Download Voxel by voxel statistics…

Contents
Contents
DISCOS
ège, 2009
Li
DISCOS SPM
SPM course,
course, CRC,
CRC, Liè
Liège,
2009
Multiple
Multiple comparison
comparison problem
problem
•• Recap
Recap &
& Introduction
Introduction
•• Inference
Inference &
& multiple
multiple
comparison
comparison
•• «
« Take
Take home
home »
» message
message
C. Phillips, Centre de Recherches du Cyclotron, ULg, Belgium
Based on slides from: T. Nichols
image data
parameter estimates
Statistical
Parametric
Map
corrected p-values
Voxel
…
Voxel by
by voxel
voxel statistics
statistics…
model specification
parameter estimation
«
«model
modelfitting
fitting
«
«statistic
statisticimage
image
Random
Randomeffect
effectanalysis
analysis
smoothing
smoothing
Dynamic
Dynamiccausal
causal
modelling,
modelling,
Functional
&
Functional &effective
effective
connectivity,
connectivity,PPI,
PPI,...
...
anatomical
reference
kernel
design
matrix
hypothesis test
correction
correctionfor
for
multiple
multiplecomparisons
comparisons
Time
normalisation
normalisation
General
GeneralLinear
LinearModel
Model
e
Ti m
realignment
realignment&&
motion
motion
correction
correction
single
single voxel
voxel
time
time series
series
statistic
Intensity
statistic image or SPM
Contents
Contents
General
General Linear
Linear Model
Model (in
(in SPM)
SPM)
Auditory words
every 20s
•• Recap
Recap &
& Introduction
Introduction
•• Inference
Inference &
& multiple
multiple
comparison
comparison
(Orthogonalised)
Gamma functions ƒi(u)
of peristimulus time u
SPM{F}
•• Single/multiple
Single/multiple voxel
voxel inference
inference
•• Family
wise
error
rate
Family wise error rate (FWER)
(FWER)
•• False
False Discovery
Discovery rate
rate (FDR)
(FDR)
Sampled every TR = 1.7s
•• SPM
SPM results
results
Design matrix, X
•• «
« Take
Take home
home »
» message
message
[ƒ1(u)⊗
(u)⊗x(t) |ƒ
|ƒ2(u)⊗
(u)⊗x(t) |...]
0
time {secs
}
{secs}
30
Inference
Inference at
at a
a single
single voxel
voxel
0.4
NULL hypothesis, H:
activation is zero
0.35
0.3
0.25
α = p(t>u|H)
0.2
0.15
0.1
0.05
0
−6
−4
−2
t-distribution
0
2
u=2
4
6
p-value:
probability of getting a
value of t at least as
extreme as u. If α is
small we reject the null
hypothesis H.
Hypothesis
Hypothesis Testing
Testing
•• Null
Null Hypothesis
Hypothesis H
H00
•• Test
statistic
T
Test statistic T
uα
–– tt observed
observed realization
realization of
of TT
•• αα level
level
––
––
––
Acceptable
Acceptable false
false positive
positive rate
rate
α
Level
>uαα || H
Level αα == P(
P( TT>u
H00 ))
Null
Distribution
of T
Threshold
αα
Threshold uuαα controls
controls false
false positive
positive rate
rate at
at level
level
•• PP-value
-value
––
––
Assessment
Assessment of
of tt assuming
assuming H
H00
P(
T
>
t
|
H
)
P( T > t | H00 )
t
•• Prob.
Prob.of
ofobtaining
obtainingstat.
stat.as
aslarge
large
or
orlarger
larger in
inaa new
new experiment
experiment
–– P(Data|Null)
) not
)
P(Data|Null
P(Null|Data
P(Data|Null)
not P(Null|Data)
P(Null|Data)
P-val
Null Distribution of T
Inference
Inference at
at a
a single
single voxel
voxel
0.4
NULL hypothesis, H:
activation is zero
0.35
0.3
0.25
α = p(t>u|H)
0.2
0.15
We can choose u to ensure
a voxel-wise significance level
of α.
