the general linear model and statistical parametric mapping l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
The General Linear Model and Statistical Parametric Mapping PowerPoint Presentation
Download Presentation
The General Linear Model and Statistical Parametric Mapping

Loading in 2 Seconds...

play fullscreen
1 / 34

The General Linear Model and Statistical Parametric Mapping - PowerPoint PPT Presentation


  • 431 Views
  • Uploaded on

The General Linear Model and Statistical Parametric Mapping. Stefan Kiebel Andrew Holmes SPM short course, May 2002. ?. SPM key concepts. …a voxel by voxel hypothesis testing approach reliably identify regions showing a significant experimental effect of interest Key concepts

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'The General Linear Model and Statistical Parametric Mapping' - obert


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
the general linear model and statistical parametric mapping

The General Linear Model andStatistical Parametric Mapping

Stefan Kiebel Andrew Holmes

SPM short course, May 2002

spm key concepts

?

SPM key concepts...

…a voxel by voxel hypothesis testing approach

      • reliably identify regions showing a significant experimental effect of interest
  • Key concepts
      • Type I error
        • significance test at each voxel
      • Parametric statistics
        • parametric model for voxel data, test model parameters
      • No exact prior anatomical hypothesis
        • multiple comparisons
  • Statistical Parametric Mapping
      • General Linear Model
      • Theory of continuous random fields
overview
Overview…

…a voxel by voxel hypothesis testing approach

      • reliably identify regions showing a significant experimental effect of interest
  • The General linear model & Statistical Parametric Mapping
      • General Linear Model
        • models & design matrices
        • model estimation
      • Generalised Linear Model
        • serial correlations
        • variance components
      • Contrasts & Statistic images
        • Statistical Parametric Maps – SPMs
      • Global effects
      • Model selection
slide4

image data

parameter

estimates

designmatrix

kernel

  • General Linear Model
    • model fitting
    • statistic image

realignment &motioncorrection

random field theory

smoothing

normalisation

StatisticalParametric Map

anatomicalreference

corrected p-values

example epoch f mri activation dataset auditory stimulation

Epoch fMRI

  • 7s RT – interscan interval
  • 84 scans – images 16–99
  • 1 session
  • 1 condition/trial: words
  • Deterministic design: Fixed SOA of 12 scans (42s)
  • Epoch design
  • First trial at time 6 scans, 6 scans / words epoch

SPM parameters(event terminology)

(50, -40, 8) — Primary Auditory Cortex

Example epoch fMRI activation dataset:Auditory stimulation
  • Single subject
    • RH male
  • Conditions
    • Passive word listening
    • Bisyllabic nouns
    • 60wpm
    • against rest
  • Epoch fMRI
    • rest & words
    • epochs of 6 scans
    • 42 second epochs
    • 7 BA cycles

experiment was 8 cycles: first pair of blocks dropped

    • BABABABABABABA
    • last 84 scans of experiment

images 16–99

    • ~10 minutes scanning time

seconds

voxel by voxel statistics
Voxel by voxel statistics…

model specification

parameter estimation

hypothesis

statistic

statistic image or SPM

f MRI time series

voxel time series

voxel statistics
Voxel statistics…
  • parametric
      • one sample t-test
      • two sample t-test
      • paired t-test
      • Anova
      • AnCova
      • correlation
      • linear regression
      • multiple regression
      • F-tests
      • etc…
  • non-parametric?  SnPM

all cases of theGeneral Linear Model

assume normality

to account for serial correlations:

Generalised Linear Model

e g two sample t test
…e.g. two-sample t-test?

t-statistic imageSPM{t}

Image intensity

compares size of effect to its error standard deviation

  • standard t-test assumes independenceignores temporal autocorrelation!

voxel time series

regression example
Regression example…

Ys =  + af(ts)+ s

f(ts) = 0 or 1

s ~ N(0,2)

=

+

a

+ error

  • t-statistic for H0: a= 0
  • correlation:test H0:  = 0 equivalent totest H0: a = 0
  • two-sample t-test:

test H0: 0 = 1equivalent totest H0: a = 0

  • can extend to account fortemporal autocorrelation!

voxel time series

box-car reference function

revisited
…revisited

=

+

a

+

error

Ys

1

a

f(ts)

s

=

+

+

general linear model
General Linear Model…
  • fMRI time series: Y1 ,…,Ys ,…,YN
      • acquired at times t1,…,ts,…,tN
  • Model: Linear combination of basis functions
    • Ys = 1f 1(ts ) + …+lf l(ts ) + … +Lf L(ts ) + s
  • f l (.):basis functions
      • “reference waveforms”
      • dummy variables
  • l: parameters(fixed effects)
      • amplitudes of basis functions (regression slopes)
  • s: residual errors: s ~ N(0,2)
      • identically distributed
      • independent, or serially correlated (Generalised Linear Model  GLM)
design matrix formulation

