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• Experimental Design • Initial GLM Intro

GLM

• General Linear Model • Single subject fMRI modeling

Single Subject fMRI Data

• Data at one voxel – Rest vs.

passive word listening

• Is there an effect?

A Linear Model r

• “Linear” in parameters 1 & 2

Intensity

Tim e = 1 2+ + err

or

x1 x2 

1 2

Linear model, in image form…

+ + = + +1 2

Y   11x 22 x

… in image matrix form…

=  + 

  1=  +

   2

Y   X 

… in matrix form.

  XY 

1 1 1p

p = +YY X

N N N

p

N: Number of scans, p: Number of regressors

Linear Model Predictors

• Signal Predictors – Block designs – Event-related responses

• Nuisance Predictors – Drift

  XY

Drift – Regression parameters

Signal Predictors

• Linear Time-Invariant system • LTI specified solely by

– Stimulus function of experiment Blocks

  XY

– Hemodynamic Response Function (HRF)

• Response to instantaneous impulse

Events

Convolution Examples

Hemodynamic

Block Design

Experimental Stimulus Function

  XYEvent-Related

Hemodynamic Response Function

Predicted Response

LTI Pet Peeve

• LTI/convolution approach implies antisymmetry – Shape of rise must match inverted shape of fall

  XY

Bump here must match...

...bump here

HRF Models

• Canonical HRF – Most sensitive

if it is correct

  XY

f – If wrong, leads to

bias and/or poor fit • E.g. True response

may be faster/slower • E.g. True response

may have smaller/ bigger undershoot

SPM’s HRF

HRF Models

• Smooth Basis HRFs – More flexible – Less interpretable

• No one parameter explains the response

  XY

– Less sensitive relative to canonical (only if canonical is correct)

Gamma Basis

Fourier Basis

HRF Models

• Deconvolution – Most flexible

• Allows any shape • Even bizarre,

non-sensical ones

L t iti l ti

  XY

– Least sensitive relative to canonical (again, if canonical is correct)

Deconvolution Basis

Drift Models

• Drift – Slowly varying – Nuisance variability – Even seen in cadavers!

• A. Smith et al, NI, 1999, 9:526-533

  XY

• Models – Linear, quadratic – Discrete Cosine Transform

Discrete Cosine Transform Basis

Some TerminologySome Terminology

• SPM (“Statistical Parametric Mapping”) is a massively univariate approach - meaning that a statistic (e.g., T-value) is calculated for every voxel - using the “General Linear Model”

• Experimental manipulations are specified in a model (“design matrix”) which is fit to each voxel to estimate the size of the ) experimental effects (“parameter estimates”) in that voxel…

• … on which one or more hypotheses (“contrasts”) are tested to make statistical inferences (“p-values”), correcting for multiple comparisons across voxels (using “Random Field Theory”)

• The parametric statistics assume continuous-valued data and additive noise that conforms to a “Gaussian” distribution (“nonparametric” version SNPM eschews such assumptions)

Some TerminologySome Terminology

• SPM usually focused on “functional specialization” - i.e. localizing different functions to different regions in the brain

• One might also be interested in “functional integration” - how different regions (voxels) interactg ( )

• Multivariate approaches work on whole images and can identify spatial/temporal patterns over voxels, without necessarily specifying a design matrix (PCA, ICA)...

• … or with an experimental design matrix (PLS, CVA), or with an explicit anatomical model of connectivity between regions - “effective connectivity” - eg using Dynamic Causal Modeling

Overview

1. General Linear Model Design Matrix Estimation/Contrasts Covariates (eg global)Covariates (eg global) Estimability/Correlation

2. fMRI timeseries Highpass filtering HRF convolution Autocorrelation (nonsphericity)

Overview

1. General Linear Model Design Matrix Estimation/Contrasts Covariates (eg global)Covariates (eg global) Estimability/Correlation

2. fMRI timeseries Highpass filtering HRF convolution Autocorrelation (nonsphericity)