0.1
0.05
0
−6
−4
−2
t-distribution
0
2
u=2
4
6
This is called an ‘uncorrected’
p-value, for reasons we’ll see
later.
We can then plot a map of
above threshold voxels.
What
What we
we need
need
•• Need
Need an
an explicit
explicit spatial
spatial model
model
•• No
No routine
routine spatial
spatial modeling
modeling methods
methods
exist
exist
–– High-dimensional mixture
High
High-dimensional
mixture modeling
modeling
problem
problem
–– Activations
’t look
don
Activations don’
don’t
look like
like Gaussian
Gaussian blobs
blobs
–– Need
realistic
shapes,
sparse
Need realistic shapes, sparse representation
representation
•• Some
, Penny
al.
Some work
work by
by Hartvig
Hartvig et
et al.,
al.,
Penny et
et al.
al.
What
’d like
What we
we’d
like
•• Don’
’t threshold,
Don
Don’t
threshold, model
model the
the signal!
signal!
–– Signal
?
location
Signal location?
location?
θˆMag.
•• Estimates
’s on
CI
Estimates and
and CI’
CI’s
on
((x,y,z)
x,y,z)
x,y,z) location
location
–– Signal
?
magnitude
Signal magnitude?
magnitude?
•• CI’
’s on
CI
CI’s
on %
% change
change
–– Spatial
?
extent
Spatial extent?
extent?
θˆLoc.
θˆExt.
space
•• Estimates
’s on
CI
Estimates and
and CI’
CI’s
on activation
activation volume
volume
•• Robust
to
choice
of
cluster
Robust to choice of cluster definition
definition
•• ...but
...but this
this requires
requires an
an explicit
explicit spatial
spatial
model
model
Real-life inference:
Real
Real-life
inference: What
What we
we get
get
•• Signal
Signal location
location
–– Local
Local maximum
maximum –– no
no inference
inference
–– Center-ofCenter
of-mass –– no
Center-of-mass
no inference
inference
•• Sensitive
-definingblob
defining-threshold
Sensitive to
to blobblob-defining-threshold
•• Signal
Signal magnitude
magnitude
–– Local
-values (&
’s)
CI
Local maximum
maximum intensity
intensity –– PP-values
(& CI’
CI’s)
•• Spatial
Spatial extent
extent
–– Cluster
-value, no
’s
CI
Cluster volume
volume –– PP-value,
no CI’
CI’s
•• Sensitive
-definingblob
defining-threshold
Sensitive to
to blobblob-defining-threshold
Voxel-level Inference
Voxel
Voxel-level
Inference
•• Retain
-level threshold
Retain voxels
voxels above
above αα-level
threshold uuαα
•• Gives
Gives best
best spatial
spatial specificity
specificity
–– The
The null
null hypothesis
hypothesis at
at aa single
single voxel
voxel can
can be
be
rejected
rejected
Cluster-level Inference
Cluster
Cluster-level
Inference
•• Two
-process
step
Two stepstep-process
––Define
Define clusters
clusters by
by arbitrary
arbitrary threshold
threshold uuclus
clus
––Retain
-level threshold
Retain clusters
clusters larger
larger than
than αα-level
threshold kkαα
uclus
uα
space
Significant
Voxels
space
No significant
Voxels
Cluster-level Inference
Cluster
Cluster-level
Inference
kα
Cluster
significant
kα
Set-level Inference
Set
Set-level
Inference
•• Count
Count number
number of
of blobs
blobs cc
•• Typically
Typically better
better sensitivity
sensitivity
•• Worse
Worse spatial
spatial specificity
specificity
–– Minimum
Minimum blob
blob size
size kk
–– The
The null
null hypothesis
hypothesis of
of entire
entire cluster
cluster is
is
rejected
rejected
–– Only
Only means
means that
that one
one or
or more
more of
of voxels
voxels in
in
cluster
cluster active
active
•• Worst
Worst spatial
spatial specificity
specificity
–– Only
Only can
can reject
reject global
global null
null hypothesis
hypothesis
uclus
uclus
space
space
Cluster not
significant
Cluster not
significant
kα
kα
Cluster
significant
k
k
Here c = 1; only 1 cluster larger than k
Sensitivity
Sensitivity and
and Specificity
Specificity
At u1
Reject
H True (o)
TN
FP
H False (x)
FN
TP
Sens=7/10=70%
Spec=9/10=90%
Eg. t-scores
from regions
that truly do and
do not activate
Sensitivity = TP/(TP+FN) = β
u1
u2
Specificity = TN/(TN+FP) = 1 - α
FP = Type I error or ‘error’
oooooooxxxooxxxoxxxx
FN = Type II error
α = p-value/FP rate/error rate/significance level
β = power
Inference
Inference for
for Images
Images
Noise
–– 1,000
1,000 multivariate
multivariate observations,
observations,
each
each with
with 100,000
100,000 elements
elements
–– 100,000
100,000 time
time series,
series, each
each
with
3
with 1,000
1,000 observations
observations
.
