11stst Level Analysis: Level Analysis:

Design matrix, contrasts and Design matrix, contrasts and inferenceinference

Rebecca Knight and Lorelei Howard

Outline

What is first level analysis?

The General Linear Model and how this relates to the Design Matrix

Regressors within the Design Matrix

Rebecca Knight

Overview

Motioncorrection

Smoothing

kernel

Spatialnormalisation

Standardtemplate

fMRI time-seriesStatistical Parametric Map

General Linear Model

Design matrix

Parameter Estimates

Once the image has been reconstructed, realigned, spatially normalised and smoothed….

The next step is to statistically analyse the data

1st level analysis – A within subjects analysis where activation is averaged across scans for an individual subject

The Between- subject analysis is referred to as a 2nd level analysis and will be described later on in this course

Design Matrix – 2D, m = regressors, n = time. A dark-light colour map is used to show the value of each variable at specific time points

The Design Matrix forms part of the General linear model, the majority of statistics at the analysis stage use the GLM

Rebecca Knight

Key Concepts

General Linear Model

Y

Generic Model

Aim: To explain as much of the variance in Y by using X, and thus reducing E

Dependent Variable (What you are measuring)

Independent Variable (What you are manipulating)

Relative Contribution (These need to be estimated)

Error (The difference between the observed data and that which is predicted by the model)

= X x β + E

Y = X1β1 + X2β2 + ....X n βn.... + E More than 1 IV ?

GLM Continued

YMatrix of BOLD signals

(What you collect)

Design matrix

(This is what is put into SPM)

Matrix parameters

(These need to be estimated)

Error matrix

(residual error for each voxel)

= X x β + E

How does this equation translate to the 1st level analysis ?

Each letter is replaced by a set of matrices (2D representations)

Time

Voxels

Time

Regressors

Regressors

Voxels

Time

Voxels

Rebecca Knight

‘Y’ in the GLM

Y = Matrix of Bold signals

Time

(scan every 3 seconds)

fMRI brain scans Voxel time course

Amplitude/Intensity

1 voxel = ~ 3mm³

Time

‘X’ in the GLM

X = Design Matrix

Time(n)

Regressors (m)

Regressors Regressors – represent hypothesised contributors in your experiment. They are represented by columns in the design matrix (1column = 1 regressor)

Regressors of Interest or Experimental Regressors – represent those variables which you intentionally manipulated. The type of variable used affects how it will be represented in the design matrix

Regressors of no interest or nuisance regressors – represent those variables which you did not manipulate but you suspect may have an effect. By including nuisance regressors in your design matrix you decrease the amount of error.

E.g. - The 6 movement regressors (rotations x3 & translations x3 ) or physiological factors e.g. heart rate

Rebecca Knight

Regressors

Time(n)

Regressors (m)

A dark-light colour map is used to show the value of each regressor within a specific time point

Black = 0 and illustrates when the regressor is at its smallest value

White = 1 and illustrates when the regressor is at its largest value

Grey represents intermediate values

The representation of each regressor column depends upon the type of variable specified

Conditions As they indicate conditions they are referred to as indicator variables

Type of dummy code is used to identify the levels of each variable

E.g. Two levels of one variable is on/off, represented as

ON = 1

OFF = 0

When you IV is presented

When you IV is absent (implicit baseline)

Changes in the bold activation associated with the

presentation of a stimulus

Fitted Box-Car

Red box plot of [0 1] doesn’t model the rise and falls

Modelling Haemodynamics

Changes in the bold activation associated with the presentation of a stimulus

Haemodynamic response function

Peak of intensity after stimulus onset, followed by a return to baseline then an undershoot

Box-car model is combined with the HRF to create a convolved regressor which matches the rise and fall in BOLD signal (greyscale)

Even with this, not always a perfect fit so can include temporal derivatives (shift the signal slightly) or dispersion derivatives (change width of the HRF response) *more later in this course

HRF Convolved

Covariates What if you variable can’t be described using conditions?

E.g Movement regressors – not simply just one state or another

The value can take any place along the X,Y,Z continuum for both rotations and translations

Covariates – Regressors that can take any of a continuous range of values (parametric)

Thus the type of variable affects the design matrix – the type of design is also important

Designs

Block design v Event- related design

Intentionally design events of interest into blocks

Retrospectively look at when the events of interest occurred. Need to code the onset time for each regressor

Separating Regressors

The type of design and the type of variables used in your experiment will affect the construction of your design matrix

Another important consideration when designing your matrix is to make sure your regressors are separate

In other words, you should avoid correlations between regressors (collinear regressors) – because correlations in regressors means that variance explained by one regressor could be confused with another regressor

This is illustrated by an example using a 2 x 3 factorial design

Example

Motion No Motion

High Medium Low

Design

IV 1 = Movement, 2 levels (Motion and No Motion)

IV 2 = Attentional Load, 3 levels (High, Medium or Low)

High Medium Low

Example Cont.

V A C1 C2 C3

M N h m l If you made each level of the variables a regressor you could get 5 columns and this would enable you to test main effects

BUT what about interactions? How can you test differences between Mh and Nl

This design matrix is flawed – regressors are correlated and therefore a presence of overlapping variance (Grey)

M N h m l

MN h ml

Orthogonal design matrix

h m l h m l

M M M N N N

If you make each condition a regressor you create 6 columns and this would enable you to test main effects

AND it enable you to test interactions! You can test differences between Mh and Nl

This design matrix is orthogonal – regressors are NOT correlated and therefore each regressor explains separate variance

M

N

h m l

Mh

Nh

MlMm

Nm Nl

h m l h m l

M M M N N N

h m l h m l

M MM N N N

Summary

YMatrix of BOLD

signals Design matrix Matrix parameters

= X x + ETime

Voxels

Time

Regressors

Regressors

Voxels

Time

Voxels

Error matrix

β

Aim: To explain as much of the variance in Y by using X, and thus reducing E

β = relative contribution that each regressor has, the larger the β value = the greater the contribution

Next: Examine the effect of regressors

Rebecca Knight

Outline

Why do we need contrasts?

