1st level analysis: Design matrix, contrasts, and inference

Stephane De Brito & Fiona McNabe

Outline What is ‘1st level analysis’?

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

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]

Rebecca Knight

Motioncorrection

Smoothing

kernel

Spatialnormalisation

Standardtemplate

fMRI time-series Statistical 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

Overview

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 –The set of regressors that attempts to explain the experimental data using the GLM

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

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

Key concepts

Y

Generic Model

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

Dependent Variable (What you are measuring)

Independent Variable (What you are manipulating)

Relative Contributionof X to the overalldata (These need tobe estimated)

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

= X x β + ε

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

General Linear Model

YMatrix of BOLDat various time points in a single voxel(What you collect)

Design matrix

(This is your model specification in SPM)

Parameters matrix

(These need to be estimated)

Error matrix

(residual error for each voxel)

= X x β + ε

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

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

Time

(rows)

Voxels (columns)

Time(rows)

Regressors (columns) Param. weights (columns)

Voxels (rows)

Time

(rows)

Voxels

GLM continued

Rebecca Knight

Y = Matrix of Bold signals

Amplitude/Intensity

Time

(scan every 3 seconds)

fMRI brain scans Voxel time course

1 voxel = ~ 3mm³

Time

‘Y’ in the GLM

Y

X = Design Matrix

Time(n)

Regressors (m)

‘X’ in the GLM

Regressors – represent hypothesised contributors to the fMRI time course 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 (2 types: Covariates and Indicators, next slides)

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, breathing or others (e.g., scanner known linear drift)

Regressors

Time(n)

Regressors (m)

Covariates = Regressors that can take any of a continuous range of values (e.g, task difficulty)

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

Regress. of Inter. (Covariates)

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

Regress. of inter. (Indicators)

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

Regr. of no inter. (Covariate)

Scanner Drift Artifact and t-test

E.g., Regress. of no inter.

Ways to improve your model: modelling haemodynamics

• 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!

Modelling haemodynamic

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

Separating regressors

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

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

Example Con’t

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

Orthogonal Design Matrix

YMatrix of BOLD

signals Design matrix Matrix parameters

= X x + εTime

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 ε

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

Next: Examine the effect of regressors and the contrasts

Interim Summary

BRIEF OVERVIEW:SPECIFY THE 1ST MODEL IN SPM

Thank you!