At u2
1,000
•• 4
-Dimensional Data
4-Dimensional
Data
.
Don’t
Reject
Sens=10/10=100%
Spec=7/10=70%
.
ACTION
TRUTH
fMRI
fMRI Multiple
Multiple Comparisons
Comparisons Problem
Problem
•• Massively
Massively Univariate
Univariate
Approach
Approach
–– 100,000
100,000 hypothesis
hypothesis
tests
tests
•• Massive
Massive MCP!
MCP!
2
1
Multiple
Multiple comparison
comparison problem
problem
Use of ‘uncorrected’ p-value, α=0.1
11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2%
Percentage of Null Pixels that are False Positives
Signal
Signal+Noise
Using an ‘uncorrected’ p-value of 0.1 will lead us
to conclude on average that 10% of voxels are
active when they are not.
This is clearly undesirable : multiple comparison
problem. To correct for this we can define a null
hypothesis for images of statistics.
9.5%
Multiple
Multiple Comparisons
Comparisons Problem
Problem
•• Which
Which of
of 100,000
100,000 voxels
voxels are
are sig.?
sig.?
Assessing
Assessing Statistic
Statistic Images
Images
Where’
’s the
Where
Where’s
the signal?
signal?
–– αα=0.05
=0.05 ⇒
⇒ 5,000
5,000 false
false positive
positive voxels
voxels
High Threshold
•• Which
Which of
of (random
(randomnumber,
number,say)
say)100
100 clusters
clusters
significant?
significant?
t > 5.5
Med. Threshold
Low Threshold
t > 0.5
t > 3.5
–– αα=0.05
=0.05 ⇒
⇒ 55 false
false positives
positives clusters
clusters
t > 0.5
t > 1.5
t > 2.5
t > 3.5
t > 4.5
t > 5.5
t > 6.5
Good Specificity
Poor Power
(risk of false
negatives)
Multiple
…
Multiple comparisons
comparisons…
Good Power
Solutions
Solutions for
for
Multiple
Multiple Comparison
Comparison Problem
Problem
••Threshold
Threshold at
at pp ?
?
t59
––expect
)% by
expect (100
(100 ×× pp)%
by
chance
chance
••Surprise
Surprise ?
?
––extreme
extreme voxel
voxel values
values
→
voxel level
→voxel
level inference
inference
p = 0.05
––big
big suprathreshold
suprathreshold
clusters
clusters
→
cluster level
→cluster
level inference
inference
––many
many suprathreshold
suprathreshold
clusters
clusters
→
set level
→set
level inference
inference
Gaussian
10mm FWHM
(2mm pixels)
Poor Specificity
(risk of false
positives)
••Power
Power &
& localisation
localisation
→
sensitivity
→sensitivity
→
spatial specificity
→spatial
specificity
•• A
False
A MCP
MCP Solution
Solution must
must control
control ““False
Positives”
”
Positives
Positives”
–– How
How to
to measure
measure multiple
multiple false
false positives?
positives?
•• Familywise
Familywise Error
Error Rate
Rate (FWER)
(FWER)
–– Chance
Chance of
of any
any false
false positives
positives
–– Controlled
, Random
Bonferroni
Controlled by
by Bonferroni,
Bonferroni,
Random Field
Field
Methods,
-parametric method
SnPM).
