourcereconstruction

2D time-frequency

2D+t scalp-time

1D time

time

SPMmip

[-10,

-92,

0]

<

< <

SPM{T11}

Positive

SPMresults:./au173801/projects/resultsHeight threshold T = 6.633283 {p<0.05 (FWE)}Extent threshold k = 2000 voxels

0.5 1 1.5

Design matrix

2

4

6

8

10

12

contrast

1

0

5

10

15

Parameter estimation

Y = Xβ + ϵY = Xβ + ϵ

y<latexit sha1_base64="/n7yg9GgrQUJNKKY+N6EmZTJ91g=">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</latexit>

X<latexit sha1_base64="NTmkPv8m/1/fjshoBDv/ETceWHs=">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</latexit>

β<latexit sha1_base64="4pizohhCa6xPp+Wavvj8IlbLaTI=">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</latexit>

ϵ<latexit sha1_base64="h/33dmTUGc6IegvVo8crjFsz0c0=">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</latexit>

ϵ ∼ N (0,σ2I)

Iff residual error is i.i.d.

Ordinary least squares

� = (X�X)�1X�y

ϵ = y− Xβ<latexit sha1_base64="jlFgDurkLkjBSlEufV/d2ysgXWo=">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</latexit>

!

iϵ2i

<latexit sha1_base64="v7BZobSndOha9WuAadY+aLrYvWs=">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</latexit>

minimise w.r.t β<latexit sha1_base64="Xyr+EIOv7EhXef+tjEUWI1ZZh/8=">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</latexit>

Parameter estimation: a geometric perspective

Ordinary least squares

Projection matrix

� = (X�X)�1X�y

x1<latexit sha1_base64="QcF302FvcDIsKZz4gwdwE0O0a/4=">AAACenicbVFdaxNBFJ1s1cZYbVMfRRiaFlpawm4r6JsFfRARqdK0hTSEu5ObZOh8LDN3bcKS3+Cr/jR/gz+gLwWdTSqYpBcGDufcrzM3zZT0FMe/KtHKg4ePVquPa0/Wnj5b36hvnnmbO4EtYZV1Fyl4VNJgiyQpvMgcgk4VnqdX70r9/Bs6L605pXGGHQ0DI/tSAAWqNeoWyaS70Yib8TT4MkjuQOPtzZ/fqx+vX55065XuZc+KXKMhocD7dhJn1CnAkRQKJ7XL3GMG4goG2A7QgEbfKabbTvhOYHq8b114hviU/b+iAO39WKchUwMN/aJWkvdp7Zz6bzqFNFlOaMRsUD9XnCwvrfOedChIjQMA4WTYlYshOBAUPmhuyrR3hmLOSTHKjRS2hwusohE5CKRH0iBN6ao4lcEy/4zX/KvVYP6poW0p776XA0n+4FO4gtlbTq6FkySLB1gGZ4fN5Kh5+CVuHG+zWVTZC7bFdlnCXrNj9oGdsBYTTLLv7Af7WbmNtqK9aH+WGlXuap6zuYhe/QU6psdV</latexit>

x2<latexit sha1_base64="Mq57byvxVfxQpS0H5ckwtwN20C0=">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</latexit>

y<latexit sha1_base64="/n7yg9GgrQUJNKKY+N6EmZTJ91g=">AAACdnicbVFdSxtBFJ2s1dq02mgfFRmaigol7NoHfVOoD32QomJUiCHcndzEwflYZu6qYfEXtI/64/pP+tbOJgom8cLA4Zz7deammZKe4vhPJZp5Mzv3dv5d9f2HhcWPtaXlM29zJ7AprLLuIgWPShpskiSFF5lD0KnC8/T6e6mf36Dz0ppTGmTY1tA3sicFUKCOB51aPW7Ew+DTIHkC9b1/q3qt9nvvqLNU6Vx2rcg1GhIKvG8lcUbtAhxJofC+epl7zEBcQx9bARrQ6NvFcNN7vh6YLu9ZF54hPmRfVhSgvR/oNGRqoCs/qZXka1orp95uu5AmywmNGA3q5YqT5aVt3pUOBalBACCcDLtycQUOBIXPGZsy7J2hGHNS3OVGCtvFCVbRHTkIpEfSIE3pqjiVwTL/ibf8xGowz2poW8qbB7IvyX89DBcwW9PJ1XCSZPIA0+Bsu5F8a2wfx/X9L2wU82yFfWabLGE7bJ/9YEesyQRD9os9sMfK32gtWo82RqlR5anmExuLKP4PC1bEYQ==</latexit>

y = Xβ<latexit sha1_base64="xt/XtxYeivoVeu0dJ1Fj2y1phX0=">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</latexit>