General Linear Model…General Linear Model…

•• Parametric statisticsParametric statistics

•• one sample one sample tt--testtest •• two sample two sample tt--testtest •• paired paired tt--testtest •• AnovaAnova •• AnCovaAnCova •• correlationcorrelation •• linear regressionlinear regression •• multiple regressionmultiple regression •• FF--teststests •• etc…etc…

all cases of the General Linear Model

General Linear ModelGeneral Linear Model

•• Equation for single (and all) voxels:Equation for single (and all) voxels:

yyjj = x= xj1j1 11 + … x+ … xjLjL LL + + jj jj ~ N(0,~ N(0,22))

yyjj : data for scan, : data for scan, j = 1…Jj = 1…J xxjljl :: explanatory variables / covariates / regressorsexplanatory variables / covariates / regressors l = 1 Ll = 1 Lxxjljl :: explanatory variables / covariates / regressors, explanatory variables / covariates / regressors, l 1…Ll 1…L l l : parameters / regression slopes / fixed effects: parameters / regression slopes / fixed effects jj : residual errors, : residual errors, independent & identically distributed (“iid”)independent & identically distributed (“iid”)

(Gaussian, mean of zero and standard deviation of (Gaussian, mean of zero and standard deviation of σσ))

•• Equivalent matrix form:Equivalent matrix form: y = Xy = X + + 

XX : “design matrix” / model: “design matrix” / model

Matrix FormulationMatrix Formulation

Equation for scan j

Simultaneous equations for scans 1.. J

…that can be solved for parameters 1.. L

scans 1.. J

Regressors

Sc an

s

Overview

1. General Linear Model Design Matrix Estimation/Contrasts Covariates (eg global)Covariates (eg global) Estimability/Correlation

2. fMRI timeseries Highpass filtering HRF convolution Autocorrelation (nonsphericity)

General Linear Model (Estimation)General Linear Model (Estimation)

^

^

• Estimate parameters from least squares fit to data, y:

 = (XTX)-1XTy = X+y (OLS estimates)

• Fitted response is:

•• Residual errorsResidual errors and estimated error variance are:and estimated error variance are:

 = y = y -- Y Y 22 = = TT / df/ df

where df are the degrees of freedom (assuming iid):where df are the degrees of freedom (assuming iid):

df = J df = J -- rank(X)rank(X) (=J(=J--L if X full rank)L if X full rank)

( R = I ( R = I -- XXXX++  = Ry= Ry df = trace(R) )df = trace(R) )

^

Y = X

^ ^ ^

Simple LS DerivationSimple LS Derivation GLM Estimation GLM Estimation –– Geometric PerspectiveGeometric Perspective

y

DATA (y1, y2, y3)

Y =   X    X

y1 x1 1  1 y2 = x2 1 + 2 y3 x3 1  3

 = (   )T^ ^ ^^ RESIDUALS

O (1,1,1)

(x1, x2, x3)X

X design space

(Y1,Y2,Y3)Y

 = ()

General Linear Model (Inference)General Linear Model (Inference)

^ • Specify contrast (hypothesis), c, a linear combination

of parameter estimates, cT 

^^ ^ ^

• Calculate T-statistic for that contrast:

T = cT / std(cT) = cT / sqrt(2cT(XTX)-1c)

c = [1 -1 0 0]T

T = cT / std(cT) = cT / sqrt(2cT(XTX) 1c)

(c is a vector), or an F-statistic:

F = [(0Tε0 – εTε) / (L-L0)] / [εTε / (J-L)]

where ε0 and L0 are residuals and rank resp. from the reduced model specified by c (which is a matrix)

• Prob. of falsely rejecting Null hypothesis, H0: cT=0 (“p-value”)

c = [ 2 -1 -1 0 -1 2 -1 0 -1 -1 2 0]

F

u

)( yft 

)0|( tp

T-distribution

ZZ--score and Tscore and T--statisticstatistic

•• The zThe z--score describes the relative score describes the relative location of a particular score (x) location of a particular score (x) when the mean (when the mean (μμ) and standard ) and standard deviation (deviation (σσ) are known) are known ThTh d b h ld b h l

  

 xZ

•• The tThe t--score describes the relative score describes the relative location of the sample mean (x) location of the sample mean (x) when the population mean is known when the population mean is known and the population standard and the population standard deviation is estimated with the deviation is estimated with the sample standard de