).
non
((SnPM).
Methods, nonnon-parametric
method (SnPM
•• False
False Discovery
Discovery Rate
Rate (FDR)
(FDR)
–– Proportion
Proportion of
of false
false positives
positives among
among rejected
rejected
tests
tests
Contents
Contents
•• Recap
Recap &
& Introduction
Introduction
•• Inference
Inference &
& multiple
multiple
comparison
comparison
•• Single/multiple
Single/multiple voxel
voxel inference
inference
•• Family
wise
error
rate
Family wise error rate (FWER)
(FWER)
Family
-wise Null
Family-wise
Null Hypothesis
Hypothesis
FAMILY-WISE NULL HYPOTHESIS:
Activation is zero everywhere
•• Family
Family of
of hypotheses
hypotheses
–– HHkk kk∈∈Ω
…,K}
{1,
Ω=={1,…
{1,…,K}
2
k
K
–– HHΩΩ==HH11∩
∩HH2…
…∩
∩HHk∩
∩HHK
If we reject a voxel null
hypothesis at any voxel, we
reject the family-wise null
hypothesis
•• Bonferroni
Random Field
correction/
Bonferroni correction/Random
correction/Random
Field Theory
Theory
•• Non-parametric approach
Non
Non-parametric
approach
•• False
False Discovery
Discovery rate
rate (FDR)
(FDR)
•• SPM
SPM results
results
•• «
« Take
Take home
home »
» message
message
FWE
FWE MCP
MCP Solutions:
Solutions:
Controlling
Controlling FWE
FWE w/
w/ Max
Max
•• FWE
FWE &
& distribution
distribution of
of maximum
maximum
FWE
FWE
== P(FWE)
P(FWE)
== P(
Tii ≥≥ uu}
} || H
P( ∪
∪ii {{T
Hoo))
== P(
P( max
maxii TTii≥≥ uu || H
Hoo))
•• 100(1-α)%ile of
100(1
100(1-α)%ile
of max
max dist
distnn controls
controls FWE
FWE
–– where
where
..
FWE
FWE == P(
P( max
maxii TTii≥≥ uuαα || H
Hoo)) == αα
1
uuαα == FF--1
(1(1-α)
max
max (1-α)
α
uα
A FP anywhere gives a Family Wise Error (FWE)
Family-wise error rate = ‘corrected’ p-value
Multiple
…
problem
Multiple comparison
comparison problem…
problem…
Example:
:
Example
Example:
Experiment
Experiment with
with 100,000
100,000 «
« voxels
voxels »
» and
and 40
40 d.f.
d.f.
type
=0.05 (5%
)⇒
risk
type II error
error αα=0.05
(5% risk)
risk)
⇒ ttαα =
= 1.68
1.68
100,000
⇒
5000
100,000 tt values
values
⇒
5000 tt values
values >
> 1.68
1.68
just
just by
by chance
chance !!
FWE
Familywise
Familywise Error
Error II test,
test, P
PFWE::
find
threshold
t
such
, in
that
find threshold tαα such that,
that,
in aa family
family of
of 100,000
100,000 tt
statistics,
, only
statistics
statistics,
only 5%
5% probability
probability of
of one
one or
or more
more tt
values
values above
above that
that threshold
threshold
Bonferroni correction:
simple method to find the new threshold
Random field theory:
more accurate for functional imaging
The
Bonferroni”
…
Bonferroni” correction…
correction
The ““Bonferroni”
correction…
Given
Given
•• aa family
family of
of N
N independent
independent voxels
voxels and
and
•• aa voxel-wise error
voxel
voxel-wise
error rate
rate vv
The
The probability
probability that
that all
all tests
tests are
are below
below the
the threshold,
threshold,
NN
i.e.
true
:
(1v)
i.e. that
that H
Hoo is
is true : (1- v)
The
-Wise Error
corrected’
Family
corrected’ error
The FamilyFamily-Wise
Error rate
rate (FWE)
(FWE) or
or ‘‘corrected’
error
rate
rate αα is
is
αα =
-v)NN
(1
= 11 –– (1(1-v)
~
~ Nv
Nv (for
(forsmall
small v)
v)
Therefore,
Therefore, to
to ensure
ensure aa particular
particular FWE
FWE we
we choose
choose
vv =
= α/N
α/N
AA Bonferroni
Bonferroni correction
correction is
is appropriate
appropriate for
for independent
independent tests.
tests.
