The General Linear Model and Statistical Parametric Mapping

Stefan Kiebel Andrew Holmes SPM short course, May 2002

…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

SPM key concepts...

?

…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

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realignment & motion

correction smoothing

normalisation

General Linear Model  model fitting  statistic image

corrected p-values

image data parameter estimates design

matrix

anatomical reference

kernel

Statistical Parametric Map

random field theory

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 SP

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• 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

Example epoch fMRI activation dataset: Auditory stimulation

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

seconds

model specification

f MRI time series statistic image or

SPM

Voxel by voxel statistics…

voxel time series

parameter estimation

hypothesis

statistic

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 the General Linear Model

assume normality to account for serial correlations:

Generalised Linear Model

( )01

11201

ˆ nn

YYt+

−= ••

σ

…e.g. two-sample t-test?

  standard t-test assumes independence ⇒ ignores temporal autocorrelation!

voxel time series

t-statistic image SPM{t}

compares size of effect to its error standard

deviation

Image intensity

Regression example…

 correlation: test H0: ρ = 0 equivalent to test H0: α = 0

 two-sample t-test: test H0: µ0 = µ1 equivalent to test H0: α = 0

box-car reference function voxel time series

Ys = µ + α f(ts) + εs

= α µ + + error

εs ~ N(0,σ2) f(ts) = 0 or 1

  t-statistic for H0: α = 0

 can extend to account for temporal autocorrelation!

…revisited

= α µ + +

εs = α µ + + f(ts) 1 Ys

error

× ×

General Linear Model…

•  fMRI time series: Y1 ,…,Ys ,…,YN –  acquired at times t1,…,ts,…,tN

•  Model: Linear combination of basis functions Ys = β1

f 1(ts ) + …+ βl f l(ts ) + … + βL f L(ts ) + εs •  f l (.): basis functions

–  “reference waveforms” –  dummy variables

•  βl : parameters (fixed effects)

–  amplitudes of basis functions (regression slopes)

•  εs : residual errors: εs ~ N(0,σ2)

–  identically distributed –  independent, or serially correlated (Generalised Linear Model → GLM)

Design matrix formulation…

Ys = β1 f 1(ts ) + …+ βl f l(ts ) + … + βL f L(ts ) + εs s = 1, …,N

Y1 = β1 f 1(t1 ) +…+ βl f l(t1 ) +…+ βL f L(t1 ) + ε1

: : : : : : : : Ys = β1

f 1(ts ) +…+ βl f l(ts ) +…+ βL f L(ts ) + εs : : : : : : : : YN = β1

f 1(tN ) +…+ βl f l(tN ) +…+ βL f L(tN ) + εN

Y1 f 1(t1 ) … f l(t1 ) … f L(t1 ) β1 ε1 : : … : … : : : Ys = f 1(ts ) … f l(ts ) … f L(ts ) βl + εs : : … : … : : : YN f 1(tN ) … f l(tN ) … f L(tN ) βL εN Y = X × β + ε

Y = X × β + ε N × 1 N × L L × 1 N × 1

data

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Box-car regression… (revisited)

= α µ + +

εs = α µ + + f(ts) 1 Ys × ×

error

Box car regression: design matrix…

α

µ

= +

ε = β + Y X

×

×

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

=

α µ β3 β4 β5 β6 β7 β8 β9

+

ε = β + Y X ×

Example: a line through 3 points…

Yi = α xi + µ + εi i = 1,2,3

Y1 = α x1 + µ × 1 + ε1 Y2 = α x2 + µ × 1 + ε2 Y3 = α x3 + µ × 1 + ε3

Y1 x1 1 α ε1 Y2 = x2 1 + ε2 Y3 x3 1 µ ε3 Y = X β + ε

simple linear regression

parameter estimates µ & α

fitted values Y1 , Y2 , Y3

residuals e1 , e2 , e3

x

Y

×(x1, Y1)

××

(x2, Y2)

(x3, Y3) µ̂

^ α 1

^ ^ ^ ^ ^

dummy variables

Geometrical perspective…

(1,1,1)

(x1, x2, x3)

(Y1, Y2, Y3)

design space

Y

O

xα

xµ

x1 1 Y = α × x2 + µ × 1 + ε x3 1

Y = α × xα + µ × xµ + ε

Estimation, geometrically…

(Y1, Y2, Y3)

design space

Y

(Y1,Y2,Y3) ^ ^ ^

e = (e1,e2,e3)T

Y ̂

^ µ × xµ α × xα ^

xα

xµ

Estimation, formally…

Residuals

Residual sum of squares

Minimised when

…but this is the lth row of

Consider parameter estimates Giving fitted values

so the least squares estimates satisfy the normal equations

giving

(obviously!)

