General Linear Model

Beatriz Calvo

Davina Bristow

Overview

Summary of regression Matrix formulation of multiple regression Introduce GLM Parameter Estimation

Residual sum of squares GLM and fMRI fMRI model

Linear Time Series Design Matrix Parameter estimation

Summary

Summary of Regression Linear regression models the linear relationship between a

single dependent variable, Y, and a single independent variable, X, using the equation:

Y = βX + c + ε

The regression coefficient, β, reflects how much of an effect X has on Y

ε is the error term and is assumed to be independently, identically, and normally distributed (mean 0 and variance σ2)

Summary of Regression Multiple regression is used to determine the effect of a

number of independent variables, X1, X2, X3 etc, on a single dependent variable, Y

The different X variables are combined in a linear way and each has its own regression coefficient:

Y = β1X1 + β2X2 +…..+ βLXL + ε

The β parameters reflect the independent contribution of each independent variable, X, to the value of the dependent variable, Y.

i.e. the amount of variance in Y that is accounted for by each X variable after all the other X variables have been accounted for

Matrix formulation

Multiplying matrices reminder:

a b cd e f

GHI

=xG x a + H x b + I x c

G x d + H x e + I x f

Matrix Formulation Write out equation for each observation of variable Y from 1 to J:

Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1

Yj = Xj1β1 +…+Xjlβl +…+ XjLβL + εj

YJ = XJ1β1 +…+XJlβl +…+ XJLβL + εJ

Y1

Yj

YJ

=X11 … X1l … X1L

Xj1 … X1l … X1L

X11 … X1l … X1L

Can turn these simultaneous equations into matrix form to get a single equation:

β1

βj

βJ

+ε1

εj

εJ

Y = X x β + ε

Observed data Design Matrix Parameters Residuals/Error

General Linear Model This is simply an extension of multiple regression

Or alternatively Multiple Regression is just a simple form of the General Linear Model

Multiple Regression only looks at ONE dependent (Y) variable

Whereas, GLM allows you to analyse several dependent, Y, variables in a linear combination

i.e. multiple regression is a GLM with only one Y variable

ANOVA, t-test, F-test, etc. are also forms of the GLM

GLM - continued.. In the GLM the vector Y, of J observations of a single Y

variable, becomes a MATRIX, of J observations of N different Y variables

An fMRI experiment could be modelled with matrix Y of the BOLD signal at N voxels for J scans

However SPM takes a univariate approach, i.e. each voxel is represented by a column vector of J fMRI signal measurements, and it processed through a GLM separately

(this is why you then need to correct for multiple comparisons)

GLM and fMRIHow does the GLM apply to fMRI experiments?

Y = X . β + ε

Observed data:

SPM uses a mass univariate approach – that is each voxel is treated as a separate column vector of 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 voxelTiming info: onset vectors, Om

j, and duration vectors, Dm

j

HRF, hm, describes shape of the expected BOLD response over timeOther regressors, e.g. realignment parameters

Parameters:

Define the contribution of each component of the design matrix to the value of YEstimated so as to minimise the error, ε, i.e. least sums of squares

Error:

Difference between the observed data, Y, and that predicted by the model, Xβ.Not assumed to be spherical in fMRI

Parameter estimation In linear regression the parameter β is estimated so that

the best prediction of Y can be obtained from X

i.e. sums of squares of difference between predicted values and observed data, (i.e. the residuals, ε) is minimised

Remember last week’s talk & graph!

The method of estimating parameters in GLM is essentially the same, i.e. minimising sums of squares (ordinary least squares), it just looks more complicated

Last week’s graph

ε

y = βx + c ε = residual error

= y i , true value

= y , predicted value

Residual Sums of Squares Take a set of parameter estimates, β

Put these into the GLM equation to obtain estimates of Y from X, i.e. fitted values, Y:

Y = X x β

The residual errors, e, are the difference between the fitted and actual values:

e = Y - Y = Y - Xβ

Residual sums of squares is: S = ΣjJej

2

When written out in full this gives:

S = ΣjJ(Yj - Xj1β1 -…- XjLβL)2

Minimising S If you plot the sum

of squares value for different parameter, β, estimates you get a curve

parameter estimates(B)

sum

s of s

quar

es (S

)

S is minimised when the gradient of this curve is zero Gradient = 0

min S

S = Σ(Y -

Xβ)2

e = Y - XβS = Σj

Jej2

Minimising S cont. so to calculate the values of β which gives you the

least sums of squares you must find the partial derivative of

S = Σj

J(Yj - Xj1β1 -…- XjLβL)2

Which is

∂S/∂β = 2Σ(-Xjl)(Yj – Xj1β1-…- XjLβL)

and solve this for ∂S/∂β = 0

In matrix form of the residual sum of squares isS = eTe

this is equivalent to ΣjJej

2

(remember how we multiply matrices)

e = Y - X β therefore S = (Y - X β )T(Y - X β )

Minimising S cont. Need to find the derivative and solve for ∂S/∂β = 0

The derivative of this equation can be rearranged to give

XTY = (XTX)β

when the gradient of the curve = 0, i.e. S is minimised

This can be rearranged to give:

β = XTY(XTX)-1

But a solution can only be found, if (XTX) is invertible because you need to divide by it, which in matrix terms is the same as multiplying by the inverse!

GLM and fMRIHow does the GLM apply to fMRI experiments?