Use of ‘uncorrected’ p-value, α=0.1
Use of ‘corrected’ p-value, α=0.1
FWE
The
Bonferroni”
…
Bonferroni” correction…
correction
The ““Bonferroni”
correction…
Experiment
Experiment with
with N
N=
= 100,000
100,000 «
« voxels
voxels »
» and
and 40
40 d.f.
d.f.
–– vv =
unknown
corrected
probability
threshold,
= unknown corrected probability threshold,
–– find
= 0.05
0.05
find vv such
such that
that family-wise
family-wise error
error rate
rate αα =
Bonferroni
Bonferroni correction:
correction:
–– probability
probability that
that all
all tests
tests are
are below
below the
the threshold,
threshold,
–– Use
Use vv =
= αα // N
N
–– Here
Here v=0.05/100000=0.0000005
v=0.05/100000=0.0000005
⇒
⇒ threshold
threshold tt =
= 5.77
5.77
Interpretation:
:
Interpretation
Interpretation:
Bonferroni
Bonferroni procedure
procedure gives
gives aa corrected
corrected pp value,
value,
i.e.
i.e. for
for aa tt statistics
statistics =
= 5.77,
5.77,
–– uncorrectd
uncorrectd pp value
value =
= 0.0000005
0.0000005
–– corrected
corrected pp value
value =
= 0.05
0.05
““Bonferroni”
Bonferroni” correction
correction &
&
independent
independent observations
observations
100 by 100 voxels, with a z value.
10000 independent measures
Fix the PFWE = 0.05, z threshold ?
100 by 100 voxels, with a z value.
100 independent measures
Fix the PFWE = 0.05, z threshold ?
Bonferroni:
v = 0.05 / 10000 = 0.000005
⇒ threshold z = 4.42
Bonferroni:
v = 0.05 / 100 = 0.0005
⇒ threshold z = 3.29
v=α/ni where ni is the number of independent observations.
““Bonferroni”
Bonferroni” correction
correction &
&
smoothed
smoothed observations
observations
Random
Random Field
Field Theory
Theory
•• Consider
Consider aa statistic
statistic image
image as
as aa lattice
lattice
representation
representation of
of aa continuous
continuous random
random field
field
•• Use
Use results
results from
from continuous
continuous random
random field
field theory
theory
100 by 100 voxels, with a z
value.
10000 independent measures
Fix the PFWE = 0.05, z threshold ?
Bonferroni:
v = 0.05/10000 = 0.000005
⇒ threshold z = 4.42
100 by 100 voxels, with a z
value.
How many independent
measures ?
Fix the PFWE = 0.05, z threshold ?
Bonferroni ?
Euler
Euler Characteristic
Characteristic (EC)
(EC)
Topological
Topological measure
measure
–– threshold
threshold an
an image
image at
at uu
–– excursion
excursion set
set AAuu
−− χχ(
(Α
Αuu)) =
=#
# blobs
blobs -- #
# holes
holes
-- At
(Α
At high
high u,
u,χχ(
Αuu)) =
=#
# blobs
blobs
Reject
Reject H
HΩΩ if
if Euler
Euler
characteristic
-zero
non
characteristic nonnon-zero
αα ≈≈ Pr(χ
χ( Α
Pr(
Pr(χ(
Αuu))>> 00 ))
Expected
Expected Euler
Euler chararcteristic
chararcteristic
≈≈ pp–value
–value
αα ≈≈ EE[[χ(
χ( Α
Αuu)]
)]]
(at
(at high
high u)
u)
Lattice
representation
Euler
…
Euler characteristic
characteristic…
Euler characteristic (EC) ≈ # blobs in a
thresholded image.
