First Level Analysis (1).ppt

1st Level Analysis Design Matrix, Contrasts & Inference

Cat Sebastian and Nathalie Fontaine University College London

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Outline  What is ‘1st level analysis’?  Design matrix  What are we testing for?  What do all the black lines mean?  What do we need to include?  Contrasts  What are they for?  t and F contrasts  Inferences  How do we do that in SPM5?

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What is 1st level analysis?  1st level analysis: activation is averaged across scans within a subject  2nd level analysis: activation is averaged across subjects (groups can be compared)  What question are we asking?: Which voxels in the brain show a pattern of activation over conditions that is consistent with our hypothesis? 4

The Design Matrix More on this

Time

Not so much on this

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The GLM in fMRI

Time

Y

=

X

x

β

ε

+

Observed data:

Design matrix:

Parameters:

Error:

Y is the BOLD signal at various time points at a single voxel

Several components which explain the observed data, i.e. the BOLD time series for the voxel

The contribution of each component of the design matrix to the value of Y (aim to minimise error)

Difference between the observed data, Y, and that predicted by the model, Xβ .

=



b1

+ b2 6

What is Y? Y is a matrix of BOLD signals



Each column represents a single voxel sampled at successive time points.

Y

Time



Intensity 7

What is X (design matrix)?  The design matrix is simply a mathematical description of your experiment E.g.: ‘visual stimulus on = 1’ ‘visual stimulus off = 0’

 It should contain ‘regressors of interest’, i.e. variables you have experimentally manipulated, and ‘regressors of no interest’ – head movement, block effects. Why?  To minimise the error term, you want to model as much of Y as possible using variables specified in X 8

What should the model look like? Regressors of interest

X=

Baseline Motion

E.g. of a regressor of no interest

Usually 6 motion regressors: 3 translations, 3 rotations

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Regressors of interest  There are different ways to specify variables, e.g.  Conditions:  'dummy' codes identify different levels of experimental factor  e.g. integers 0 or 1: 'off' or 'on'

on

off

off

on

 Covariates:  parametric modulation of independent variable  e.g. task-difficulty 1 to 6 10

Block vs. event related designs

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Modelling the baseline This a column of ‘ones’ modelling the constant, or mean signal (the signal is not zero even without any stimuli or task)

SPM will model this automatically Two eventrelated conditions

Baseline often used as a reference (not the same as baseline fixation)

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From design to a design matrix: an example  Imaging a 2x3 factorial design with factors Modality (Auditory, Visual) and Condition (Concrete, Abstract, Proper) V A C1 C2 C3

C1: Concrete nouns Visual

Auditory

You can model it like this…but is it the best way?

C2: Abstract nouns C3: Proper nouns C1: Concrete nouns C2: Abstract nouns C3: Proper nouns 13

What can we test with this design matrix? V A C1 C2 C3

• We can test for main effects: - Visual > Auditory? - Concrete > Abstract? • But we can’t test for interactions or simple main effects: Visual/concrete > Visual/Abstract? etc The design is not orthogonal… 14

An orthogonal design matrix C1

C1 C1 C2 C2 C3 C3

C2

C3

VAVAVA

V A

Just like in SPSS, you need to cross your variables in order to model interactions

SPM will do this for you automatically if you have a factorial design – just input the factors and the number of levels 15

Ways to improve your model: modelling haemodynamics  The brain does not just switch on and off.

HRF basic function

 Reshape (convolve) regressors to resemble HRF

More on this next week!

Original HRF Convolved 16

To return to the GLM…

Time

Y

=

=

X

x



β

ε

+

b1

+ b2

• We calculate beta values for each regressor in the design matrix • We can then perform contrasts to see which regressors make a significant contribution to the model

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Interim summary: design matrix  We want X to model as much of Y as possible, making the error term small – therefore model everything!  This will ensure that the beta values associated with your regressors of interest are as accurate as possible  Make sure you specify a new regressor for each crossed variable of interest (orthogonality)  Additional complications (basis functions and correlated regressors) will be covered next week  Contrasts can then be performed...over to Nathalie 18

Outline  What is ‘1st level analysis’?  Design matrix  What are we testing for?  What do all the black lines mean?  What factors do we need to include?  Contrasts  What are they for?  t and F contrasts  Inferences  How do we do that in SPM5?

A

[1

B

C

D

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What are they for?  General Linear Model (GLM) characterises relationships between our experimental manipulations and the observed data  Multiple effects all within the same design matrix

 Thus, to focus on a particular characteristic, condition, or regressor we use contrasts 20

What are they for?  A contrast is used by SPM to test hypotheses about the effects defined in the design matrix, using t-tests and Ftests  Contrast specification and the interpretation of the results are entirely dependent on the model specification which in turn depends on the design of the experiment 21

Some general remarks • Clear hypothesis / question • Clear design to answer the research question • The contrasts and inferences made are dependent on choice of experimental design • Most of the problems concerning contrast specification come from poor design specification • Poor design: • Unclear about what the objective is • Try to answer too many questions in a single model

We need to think about how the experiment is going to be modelled and which comparisons we wish to make BEFORE acquiring the data 22

Contrasts E.g.: Contrasts with conditions:  The conditions that we are interested in can take on a positive value, such as 1

 The conditions that we want to subtract from these conditions of interest can take on a negative value, such as -1

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Contrasts Condition 1: Language task Condition 2: Memory task Condition 3: Motor task Condition 4: Control

   

Contrast 1: Language minus Control: 1 0 0 -1 Contrast 2: Motor minus Memory: 0 -1 1 0 Contrast 3: Control minus Motor: 0 0 -1 1 Contrast 4: (Language + Memory) minus Control: 1 1 0 -2  This contrast will measure areas of the brain that have significantly increased activity in the average of the language and memory conditions, compared with the control condition – another way of looking at this contrast is the sum of the individual condition contrasts of 1 0 0 -1 and 0 1 0 -1.