Y = X . β + ε

Observed data:

SPM uses a mass univariate approach – that is each voxel is treated as a separate column vector of 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 voxelTiming info: onset vectors, Om

j, and duration vectors, Dm

j

HRF, hm, describes shape of the expected BOLD response over timeOther regressors, e.g. realignment parameters

Parameters:

Define the contribution of each component of the design matrix to the value of YEstimated so as to minimise the error, ε, i.e. least sums of squares

Error:

Difference between the observed data, Y, and that predicted by the model, Xβ.Not assumed to be spherical in fMRI

fMRI models

Completed the experiment, after preprocessing, the data are ready for STATS.

STATS: (estimate parameters, β, inference) indicating evidence against the Ho of no effect at

each voxel are computed->an image of this statistic is produce

This statistical image is assessed (other talk will explain that)

Example: 1 subject. 1 sessionMoving finger vs rest7 cycles of rest and moving

Time series of BOLD response in one voxel

Time seconds

Res

po

nse

s at

vo

xel

(x,

y, z

)

Question: Is there any change in the BOLD response between moving and rest?

Each epoch 6 scansWhole brain acquisition data

TIME SERIES: consist on the sequential measures of fMRI data signal intensities over the period of the experiment

The same temporal model is used at each voxel

Mass-univariated model and perform the same analysis at each voxel

Therefore, we can describe the complete temporal model for fMRI data by looking at how it works for the data from a voxel. Single Voxel Time Series

Time

Linear Time Series Model

Y: My data/ observations

Single Voxel Time Series

My Data

TimeY1

Ys

YN

Time series of N observations

Y1,…,Ys,…,Yn.

N= scan number

Acquired at one voxel

at times ts, where S=1:N

Model specification The overall aim of regressor generation is to come

up with a design matrix that models the expected fMRI response at any voxel as a linear combinations of columns.

Design matrix – formed of several components which explain the observed data.

Two things SPM need to know to construct the design matrix:

Specify regressors Basis functions that explain my data

Model specification … Specify regressors X

Timing information consists of onset vectors Om

j and duration vectors Dm

Other regressors e.g. movement parameters Include as many regressors as you consider

necessary to best explain what’s going on.

Basis functions that explain my data (HRF)Expected shape of the BOLD response due to

stimulus presentation

GLM and fMRI data Model the observed time series at each voxel as a linear

combination of explanatory functions, plus an error term

Ys= β1 X1(tS)+ …+ βl Xl

(tS)+ …+ βL XL(tS) + εs

Here, each column of the design matrix X contains the values of one of the continuous regressors evaluated at each time point ts of fMRI time series

That is, the columns of the design matrix are the discrete regressors

Consider the equation for all time points, to give a set of equations

Y1

Ys

YN

β1

βl

βL

εN

εs

εN

+=

X1(t1) Xl

(t1) XL(t1)

X1(tS) Xl

(tS) XL(tS)

X1(tN) Xl

(tN) XL(tN)

Y1= β1 X1(t1)+ …+ βl Xl

(t1)+ …+ βL XL(t1) + ε1

Ys= β1 X1(tS)+ …+ βl Xl

(tS)+ …+ βL XL(tS) + εs

YN= β1 X1(tN)+ …+ βl Xl

(tN)+ …+ βL XL(tN) + εN

Y = X β + εIn matrix notation:

GLM and fMRI data …

In matrix form:

Getting the design matrixRegressors

εErrors are normally and independently and identical distributed

Intensity

Tim

e

= β1 β2+ +

Observations

y = x1 x2

Getting the design matrix …Regressors

Intensity

Tim

e

= β1β2+ +

Observations

y = β1x1 + β2x2 + ε

Error

Design matrixRegressors

= β2

β1+

Observations

Y = X β + ε

Error

x

Design matrixRegressors

β1

β2

Observations Error

Y = X β + ε

N N N

l L

l

l

L

N: nuber of scansP: number of regressors Y = X β + ε

Model is specified by:

•design matrix

•Assumptions about ε Y1

Ys

YN

X1(t1) Xl

(t1) XL(t1)

X1(tS) Xl

(tS) XL(tS)

X1(tN) Xl

(tN) XL(tN)

β1

βl

βL

εN

εs

εN

Parametric estimation

=

β1

β2+ +

Y X ε

Estimate parameters

β

The error is minimal when

Least squaresParameter estimates

β = XTY(XTX)-1

ε = Y - Y = Y - Xβ

S = ΣtJεt

2

(Get this by putting into matrix form and finding derivative)

Summary The General Linear Model allows you to find the

parameters, β, which provide the best fit with your data, Y

The optimal parameters estimates, β, are found by minimising the Sums of Squares differences between your predicted model and the observed data

The design matrix in SPM contains the information about the factors, X, which may explain the observed data

Once we have obtained the βs at each voxel we can use these to do various statistical tests

but that is another talk….

THE END

Thank you toLucy, Daniel and Will

and toStephan for his chapter and slides about GLM

and to Adam for last year’s presentation

Links:http://www.fil.ion.ucl.ac.uk/~wpenny/notes03/slides/glm/slide1.htm

http://www.fil.ion.ucl.ac.uk/spm/HBF2/pdfs/Ch7.pdfhttp://www.fil.ion.ucl.ac.uk/~wpenny/mfd/GLM.ppt