Modelling Neuroimaging Data Using the General Linear Model (GLM ©Karl ) Jesper Andersson KI, Stockholm & BRU, Helsinki

Modelling Neuroimaging Data Using the General

Linear Model (GLM©Karl)

Jesper AnderssonKI, Stockholm & BRU, Helsinki

No escaping this one

MotionCorrection

Smoothing

SpatialNormalisation

General Linear Model

Statistical Parametric Map

fMRI time-seriesParameter Estimates

Design matrix

Anatomical Reference

Overview

• The Experiment

• The Data

• The Model

• The Estimation

• The Inquisition (of the data)

• A Better Model

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The Experiment:fMRI adaptation of classic PET experiment

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Bed

Healthy Volunteer

Screen

• Three Conditions in 21 second epochs• 1st Conditon: Word Generation

Jellyfish


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Bed

Healthy Volunteer

Screen


Noun is presented

Catch

Verb is generated

Burger


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Bed

Healthy Volunteer

Screen


Noun is presented

Fry

Verb is generated

Swim


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Bed

Healthy Volunteer

Screen

• Three Conditions in 21 second epochs• 1st Conditon: Word Generation• 2nd Condition: Word Shadowing

Verb is presented

Swim

Verb is repeated

Strut


Scanner

Bed

Healthy Volunteer

Screen

• Three Conditions in 21 second epochs• 1st Conditon: Word Generation• 2nd Condition: Word Shadowing

Verb is presented

Strut

Verb is repeated

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Scanner

Bed

Healthy Volunteer

Screen

• Three Conditions in 21 second epochs• 1st Conditon: Word Generation• 2nd Condition: Word Shadowing• 3rd Condition: Baseline

Hair-cross is shown

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Scanner

Bed

Healthy Volunteer

Screen

• Three Conditions in 21 second epochs• 1st Conditon: Word Generation• 2nd Condition: Word Shadowing• 3rd Condition: Baseline

Hair-cross is shownzzzz z z z

zzz

The Data:Set of Volumes or Set of Time-series

Volunteer

Time

Serial Snapshots of Volunteers

brain

Generation Shadowing

BaselineTime

Time

• A model consists of a set of assumptions of the type:

• and

The Model:A Set of Hypothetical Time-series

Generation Shadowing Baseline

”I think a voxel that is into generating words might have a time-series looking like this”

”A voxel that is into repeating, like this”

and”A voxel that just

doesn’t care, like this”

The Model:A Set of Hypothetical Time-series

Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙

• For a given voxel (time-series) we try to figure out just what type that is by ”modelling” it as a linear combination of the hypothetical time-series.

Measured ”Known” Unknown ”parameters”

The Estimation:Finding the ”best” parameter values

Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙

• The estimation entails finding the parameter values such that the linear combination ”best” fits the data.

2 3 4 0 1 0 1 0 1 2


Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙


2 3 4 0 1 0 1 0 1 2

0 0 3Not brilliant


Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙


2 3 4 0 1 0 1 0 1 2

1 0 4Neither that


Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙


2 3 4 0 1 0 1 0 1 2

0.83 0.16 2.98Cool!


Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙

• And the nice thing is that the same model fits all the time-series, only with different parameters.

1 2 3 0 1 0 1 0 1 2

0.68 0.82 2.17Into words


Generation S

hadowing B

aseline

≈ β1∙ + β2∙ + β3∙

• And the nice thing is that the same model fits all the time-series, only with different parameters.

1 2 3 0 1 0 1 0 1 2

0.03 0.06 2.04Doesn’t care

The Estimation:The format of data, model and parameters

• Same model for all voxels.

• Different parameters for each voxel.

98.2

16.0

83.0

β

04.2

06.0

03.0

β

17.2

82.0

68.0

β

beta_0001.img

beta_0002.img

beta_0003.img...

...

Time-

serie

s

The model revisited.• And, of course, the way we are used to see

the model is like this.

β1∙ +β2∙ +β3∙≈

3

2

1

β1∙ +β2∙ +β3∙

y ≈ X β

The estimation revisitedWhat do I mean by ”best” fit

98.2

16.0

83.0

β

31.3

0

0

β – Error

– Data– Best fit

– Data– Some fit

– Error

ei

n

iie

1

0

n

iie

1

0

n

iie

1

2 16.17

n

iie

1

2 47.9

Model revisited – again

≈

3

2

1

≈

Remember?

Now, what’s that all about?

3

2

1

= +

y = Xβ+e

e ~ N(0,σ2I)

Observed Known Unknown

We need a model for the error!

Format of data revisitedbeta_0001.img

beta_0002.img

beta_0003.img

...

...

Time-

serie

s

ResMS.img

98.2

16.0

83.0

β

47.91

2

n

iie

But why do we need the error?It is about trust.