(True only for high threshold)
EC = function of
• threshold used
• number of resels
where resels (« resolution
elements »)~ number of
independent obsevations
⇒ E[EC] ≈ PFWE
Euler
…
Euler characteristic
characteristic…
Euler
…
Euler characteristic
characteristic…
Euler characteristic (EC) ≈ # blobs in a
thresholded image.
(True only for high threshold)
EC = function of
• threshold used
• number of resels
Threshold z-map
at 2.75
Threshold z-map
at 2.50
EC = 1
EC = 3
where resels (« resolution
elements »)~ number of
independent obsevations
⇒ E[EC] ≈ PFWE
For a threshold zt
at 2.50, E[EC] = 1.9
at 2.75, E[EC] = 1.1
Expected
…
characteristic
Expected Euler
Euler characteristic…
characteristic…
EE[[χ(
(Ω) √√ ||Λ|
Λ| (u
u 22--1)
χ( Α
1) exp(-u 22/2)
2π)22
exp
/2
Αuu)]
)]] ≈≈ λλ(Ω)
(u
exp(-u
/2) // (2
(2π)
33
–– Ω
→
large
search
region
Ω
⊂
R
Ω
→ large search region
Ω⊂R
–– λλ(Ω)
(Ω) →
→ volume
volume
–– √√|Λ|
|Λ| →
→ smoothness
smoothness
3
–– AAuu
→
AAuu == {x
→ excursion
excursion set
set
{x ∈
∈ RR3 :: Z(x)
Z(x) >> u}
u}
3
–– Z(x)
Z(x) →
→ Gaussian
Gaussian random
random field
field xx ∈
∈ RR3
+
+ Multivariate
Multivariate Normal
Normal Finite
Finite Dimensional
Dimensional distributions
distributions
+
+ continuous
continuous
+
+ strictly
strictly stationary
stationary
Au
+
(0,1)
N
0,1)
+ marginal
marginal N(
N(0,1)
+
+ continuously
continuously differentiable
differentiable
+
+ twice
twice differentiable
differentiable at
at 00
+
+ Gaussian
Gaussian ACF
ACF
Ω
instead of 3 and 1 as in example
Unified
Unified Theory
Theory
•• General
General form
form for
for expected
expected Euler
Euler characteristic
characteristic
•• χχ22,,FF,, &
& ttfields
fields
Au
•• restricted
restricted search
search regions
regions
αα == Σ R
Ω) ρρdd ((u)
u)
Rdd ((Ω)
Rd (Ω), RESEL count
depends on :
• the search region
Ω
ρd (υ): EC density
depends on :
• type of field (eg. Gaussian, t)
– how big, how smooth, • the threshold, u.
what shape ?
(at
(atleast
leastnear
nearlocal
localmaxima)
maxima)
Worsley et al. (1996), HBM
Unified
Unified Theory
Theory
Estimated
Estimated component
component fields
fields
Au
ρ0(u)
ρ1(u)
ρ2(u)
ρ3(u)
ρ4(u)
• R3(Ω)=resel volume
?
^
β
parameter
« estimate
E.g. Gaussian RF:
• R2(Ω)=resel surface area
errors
scans
d-dimensional EC
density :
• R1(Ω)=resel diameter
× parameters? +
Ω
ρd (u),
• R0(Ω)=χ(Ω) Euler
characteristic of Ω
=
data matrix
αα == Σ R
Ω) ρρdd ((u)
u)
Rdd ((Ω)
Rd (Ω), d-dimensional
Minkowski functional
of Ω
voxels
design
matrix
•• General
General form
form for
for expected
expected Euler
Euler characteristic
characteristic
•• χχ22,,FF,, &
& ttfields
fields •• restricted
restricted search
search regions
regions
« estimates «
residuals
= 1- Φ(u)
= (4 ln2)1/2 exp(-u2/2) / (2π)
= (4 ln2) exp(-u2/2) / (2π)3/2
Each row is
an estimated
component field
= (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2π)2
= (4
ln2)2
(u3
-3u)
exp(-u2/2)
/
(2π)5/2
÷ estimated variance
=
estimated
component
fields
Worsley et al. (1996), HBM
Random
Random Field
Field Theory
Theory
Smoothness
Smoothness Parameterization
Parameterization
••Smoothness
|Λ|
Smoothness √√|Λ|
•• RESELS
RESELS
––
––
––
Smoothness,
...