ϵ<latexit sha1_base64="h/33dmTUGc6IegvVo8crjFsz0c0=">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</latexit>

Design space of X

Residual forming matrix

ϵ = Ry<latexit sha1_base64="qzfBtFSfZBTTlVR2+JLolu8vSeA=">AAACgnicbVFdTxNBFJ0uIFhRCz5qyMRigsY0u/AADxhI9MEHYpBQIGk3zd3pbZkwH8vMXaTZ8A9891V/lr9GZ1tMbMtNbnJyzv2+Wa6kpzj+XYsWFpceLa88rj9ZffrseWNt/czbwglsC6usu8jAo5IG2yRJ4UXuEHSm8Dy7+ljp5zfovLTmlEY5phqGRg6kAApU2sXcS2UN/8BPRr1GM27FY+PzILkHzYM/r/RG4/vBcW+t1uv2rSg0GhIKvO8kcU5pCY6kUHhX7xYecxBXMMROgAY0+rQcT33H3wSmzwfWBTfEx+z/GSVo70c6C5Ea6NLPahX5kNYpaLCXltLkBaERk0aDQnGyvDoB70uHgtQoABBOhlm5uAQHgsKhprqMa+copjYpbwsjhe3jDKvolhwE0iNpkKbaqjyVYWX+Bb/xE6vB/FND2Ure+iSHkvz7o/AN83Y+uB5eksw+YB6cbbeSndb217h5uMkmtsJestdsiyVslx2yz+yYtZlg1+wH+8l+RYvRuyiJdiahUe0+5wWbsmj/L1lhx/8=</latexit>

R = I � P<latexit sha1_base64="KyeGmzzWKn8qQDTq3UNcPPO+l3Y=">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</latexit>

y = Py<latexit sha1_base64="fkJ7AzlirczpaLtZybvM/JRusak=">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</latexit>

P = X(X�X)�1X�<latexit sha1_base64="dtTPOaju8brr7d4AuzFgCUL6h/k=">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</latexit>

Channels to voxels transformation

Spatio-temporalinterpolation

3D image (Nifti)M/EEG channels

time

Mass-univariate statistical framework

mm

mm

time

mm

mm

time

−100 −50 0 50 100 150 200 250 300 350 400

−80

−60

−40

−20

0

20

40

60

80

time

Avoid selection bias (“cherry picking”)

Mass-univariate statistical framework




Overview

Hypothesis testing in classical inference

T-10 -5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

p(T ≥ t|H0)

The null hypothesis H0

No effect under null distribution

Alternative hypothesis HA

Unlikely or surprising effect under null distributions

Contrast and t-test

c⊤ = [1 − 1]<latexit sha1_base64="Zb+hqq6q+yMyp72Cua7VGuBB+iQ=">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</latexit>

SPMmip

[4.2

5, 7

2, �

100]

<

< <

SPM{T11}

negative



2

4

6

8

10

12

contrast(s)

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

00

50

00

50

00



qFDR−corr

T (Z�) p

uncorrmm mm ms



0.023 0.056 1793 0.011 0.156 0.451 6.50 4.08 0.000 17 −6 148

SPMmip

[4.2

5, 7

2, �

100]

<

< <

SPM{T11}

negative



2

4

6

8

10

12

contrast(s)

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

00

50

00

50

00



qFDR−corr

T (Z�) p

uncorrmm mm ms



0.023 0.056 1793 0.011 0.156 0.451 6.50 4.08 0.000 17 −6 148

SPM{T138}

t =c��

�2(c�(X�X)�1c)

SPM-t over time-spaceContrast vector Linear combination of parameters

β1 > β2<latexit sha1_base64="wORoxTASs+fn5tTvAY5zTuFMF3c=">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</latexit>

(1× β1) + (−1× β2) > 0<latexit sha1_base64="MgPEMdKJiKe9En3e1V1LFtStBFI=">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</latexit>

c⊤β > 0<latexit sha1_base64="aT6LcCx6O9pJ71wLNXgxi8PgMKE=">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</latexit>

t-contrast

Classical inference

Constrast

t statistic

Analytic T null distribution

c = [1� 1]�

t =c��

�2(c�(X�X)�1c)

p value

p(T ≥ t|H0)

T-10 -5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

p(T ≥ t|H0)

False-positive rate

α = p(T ≥ uα|H0)