Inference, geometrically… Inference, geometrically…

(Y1, Y2, Y3)

design space

Y

(Y1,Y2,Y3) ^ ^ ^

e = (e1,e2,e3)T

^ µ × xµ α × xα ^

xα

xµ

model: Yi = α xi + µ + εi

null hypothesis: e.g. H0: α = 0 (zero slope…)

i.e. does xα explain anything? (after xµ)

Inference, formally…

For any linear compound of the parameter estimates:

So hypotheses can be assessed using:

Further (independently):

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

Suppose the model can be partitioned…

“Extra sum-of-squares”

Multivariate perspective…

ε = β + Y X

data matrix

desi

gn m

atri

x

parameters errors + ? = × ? voxels

scans

 estimate

β̂

% residuals

estimated component

fields

parameter estimates

“Image regression”

variance σ2

estimated variance

%

÷%=%

fMRI box car example…

=

α µ β3 β4 β5 β6 β7 β8 β9

+

ε = β + Y X

…fitted

raw fMRI time series

scaled for global changes

adjusted for global & low Hz effects

residuals fitted “high-pass filter”

fitted box-car

hæmodynamic response

power spectrum

•  Drifts –  physical –  physiological

•  Aliased high frequency effects e.g.

–  cardiac (~1 Hz) –  respiratory (~0.25 Hz)

Serial correlations...

Y = Xβ + ε ε ~ N(0,σ2V) –intrinsic autocorrelation V

Problem: Estimate σ2V at each voxel and make inference about cTβ

Model: Model V as linear combination of m variance components

V = λ1 Q1 + λ2 Q2 + … + λm Qm Assumptions: V is the same at each voxel σ2

is different at each voxel

Q1 Q2 Example: For one fMRI session, use 2 variance components. Choice of Q1 and Q2 motivated by autoregressive model of order 1 plus white noise (AR(1)+wn)

Serial correlations…estimation

Estimation: β = (XTX)– XT Y – unbiased, ordinary least squares estimate

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

Important: Data Yk must be high-pass filtered.

Model CY as CY = X β βTXT + Σ λiQi and estimate hyperparameters λi using Restricted Maximum Likelihood (ReML)

Estimate V by V = N Σ λiQi /trace(Σ λiQi ) Estimate σ2 at each voxel in the usual way by

σ2 = (RY) T(RY) / trace(R V) – unbiased where R = I – X(XTX)– X

CY

CY

^

^

^ ^

^

Serial correlations…inference

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

standard deviation: t = cTβ / std[cTβ] where std[cTβ] = sqrt(σ2 c' (X TX)– XT V X (XTX)– c ) … but … std[cTβ] is not a χ2 variable because of V Find approximating χ2 distribution using Satterthwaite approximation:

Var[σ2] = 2σ4 trace(R V R V ) / trace(R V )2

ν = 2E[σ2]2/Var[σ2] = trace(RV)2 / trace(RVRV) – effective degrees of freedon –  Worsley & Friston (1995) “Analysis of fMRI time series revisited – again”

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

^ ^

^ ^

^

Inference — contrasts — SPM{t}

SPM{t}

contrast — linear combination of parameters: cTβ

c = +1 0 0 0 0 0 0 0

T =

contrast of estimated

parameters

variance estimate

t-test H0: cTβ = 0 e.g. activation: box-car amplitude > 0 ?

correct variance estimate & degrees of freedom for temporal autocorrelation

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

Inference — F-tests — SPM{F}

SPM{F}

H0: β2 = 0

F = error

variance estimate

e.g. Does HPF basis set model anything ?

multiple linear hypotheses

additional variance

accounted for by effects of

interest

X2 (β2)

correct variance estimate & degrees of freedom for temporal autocorrelation

0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

H0: cTβ = 0 c =

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

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

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” NeuroImage 12: 196-208

Zarahn E, Aguirre GK, D'Esposito M (1997) “Empirical Analyses of BOLD fMRI Statistics” NeuroImage 5:179-197

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

The General Linear Model and Statistical Parametric Mapping

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