signal processing for functional brain imaging:...
TRANSCRIPT
Signal Processing for Functional Brain Imaging:General Linear Model (2)
Dimitri Van De VilleMedical Image Processing Lab, EPFL/[email protected]
March 7, 2013
Overviewn GLM method (part 1, last week 28.02.13)
n intuitive explanationn matrix algebra explanation
n model generationn parameter estimationn hypothesis testing
2
n hypothesis testing continuedn t-test and F-test
n multiple comparisonsn enriching the model
n accounting for imaging artifacts, physiological noisen from single-subject to group-level analysis
n GLM method (part 2, today 07.03.13 )
get error
significant?
Null hypothesis H0: cT� = 0
then cT � is asymptotically
normal N (cT�,�2cT (XTX)�1c)
and t = cT �
�p
cT (XTX)�1cfollows
Student t-distribution with N � L
degrees of freedom
From GLM fitting to hypothesis testing
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get data
model get effect size
��� =
�
⇧⇤0.830.162.98
⇥
⌃⌅
t = = 6.42
contrast
cT = [1 0 0]
time
Hypothesis testing: t-test
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Null hypothesis expresses “no effect” (i.e., true cT� is 0)
H0 : cT� = 0
t = cT �/q
⇥2 cT (XTX)�1c follows Student t-distribution assuming H0
reject H0 if t � T , where the �-level is the acceptable false positive rate:
� = P (t0 � T ) (one-sided t-test)
p-value indicates the assessment of t assuming H0:
p = P (t0 � t)
specificity: risk of false positives (type I errors)
sensitivity: risk of false negatives (type II errors)
Null hypothesis acceptation/rejection controls specificity only
useful as “evidence of presence”, not “evidence of absence” (neurosurgeon!)
RejectH0
H0false
H0true
F-test - putting the same question...
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fit model
��� =
�
⇧⇤0.830.162.98
⇥
⌃⌅
2=
F = = 41.2-
fit reduced model
��� =
��0.253.40
⇥
2=
Partitioning into two blocks of regressors
Consider reduced model by design matrix X0
y = X� + e
y0 = X0�0 + e0
Null hypothesis H0 expresses “no improvement of X over X0”
F =eT e�e0
T e0L�L0
eT eN�L
follows F-distribution (L� L0, N � L) assuming H0
reject H0 if F � T , where the ↵-level is the acceptable false positive rate
two-sided test if reduced model removes one regressor
Hypothesis testing: F-test
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... or putting more general questions
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fit model
fit reduced model��� =
�2.87
⇥
F-contrast
cT =
�1 0 00 1 0
⇥
F2,57 = 60.8
��� =
�
⇧⇤0.74�0.712.86
⇥
⌃⌅
More flexible by contrast matrix
reduced model can be made up by linear combinations of regressors
avoid reparametrization of model
F =eT e�e0
T e0L�L0
eT eN�L
=yTMy
yTRy
N � L
L� L0=
�TXTMX�
yTRy
N � L
L� L0⇠ F (L�L0, N�L)
model to remove (specified by C): Xc = XC
reduced model: X0 = XC0 where C0 = IL � CC+(residual forming of
contrast matrix)
R = IN �XX+
R0 = IN �X0X+0
M = R0 �R
Hypothesis testing: F-test (2)
8X
y
X0
Xc
RyR0y
My
can be computed efficiently
only needs to be computed once
Hypothesis testing: F-test (3)
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✔
✘
✔
Use of F-testinterested in activation for “faces or objects”; some arrogant voxel is activating
during “faces”, deactivating during “objects”
– t-contrast: cT = [1 1 0]
– F -contrast: CT =
"1 0 0
0 1 0
#
any difference between three conditions (like ANOVA)
– F -contrast: CT =
"1 �1 0
1 0 �1
#
Multiple comparisons
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Mass univariate testing (V =10K-100K intracranial voxels)
E[FP] = �V , so false positives should be controlled adequately!