Smoothness, PRF,
PRF, resels
resels...
–– variance-covariance matrix
variance
variance-covariance
matrix of
of
partial
(possibly location
location
partial derivatives
derivatives (possibly
Resolution
olution Elements
ements
Res
El
Resolution
Elements
11 RESEL
RESEL ==FWHM
FWHMxx ××FWHM
FWHMyy ××FWHM
FWHMzz
RESEL
RESELCount
CountRR
dependent)
dependent)
3/2 (Ω) / ( FWHM × FWHM × FWHM )
•• RR==λλ(Ω)
(Ω) √√|Λ
|Λ|| ==(4log2)
(4log2)3/2λλ(Ω)
/ ( FWHMxx × FWHMyy × FWHMzz )
•• Volume
Volumeof
ofsearch
searchregion
regionin
inunits
unitsof
ofsmoothness
smoothness
–– Eg:
: 10
Eg
voxels
Eg:
10 voxels,
voxels,, 2.5
2.5 FWHM
FWHM44RESELS
RESELS
1
2
1
3
4
5
6
2
7
3
8
9
10
Λ
[
⎛ var[∂∂xe ] cov ∂∂ex , ∂∂ye
⎜
var ∂∂ye
Λ = ⎜ cov ∂∂ex , ∂∂ye
⎜
∂e ∂e
∂e ∂e
⎝ cov[∂x , ∂z ] cov ∂y , ∂z
[
]
[
[]
]
]
cov[∂∂ex , ∂∂ez ]⎞
⎟
cov ∂∂ey , ∂∂ez ⎟
⎟
var[∂∂ez ] ⎠
[
]
•• Point
Point Response
Response Function
Function PRF
PRF
4
–– ΣΣ–– kernel
kernel var/cov
var/cov matrix
matrix
–– ACF
Σ
ACF 22Σ
1
–– ΛΛ ==(2Σ
Σ) -1
(2
(2Σ)
(8ln(2))
⇒
FWHM ff== σσ √√(8ln(2))
⇒FWHM
ffxx 00 00
–– ΣΣ== 00 ffyy 00
11
00 00 ffzz 8ln(2)
8ln(2)
ignoring
ignoringcovariances
covariances
3/2 // (f
⇒
|Λ| == (4ln(2))
fxx ×× ffyy×× ffzz))
((f
⇒ √√|Λ|
(4ln(2))3/2
••Resolution
olution Element
ement ((RESEL
Res
El
Resolution
Element
RESEL))
–– Resel
fxx×× ffyy×× ffzz))
Resel dimensions
dimensions ((f
–– RR33((Ω)
Ω) ==λλ(Ω)
(Ω) //(f
fxx×× ffyy×× ffzz))
((f
•• Beware
Beware RESEL
RESEL misinterpretation
misinterpretation
–– RESEL
number of
things’
”
things’ in
image
RESEL are
arenot
not““number
of independent
independent ‘‘things’
in the
the image”
image”
•• See
. in
Meth
SeeNichols
Nichols&
&Hayasaka,
Hayasaka,2003,
2003,Stat.
Stat.Meth.
Meth.
inMed.
Med.Res.
Res.
..
••Gaussian
Gaussian PRF
PRF
ififstrictly
strictlystationary
stationary
•• Full
Full Width
Width at
at Half
Half Maximum
Maximum
FWHM.