What are contrasts?

T contrasts

F contrasts

Rebecca Knight

Why use contrasts

GLM:

- Specify design matrix

- Determine β’s for each voxel for each regressor

Use contrasts to:

- Specify effects of interest

- Perform statistical evaluation of hypotheses

Contrasts used and their interpretation depends on the model specification, which in turn depends on the design of the experiment

Rebecca Knight

What is a contrast?

cT = [1 0 0 0 0 …]

Contrast vector of length p

cT β = 1xb1 + 0xb2 + 0xb3 + 0xb4 + 0xb5 + . . .

Contrast = statistical assessment of cT β

p

Rebecca Knight

Different contrasts

T contrasts

- Unidimensional (vectors)

- Directional

- Assess effect of one parameter OR compare specific

combinations of parameters

F contrasts

- Multidimensional (matrix)

- Non-directional

- Collection of T contrasts

Example

Two event-related conditions

The subjects press a button with either their left or right hand, depending on visual instruction

Left Right

Rebecca Knight

T contrasts

Left Right

cT = [1 0 0 …]

cTβ = 1xb1 + 0xb2 + 0xb3 + . . .

identifies voxels whose activation increases in response to Left button presses

Question: Which brain regions respond to Left button presses?

cT = [-1 0 0 …]

cTβ = -1xb1 + 0xb2 + 0xb3 + . . .

identifies voxels whose activation decreases in response to Left button presses

Rebecca Knight

T contrasts

H0 : cTβ = 0

Experimental Hypotheses:

- H1: cTβ > 0 ?

- H1: cTβ < 0 ?

T-test is a signal-to-noise measure

Test Statistic:

T df =

cT β

Contrast of estimated

parameters

Variance estimate

SD (cTβ) =

Rebecca Knight

T contrasts

Subtractive Logic:

“ The direct comparison of two regressors that are assumed to differ only in one property, the IV ”

Question: Which brain regions respond more to Left than to Right button presses?

Left Right

cT = [1 -1 0 …]

cTβ = 1xb1 + -1xb2 + 0xb3 + . . .

cT = [1 -1 0 …] ≠ cT= [-1 1 0 …]

must ensure sum of the weights = 0

Rebecca Knight

T contrasts

SPM-t image

Clearly see contralateral motor cortex response

The map of T-values:

spmT_*.img

The contrast itself (cTβ; ie, numerator):

con_*.img

* = number in Contrast Manager

2nd Level

Rebecca Knight

F contrasts

Left Right

cT = 1 0 0 … 0 1 0 …

Question: Which brain regions respond to Left and/or Right button presses?

Matrix of T contrasts

Non-directional

Identify voxels showing modulation in response to experimental task, ahead of more specific contrasts

Rebecca Knight

F contrasts

Rebecca Knight

F = Explained variability

Error variance estimate

Determines whether any one regressor OR combination of regressors explains a significant amount of the variance in Y

NOT which regressor the effect can be attributed to

H0 : β1 = β2 = 0

H1: at least one β ≠ 0

Test Statistic:

Rebecca Knight

F contrasts

Rebecca Knight

SPM-F image

Clearly see motor cortex response

The map of F-values:

spmF_*.img

Also outputs:

ess_*.img

* = number in Contrast Manager

Rebecca Knight

Factorial e.g.

IV 1 = Movement, 2 levels (Motion and No Motion)

IV 2 = Attentional Load, 3 levels (High, Medium or Low)

M M M N N Nh m l h m l

ME Movement

• Stack of M > N contrasts for each level of Load

• Shows voxels which are more active in M than N (regardless of attentional load)

Rebecca Knight

Factorial e.g.

IV 1 = Movement, 2 levels (Motion and No Motion)

IV 2 = Attentional Load, 3 levels (High, Medium or Low)

M M M N N Nh m l h m l

ME Attention

• First row = h > m• Second row = m > l

• Shows voxels which are more active in h than m AND/OR m than l (regardless of movement level)

Rebecca Knight

Factorial e.g.

IV 1 = Movement, 2 levels (Motion and No Motion)

IV 2 = Attentional Load, 3 levels (High, Medium or Low)

M M M N N Nh m l h m l

Interaction

• Shows voxels where the attentional load elicits a brain response that is different when there is motion, or not

Rebecca Knight

Inference

We’ve talked about 1st level so far… examining within subject

variability.

However, we can’t use a sample of one to extrapolate our findings to

the general population

2nd level analyses to look for effects at the group level… discussed

later in course

Rebecca Knight

Summary

Contrasts are statistical (t or F) tests of specific hypotheses

T contrasts:

- Compare effect of one regressor with 0

- Compare 2 or more regressors

F contrasts:

- Multidimensional contrasts

Rebecca Knight

Resources

Huettel. Functional magnetic resonance imaging (Chap 10)

MfD Slides 2007

Human Brain Function (Chap 8)

Rik Henson and Guillaume Flandin’s slides from SPM courses