Family-wise error rate: �FWE = P ([Vk=1t
0k � T )
Bonferroni correction: assuming independent observations
�FWE = 1� (1� �)V ⇡ �V
to obtain �FWE, use �FWE/V at the individual tests
high specificity, low sensitivity since neglecting spatial correlation
therefore, too conversative
can be applied locally (if ROI is chosen a priori)
Glance at Gaussian random field theory
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≈
1.8mm x 1.8mm FWHM=6mm
Consider contrast as lattice representationof continuous Gaussian random field
spatially smooth data with 3D Gaussian kernel,typical full-width-at-half-maximum (FWHM) about 6–12 mm
Euler characteristic:�T =topological measure #blobs � #holes
Assuming H0 and high T , we have
P (⇧Vk=1tk > T ) = P (maxk(tk) > T )
= P (one or more blobs)⇤ P (�T ⇥ 1) (no holes)⇤ E[�T ] (one blob)
can be further approximated assuming(sufficient) spatial smoothness
Gaussian random field theory
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Advantages:
increased sensitivity
decreased inter-subject variability (group studies!)
Limitations:
requires sufficient smoothness: typically FWHM like 3� 4⇥ voxel size
smoothness needs to be estimated
– bias if not sufficiently smooth
several approximations in cascade (high T )
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SPM FWE
[VDV et al., IEEE JSTSP, 2008]
5% corrected
Bonferroni
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Bonferroni
SPM FWE
5% corrected
[VDV et al., IEEE JSTSP, 2008]
To correct or not to correct?
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To correct or not to correct?
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To correct or not to correct?
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Model specificationn From stimuli to modeled BOLD response
n blocks (epochs)
n events
n Convolution is performed in microtime (see exercise)18
⊗ =
⊗ =h(t;w)
Enriching the model
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⊗ =
Hemodynamic variations
subject-dependent, regional changes, habituation and anticipation effects
approximate h(t + �t;w + �w), where w is dispersion
h(t + �t; w + �w) � h(t; w) + �t�h
��t+ �w
�h
��w
More involved techniques using Volterra kernels
Enriching the model (2)n Low-frequency components
n truncated DCT-basisn act like a high-pass filtern scanner driftsn physiological fluctuations (aliased)n intrinsic brain activity
n Add nuissance regressorsn realignment
parametersn spike-
regressorsto “cancel”bad scans
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Parameter estimation (revisited)
21[Bullmore et al., 1996]
GLM with correlated noise, introduce filter S
y = X� + e, where e : N (0,�2V )
Sy = SX� + Se|{z}e0
, where e0 : N (0,�2SV ST )
normal equations: (SX)TSy = (SX)T (SX)�
estimate: � = (SX)+Sy
assume known covariance matrix V = KKT
– e.g., parametric form of noise model (1/f-noise, autoregressive model)
then BLUE is obtained for S = K�1
Time
n Analysis of (many) timecourses
From single-subject analysis...
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... to group-level analysisn Fixed effects analysis
n concatenate data and design matrices of subjectsn inference on the observed group
n Random effects analysisn estimate contrast of interest for individual subjects (1st level)n enter contrast in “basic model” and re-estimate (2nd level)n inference on the population from which group is sample
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Subjec
ts
n Analysis of (many) subjects
... to group-level analysis (RFX)
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...
One-sample t-test!voxel’s intensity
subject
significant?
one-
sam
ple
t-tes
t
two-
sam
ple
t-tes
t
paire
d t-t
est
Conclusionn General linear model
n many ways of testing the fitted parametersn t-test, F-test
n conceptually simple, yet powerful and flexiblen many tricks to “enrich” the modeln generalizes “basic” models
n Multiple comparisons problemn Gaussian smoothing is state-of-the-art
n degrades spatial resolution, n improves sensitivity, reduces inter-subject variability
n Alternatives n FP rates (e.g., false discovery rate)n Spatial modeling (e.g., wavelets,... )n Bayesian inferences (alternative hypothesis made explicit)n ...
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