But why do we need the error?Would you trust this time-series?

But why do we need the error?Or this (why/why not)?

But why do we need the error?What about this?

But why do we need the error?This?

But why do we need the error?Surely this one?

But why do we need the error?Would you trust these?

β1=1σ=0.2n=60

β1=1σ=0.5n=60

β1=0.3σ=0.2n=60

β1=1σ=0.2n=15

Age: 52Good hair

Scary smile

But why do we need the error?In conclusion:

• We trust long series with large effects and small error.

cXXc

βc1)(

TT

T

t 1)( XXT

βcT

Effect size

Uncertainty of effect size

But why do we need the error?In conclusion:

• We trust: Long series with large effects and small error.

cXXc

βc1)(

TT

T

t 1)( XXT

βcT

beta_0001.img beta_0002.img beta_0003.img

ResMS.img ?

Asking questions of your datat-contrasts

98.2

16.0

83.0

β

04.2

06.0

03.0

β

17.2

82.0

68.0

β

...

...

Time-

serie

s

• Can we find voxels that are active in word-generation taks?


98.2

16.0

83.0

β

04.2

06.0

03.0

β

17.2

82.0

68.0

β

...

...

Time-

serie

s


Hmm, seem to have large

values for β1


**42.6

32.0*41.0

83.0

32.0*41.0

98.2

16.0

83.0

001

t

...

...

Time-

serie

s


44.032.0*19.0

03.0

32.0*19.0

04.2

06.0

03.0

001

t

**41.5

32.0*40.0

68.0

32.0*40.0

17.2

82.0

68.0

001

t

c 001


**16.5

32.0*41.0

67.0

32.0*41.0

98.2

16.0

83.0

011

t

...

...

Time-

serie

s

• Voxels that are more active in generation than shadowing?

58.032.0*19.0

03.0

32.0*19.0

04.2

06.0

03.0

011

t

12.132.0*40.0

14.0

32.0*40.0

17.2

82.0

68.0

011

t

c 011

But why do we need the error?Now we know who to trust!

β1=1σ=0.2n=60

t=6.42

β1=1σ=0.5n=60

t=3.11

β1=0.3σ=0.2n=60

t=2.18

β1=1σ=0.2n=15

t=1.34

Age: 52Good hair

Scary smileJury still out

t-contrasts revisited

c

98.2

16.0

83.0

β

Get data

Fit model Get effect size

Get error

cf.

t ~

(sort of)

=6.42

I’m sorry, can you pose that question differently?

F-contrasts

98.2

16.0

83.0

β

40.3

25.0β

Get data

Fit model

Fit reduced model

Estimate error

Estimate error

2

2

=

=

cf.

F ~(sort of)

=41.21

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But why ask the same question twice?Isn’t that like nagging?

QUIZ: Let us say you wanted to find ares that changed its activity as a result of word-generation or word-shadowing, or both. How would you construct your t-contrast?

001

010

011

?

?

?

Quite arrogant voxel that likes to generate words, but positively disdains

repeating.

Let us try [1 1 0]

c

86.2

71.0

74.0

β

Fit model

Get effect size

Get error

t = 0.10

86.2

71.0

74.0

β

87.2β

Get data

Fit model

Fit reduced model

Estimate error

Estimate error

Where as

010

001

SS=40.5

SS=19.6

F2,57=60.8

F-contrasts made easyA stack of t-contrasts

Is any condition different from the other?

Is there any difference between the two conditions?

0011

Is condition A different from condition B?




0101

Is condition A different from condition C?

0011




1001

Is condition A different from condition D?

0011 0101




0110

Is condition B different from condition C?

0011 0101 1001




0110

Is condition B different from condition C?

0011 0101 1001

Wait a second, do we really need that?




0011 0101 1001

Nah, too messy!




0011 0101 1001

That’s much nicer.




Amplitude of response

Latency of response

Duration of response

This is actually ”our” study.

Promise!




001001

Is the amplitude response different between generation and shadowing?




010010

Is the latency of response different between generation and shadowing?

001001




100100

Is the duration of response different between generation and shadowing?

001001 010010




100100

001001 010010

aka

And now all this is perfectly clear?(And yes, there really was such an

experiment.)

Summary

• A model is used to summarise data in a few (sometimes a big few) parameters that are pertinent to the experiment.

• It consists of a set of hypotheses about how BOLD activity might change as a result of the experiment.

• Specific questions may be asked of the data, via the model, through contrasts.

• Contrasts may be t- or F-contrasts, depending on the nature of the question.

Documents

Modelling Neuroimaging Data Using the General Linear Model (GLM ©Karl ) Jesper Andersson KI, Stockholm & BRU, Helsinki