Y1f 1(t1) … f l(t1 ) … f L(t1 )11

:: … : … : : :

Ys = f 1(ts ) … f l(ts ) … f L(ts )l + s

::… :… :: :

YNf 1(tN ) … f l(tN ) … f L(tN )LN

Y = X×b + e

Design matrix formulation…

Ys = 1f 1(ts ) + …+lf l(ts ) + … +Lf L(ts ) + ss = 1, …,N

Y1 = 1f 1(t1) +…+lf l(t1 ) +…+Lf L(t1 ) + 1

: :: ::: ::

Ys = 1f 1(ts ) +…+ lf l(ts ) +…+Lf L(ts ) + s

: :::::::

YN = 1f 1(tN ) +…+lf l(tN ) +…+Lf L(tN ) + N

vector of parameters

error vector

design matrix

data vector

Y=X×b +e

N 1NL L 1N 1

box car regression revisited
Box-car regression… (revisited)

=

+

a

+

error

Ys

1

a

f(ts)

s

=

+

+

box car regression design matrix
Box car regression: design matrix…

data vector

(voxel time series)

parameters

error vector

design matrix

a

=

+

Y

=

X

+

low frequency nuisance effects
Low frequency nuisance effects…
  • Drifts
    • physical
    • physiological
  • Aliased high frequency effects
    • cardiac (~1 Hz)
    • respiratory (~0.25 Hz)
  • Discrete cosine transform basis functions
    • r = 1,…,R
    • Rcut-off period
design matrix
…design matrix

parameters

error vector

design matrix

data vector

a

m

3

4

5

6

7

8

9

=

+

=

+

Y

X

example a line through 3 points

^

a

Y

^

m

x

^

^

Y1x11ae1

Y2 = x21 + e2

Y3x31me3

Y = Xb + e

^

^

^

Example: a line through 3 points…

simple linear regression

parameter estimates m&a

fitted values Y1 ,Y2 ,Y3

residuals e1 , e2 , e3

Yi = axi + m + eii = 1,2,3

(x2, Y2)

1

(x3, Y3)

Y1 = ax1 + m × 1 + e1

Y2 = ax2 + m × 1 + e2

Y3 = ax3 + m × 1 + e3

(x1, Y1)

dummy variables

geometrical perspective

x11

Y = ax2 + m1 + e

x31

Y= axa + mxm + e

Geometrical perspective…

(Y1, Y2, Y3)

Y

(x1,x2, x3)

xa

(1,1,1)

xm

O

design space

estimation geometrically

^

Y

^

a xa

^

m xm

^

^

^

(Y1,Y2,Y3)

Estimation, geometrically…

(Y1, Y2, Y3)

Y

e = (e1,e2,e3)T

xa

xm

design space

estimation formally
Estimation, formally…

Consider parameter estimates

Giving fitted values

Residuals

Residual sum of squares

Minimised when

…but this is the lth row of

(obviously!)

so the least squares estimates satisfy the normal equations

giving

inference geometrically

a xa

^

^

m xm

^

^

^

(Y1,Y2,Y3)

Inference, geometrically…

Inference, geometrically…

model:

Yi = axi + m + ei

null hypothesis:

e.g.H0: a = 0 (zero slope…)

i.e.does xa explain anything?(after xm)

(Y1, Y2, Y3)

Y

e = (e1,e2,e3)T

xa

xm

design space

inference formally
Inference, formally…

For any linear compound of the parameter estimates:

Further (independently):

So hypotheses can be assessed using:

a Student’s t statistic, giving an SPM{t}

Suppose the model can be partitioned…

“Extra sum-of-squares”

multivariate perspective

Y

=

X

+

^

Multivariate perspective…

voxels

?

?

=

+

parameters

design matrix

errors

data matrix

scans

s2

variance

parameterestimates

  • estimate

residuals

estimated variance

=

estimatedcomponentfields

“Image regression”

f mri box car example
fMRI box car example…

parameters

error vector

design matrix

data vector

a

m

3

4

5

6

7

8

9

=

+

Y

=

X

+

fitted
…fitted

raw fMRI time series

adjusted for global & low Hz effects

fitted box-car

scaled for global changes

fitted “high-pass filter”

residuals

slide27

hæmodynamic response

power spectrum

  • Drifts
    • physical
    • physiological
  • Aliased high frequency effects

e.g.