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Contrasts - Factorial design 

SIMPLE MAIN EFFECT  A–B  Simple main effect of motion (vs. no motion) in the context of low load  [ 1 -1

0

0]

LOW A



MOTION

B

C

D

A

B

C

D

LOAD

MAIN EFFECT  (A + B) – (C + D)

HIGH

 The main effect of low load (vs. high load) irrelevant of motion   Main effect of load  [ 1 1 -1 -1] A



NO MOTION

B

C

D

INTERACTION  (A - B) – (C - D)  The interaction effect of motion (vs. no motion) greater under low (vs. high) load  [ 1 -1 -1 1]

A

B C

D

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Contrasts  t-test: is there a significant increase or is there a significant decrease in a specific contrast (between conditions) – directional  F-test: is there a significant difference between conditions in the contrast – non-directional

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Example  Two event-related conditions  The subjects press a button with either their left or right hand depending on a visual instruction (involving some attention)  We are interested in finding the brain regions that respond more to left than right motor movement

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t-contrasts Left

Right

Mean

 t-contrasts are directional  To find the brain regions corresponding more to left than right motor responses we use the contrast: T = [1 -1 0]

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t-contrasts  A one dimensional contrast contrast of estimated parameters t=

c’b t=

variance estimate

s2c’(X’X)+c

So, for a contrast in our model of 1 -1 0: t = (ß1x1 + ß2x-1 + ß3x0) Estimated variance 29

Brain activation: Left motor responses This shows activation of the contralateral motor cortex, ipsilateral cerebellum, etc.

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F-contrasts Left

Right Mean

 F-contrasts are non-directional  To test for the overall difference (positive or negative) from the left and right responses we use: [1 0 0; 0 1 0]

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F-test  To test a hypothesis about general effects, independent of the direction of the contrast  A collection of t-contrasts that you want to test together Additional variance accounted for by tested effects F=

Error variance estimate 32

Brain activation Areas involved in the overall difference (positive or negative) from the left and right responses (non-directional)

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Test

Design and contrast

SPM(t) or SPM(F)

[1 -1 0]

t-test

[1 0 0; 0 1 0]

F-test

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Inferences about subjects and populations  Inference about the effect in relation to:  The within-subject variability (1st level analysis)  The between subject variability (2nd level analysis)

 This distinction relates directly to the difference between fixed and random-effect analyses  Inferences based on fixed effects analyses are about the particular subject(s) studied  Random-effects analyses are usually more conservative but allow the inference to be generalized to the population from which the subjects were selected More on this in few weeks! 35

One voxel = One test (t, F) amplitude General Linear Model fitting statistical image

Statistical image (SPM) Temporal series fMRI

voxel time course

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Choosing a statistical threshold  Important consideration in neuroimaging = the tremendous number of statistical tests computed for each comparison  E.g.: if 100,000 voxels are tested at a probability threshold of 5%, we should expect:  5000 voxels will incorrectly appear as significant activations = Apparent activations by chance; FALSE POSITIVE

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Choosing a statistical threshold  Uncorrected threshold of p < .001

 Familywise Error (FWE)  Bonferroni correction  E.g.: .05/100,000 = .0000005

 False Discovery Rate (FDR)  Adjusts the criterion used based on the amount of signal present in the data

 Reduce the number of comparisons  E.g.: Instead of examining the entire brain, examine just a small region

IMPORTANCE of taking into account the multiple comparisons across voxels BUT also the multiple comparisons across contrasts (i.e., the number of contrasts tested) 38

How do we do that in SPM5?

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Summary  Contrasts are statistical (t or F) tests of specific hypotheses  t-contrast looks for a significant increase or decrease in a specific contrast (directional)  F-contrast looks for a significant difference between conditions in the contrast (non-directional)  Importance of having a clear design  Inferences about subjects (1st level) and populations (2nd level)  Importance of considering the multiple comparisons 40

References  Human Brain Function 2, in particular Chapter 8 by Poline, Kherif, & Penny (http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch8.pdf)  Introduction: Experimental design and statistical parametric mapping, by Friston  Linear Models and Contrasts, PowerPoint presentation by Poline (April, 2005), SPM short course at Yale  Previous years’ slides  CBU Imaging Wiki (http://imaging.mrccbu.cam.ac.uk/imaging/PrinciplesStatistics) (http://imaging.mrccbu.cam.ac.uk/imaging/SpmContrasts)  SPM5 Manual, The FIL Methods Group (2007)  An introduction to functional MRI by de Haan & Rorden 41

Thank you!

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