Hight threshold

-10 -5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18uα

T

u� = T � �

� = 0.05

Classical inference

Analytic T null distribution

t-test summary

ContrastLinear combination of parameters

t-testSignal-to-noise ratio measure

t-statisticNo dependence on scaling of regressors or contrast

Unilateral test

c⊤β = 0<latexit sha1_base64="GAvoCzSyMZqqj/Xfocn92ZcP2P8=">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</latexit>

c⊤β > 0<latexit sha1_base64="aT6LcCx6O9pJ71wLNXgxi8PgMKE=">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</latexit>

H0: HA:

-10 -5 0 5 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

t =contrast√variance

<latexit sha1_base64="0Tnh0Pot27PK3atwHOOk5Df65mA=">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</latexit>

F-test and extra sum-of-squares

RSSRSSRSSF -

µ 0

21 ,~ nnFRSS

ESSF µ

Test statistic: ratio of explained and unexplained

variability (error)

n1 = rank(X) – rank(X0)n2 = N – rank(X)

RSSå 2ˆ fulle

RSS0å 2ˆreducede

Full model ?

X1X0

Or reduced model?

X0

RSSRSSRSSF -

µ 0

21 ,~ nnFRSS

ESSF µ


variability (error)


RSSå 2ˆ fulle

RSS0å 2ˆreducede

Full model ?

X1X0

Or reduced model?

X0Full model Reduced model

RSSRSSRSSF -

µ 0

21 ,~ nnFRSS

ESSF µ


variability (error)


RSSå 2ˆ fulle

RSS0å 2ˆreducede

Full model ?

X1X0

Or reduced model?

X0 Ratio of explained andunexplained variance

F-statistic

X0<latexit sha1_base64="DPJtx/fozWJ5gaBx1SnyBuHVKTw=">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</latexit>


X1<latexit sha1_base64="vIfLteJ0ClNGP95Gp71ivuOP5C0=">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</latexit>

F-test and multi-dimensional contrasts

H0: True model is X0

Full or reduced model?

X1 (b3-4) X0 X00 0 1 0 0 0 0 1

cT =

H0: b3 = b4 = 0 test H0 : cTb = 0 ?β3 = β3 = 0

<latexit sha1_base64="jtwWFdGgcNJ6bVlyO5/X5NtPrT0=">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</latexit>

H0: c⊤β > 0<latexit sha1_base64="aT6LcCx6O9pJ71wLNXgxi8PgMKE=">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</latexit>

HA:



X1<latexit sha1_base64="vIfLteJ0ClNGP95Gp71ivuOP5C0=">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</latexit>

c⊤ =<latexit sha1_base64="w9LQrCkPUsEUTW7AvKqghj97L30=">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</latexit>

0 0 1 00 0 0 1

F-test summary

F-test

Additional variance explained by full modelw.r.t. a reduced (nested) model

Uni-dimensional contrast

Testing b1 – b2 is the same as testing b2 – b1 It is exactly the square of the t-test, testing for both positive and negative effects

Multi-dimensional contrast

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0000010000100001 0 : Hypothesis Null 3210 === bbbH

0 oneleast at : Hypothesis eAlternativ ¹kAH búúúú

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RSSRSSRSSF -

µ 0

21 ,~ nnFRSS

ESSF µ


variability (error)


RSSå 2ˆ fulle

RSS0å 2ˆreducede

Full model ?

X1X0

Or reduced model?

X0

Correlated and orthogonal regressors

x1x2x2*

x2 orthogonalized w.r.t. x1Þ only the parameter estimate

for x1 changes, not that for x2!

Correlated regressors Þ explained variance

shared between regressors

121

2211

==++=

bbbb exxy

1;1 *21

*2

*211

=>

++=

bb

bb exxy

Correlated regressors

Explained variance shared between regressors

Orthogonalised regressors

Parameter estimate for x1 changes, not for x2

Orthogonal regressors

Variability explained by x1 Variability explained by x2


Varia

bilit

y ex

plai

ned

by x

1 Variability explained by x2



Testing for x1

Varia

bilit

y ex

plai

ned

by x




Testing for x2

Varia

bilit

y ex

plai

ned

by x



Summary

Y = Xβ + ϵSpecial cases of the GLM

• t-test

• F-test

• Analysis of variance (ANOVA)

• Analysis of covariance (AnCova)

• multiple regression

Thank you!

Any questions ?

Thanks to Christophe, Guillaume, Will, Rik, Stefan and Karl!

general linear model and classical inference...−40 −20 0 20 40 60 80 parameter estimates general...

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