    • cardiac (~1 Hz)
    • respiratory (~0.25 Hz)
serial correlations
Serial correlations...

Y = Xb+ ee ~ N(0,s2V)–intrinsic autocorrelationV

Problem: Estimate s2Vat each voxel and make inference about cTb

Model:

Model Vas linear combination of mvariance components

V= l1 Q1 + l2 Q2 + … + lm Qm

Assumptions:

Vis the same at each voxel

s2is different at each voxel

Example:

For one fMRI session, use 2 variance components. Choice of Q1and Q2motivated by autoregressive model of order 1 plus white noise (AR(1)+wn)

Q1

Q2

serial correlations estimation
Serial correlations…estimation

Estimation:

b = (XTX)–XTY – unbiased, ordinary least squares estimate

Compute sample covariance matrix of data at all activated voxels: CY = SkYk YkT /K

Important: Data Yk must be high-pass filtered.

Model CY as CY = XbbTXT + SliQi and estimate hyperparameters li using

Restricted Maximum Likelihood (ReML)

Estimate VbyV= N SliQi /trace(SliQi )

Estimate s2 at each voxel in the usual way by

s2 = (RY) T(RY) / trace(R V)– unbiased

where R = I – X(XTX)–X

^

^

^

CY

^

CY

^

serial correlations inference
Serial correlations…inference

Inference:

To test null hypothesis cTb= 0, compute t-value by dividing size of effect by its

standard deviation: t =cTb/ std[cTb]

where std[cTb] = sqrt(s2c'(X TX)–XTVX(XTX)–c )

… but … std[cTb]is not a c2variable because of V

Find approximating c2distribution using Satterthwaite approximation:

Var[s2] = 2s4 trace(R V R V) / trace(R V)2

n = 2E[s2]2/Var[s2] = trace(RV)2/ trace(RVRV)– effective degrees of freedon

  • Worsley & Friston (1995) “Analysis of fMRI time series revisited – again”

Use t-distribution with n degrees of freedom to compute p-value for t

^

^

^

^

^

inference contrasts spm t
Inference — contrasts — SPM{t}

is there an effect of interest after other modelled effects have been taken into account

contrast— linear combination of parameters: cT

c = +1 0 0 0 0 0 0 0

t-test H0: cT = 0

e.g.activation: box-car amplitude > 0 ?

contrast ofestimatedparameters

T =

varianceestimate

SPM{t}

correct variance estimate & degrees of freedom for temporal autocorrelation

inference f tests spm f
Inference — F-tests — SPM{F}

is there an effect of interest after other modelled effects have been taken into account

multiple linear hypotheses

0 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

H0: cT = 0

c =

e.g.Does HPF basis set model anything ?

additionalvarianceaccounted forby effects ofinterest

F =

errorvarianceestimate

SPM{F}

correct variance estimate & degrees of freedom for temporal autocorrelation

H0: 2 = 0

X2()

conclusions
Conclusions…
  • General Linear Model
    • (simple) standard statistical technique
      • temporal autocorrelation – a Generalised Linear Model
    • single general framework for many statistical analyses
      • flexible modelling  basis functions
    • design matrix visually characterizes model
      • fit data with combinations of columns of design matrix
    • statistical inference: contrasts…
      • t–tests: planned comparisons of the parameters
      • F–tests: general linear hypotheses, model comparison
the general linear model and statistical parametric mapping34

Ch4

Ch3

The General Linear Model and Statistical Parametric Mapping

Friston KJ, Holmes AP, Worsley KJ, Poline J-B, Frith CD, Frackowiak RSJ (1995)“Statistical parametric maps in functional imaging: A general linear approach”Human Brain Mapping 2:189-210

Worsley KJ & Friston KJ (1995)“Analysis of fMRI time series revisited — again” NeuroImage 2:173-181

Friston KJ, Josephs O, Zarahn E, Holmes AP, Poline J-B (1999)

“To smooth or not to smooth”NeuroImage12: 196-208

Zarahn E, Aguirre GK, D'Esposito M (1997)

“Empirical Analyses of BOLD fMRI Statistics” NeuroImage5:179-197

Aguirre GK, Zarahn E, D'Esposito M (1997)

“Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5:199-212

Holmes AP, Friston KJ (1998)

“Generalisability, Random Effects & Population Inference” NeuroImage 7(4-2/3):S754