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1st Level Analysis1st Level AnalysisDesign Matrix, Contrasts & InferenceDesign Matrix, Contrasts & Inference

Cat Sebastian and Nathalie FontaineCat Sebastian and Nathalie FontaineUniversity College LondonUniversity College London

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OutlineOutline 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?

A B C D

[1 -1 -1 1]

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What is 1st level analysis?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?

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The Design MatrixThe Design Matrix

Not so much on this

More on this

Time

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The GLM in fMRIThe GLM in fMRI Y = X x β + ε

Observed data:

Y is the BOLD signal at various time points at a

single voxel

Design matrix:

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

for the voxel

Parameters:

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

Error:

Difference between the observed data,

Y, and that predicted by the model, Xβ.

= +Tim

e

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What is Y?What is Y?

IntensityT

ime

Y Y is a matrix of BOLD signals

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

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What is X (design matrix)?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

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What should the model look like?What should the model look like?

Regressors of interest Motion

X =

Baseline

Usually 6 motion regressors: 3 translations, 3

rotations

E.g. of a regressor of no interest

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Regressors of interestRegressors 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'

Covariates: parametric modulation of independent

variable e.g. task-difficulty 1 to 6

on off

off on

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Block vs. event related designsBlock vs. event related designs

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Modelling the baselineModelling the baseline

Two event-related

conditions

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

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

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From design to a design matrix: an exampleFrom design to a design matrix: an example

Imaging a 2x3 factorial design with factors Modality (Auditory, Visual) and Condition (Concrete, Abstract, Proper)

Visual

Auditory

C1: Concrete nouns

C2: Abstract nouns

C3: Proper nouns

C2: Abstract nouns

C3: Proper nouns

C1: Concrete nouns

V A C1 C2 C3You can model it like this…but is it

the best way?

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What can we test with this design matrix?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…

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An orthogonal design matrixAn orthogonal design matrix

V A V A V AC1 C1 C2 C2 C3 C3

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

V

A

C1 C2 C3

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Ways to improve your model: modelling Ways to improve your model: modelling haemodynamicshaemodynamics

The brain does not just switch on and off.

Reshape (convolve) regressors to resemble HRF

HRF basic function

Original

HRF Convolved

More on this next

week!

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To return to the GLM…To return to the GLM… Y = X x β + ε

= +Tim

e

• 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 matrixInterim 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

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OutlineOutline 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 B C D

[1 -1 -1 1]

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What are they for?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

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What are they for?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 F-tests

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

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Some general remarksSome 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 BEFOREBEFORE acquiring the data

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ContrastsContrasts

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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ContrastsContrastsCondition 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 designContrasts - Factorial design

A B

C D

LOW

LOAD

HIGH

MOTION NO MOTION

SIMPLE MAIN EFFECT A – B Simple main effect of motion (vs. no motion) in

the context of low load [ 1 -1 0 0]

MAIN EFFECT (A + B) – (C + D) The main effect of low load (vs. high load)

irrelevant of motion Main effect of load [ 1 1 -1 -1]

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

A B C D

A B C D

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ContrastsContrasts

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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ExampleExample

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-contrastst-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-contrastst-contrasts A one dimensional contrast

t =

contrast ofestimated

parameters

varianceestimate

t =

ss22c’(X’X)c’(X’X)++cc

c’bc’b

So, for a contrast in our model of 1 -1 0:

t = (ß1x1 + ß2x-1 + ß3x0) Estimated variance

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Brain activation: Left motor responsesBrain activation: Left motor responses

This shows activation of the contralateral motor cortex, ipsilateral cerebellum, etc.

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F-contrastsF-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-testF-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

F = Error

varianceestimate

Additionalvariance

accounted forby tested effects

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Brain activationBrain activation

Areas involved in the overall difference (positive or negative) from the left and right responses

(non-directional)

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Test Design andcontrast

SPM(t) orSPM(F)

t-test

F-test

[ 1 0 0 ; 0 1 0 ]

[1 -1 0]

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Inferences about subjects and Inferences about subjects and populations 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!

36Temporal series fMRI

Statistical image(SPM)

voxel time course

One voxel = One test (t, F)One voxel = One test (t, F)amplitude

time

General Linear Modelfittingstatistical image

From Poline (2005)

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Choosing a statistical thresholdChoosing 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 thresholdChoosing 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)

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How do we do that in SPM5?How do we do that in SPM5?

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SummarySummary

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

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ReferencesReferences 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.mrc-cbu.cam.ac.uk/imaging/PrinciplesStatistics) (http://imaging.mrc-cbu.cam.ac.uk/imaging/SpmContrasts)

SPM5 Manual, The FIL Methods Group (2007) An introduction to functional MRI by de Haan & Rorden

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Thank you!Thank you!