The General Linear Model and Statistical Parametric...
Transcript of The General Linear Model and Statistical Parametric...
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
M p
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s (e
vent
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ogy)
• 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
vect
or
desig
n m
atrix
ve
ctor
of p
aram
eter
s er
ror v
ecto
r
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
Ch4 Ch3