FWHM.Approximate
Approximate the
the peak
peak of
of
the
the Covariance
Covariance function
function with
with aa
Gaussian
Gaussian
EE[χ(Α
[χ(Α
Αu)])]
u 22--1)
1) exp(-u 22/2)
π)22
] = R (Ω) (4ln(2))3/2
3/2((u
exp(
(2
exp(-u
/2)//(2π
(2π)
u = R33(Ω) (4ln(2))
≈≈RR3((Ω)
Ω) ((11––ΦΦ(u)
(u))) for
forhigh
highthresholds
thresholds uu
3
RFT
RFT Assumptions
Assumptions
••Model
Model fit
fit &
& assumptions
assumptions
–– valid
valid distributional
distributional
results
results
••Multivariate
Multivariate normality
normality
–– of
of component
component images
images
••Covariance
Covariance function
function of
of
component
component images
images must
must
be
be
-- Can
Can be
be nonstationary
nonstationary
-- Twice
Twice differentiable
differentiable
Smoothness
Smoothness
smoothness
smoothness »» voxel
voxel size
size
Random
Random Field
Field Intuition
Intuition
•• Corrected
-value for
P
Corrected PP-value
for voxel
voxel value
value tt
PPcc == P(max
P(max TT >> tt))
≈≈ E(χ
χtt))
E(
E(χ
1/2 tt22exp(-t22/2)
≈≈ λλ(Ω)
(Ω) |Λ
Λ|1/2
exp(
||Λ|
exp(-t
/2)
lattice
lattice approximation
approximation
smoothness
smoothness estimation
estimation
practically
practically
FWHM
FWHM ≥≥ 33 ×× VoxDim
VoxDim
otherwise
otherwise
conservative
conservative
““Typical”
Typical”
Typical” applied
applied smoothing:
smoothing:
Single
Single Subj
Subj fMRI:
fMRI: 6mm
6mm
PET:
PET: 12mm
12mm
Multi
-12mm
88-12mm
Multi Subj
Subj fMRI:
fMRI: 8PET:
PET: 16mm
16mm
Level
Level of
of smoothing
smoothing should
should
actually
actually depend
depend on
on what
what
you’
’
re
looking
for…
…
you
for
you’re looking for…
Small
Small Volume
Volume Correction
Correction
•• Statistic
Statistic value
value tt increases
increases
–– PPcc decreases
decreases (but
(but only
only for
for large
large tt))
•• Search
)
Search volume
volume increases
increases (bigger
(biggerΩ
Ω)
–– PPcc increases
increases (more
(more severe
severe MCP)
MCP)
•• Smoothness
Smoothness increases
increases (roughness
(roughness|Λ
|Λ||
1/2
1/2decreases)
decreases)
–– PPcc decreases
decreases (less
(less severe
severe MCP)
MCP)
Resel
Resel Counts
Counts for
for Brain
Brain Structures
Structures
SVC
SVC =
= correction
correction for
for multiple
multiple comparison
comparison in
in aa
user’
’s defined
of interest’
’.
user
interest
user’s
defined volume
volume ‘‘of
interest’.
Shape and size of
volume become
important for small or
oddly shaped volume !
Example of SVC (900 voxels)
• compact volume: samples
from maximum 16 resels
• spread volume: sample
from up to 36 resels
⇒ threshold higher for
spread volume than
compact volume.
FWHM=20mm
(1) Threshold depends on Search Volume
(2) Surface area makes a large contribution
Contents
Contents
Summary
Summary
•• We
We should
should correct
correct for
for multiple
multiple comparisons
comparisons
–– We
We can
can use
use Random
Random Field
Field Theory
Theory (RFT)
(RFT) or
or other
other methods
methods
•• RFT
RFT requires
requires
–– aa good
good lattice
lattice approximation
approximation to
to underlying
underlying multivariate
multivariate
Gaussian
Gaussian fields,
fields,
–– that
that these
these fields
fields are
are continuous
continuous with
with aa twice
twice differentiable
differentiable
correlation
correlation function
function
•• Recap
Recap &
& Introduction
Introduction
•• Inference
Inference &
& multiple
multiple
comparison
comparison
•• Single/multiple
Single/multiple voxel
voxel inference
inference
•• Family
wise
error
rate
Family wise error rate (FWER)
(FWER)
•• To
To aa first
first approximation,
approximation, RFT
RFT is
is aa Bonferroni
Bonferroni correction
correction
using
using RESELS.
RESELS.
•• We
We only
only need
need to
to correct
correct for
for the
the volume
volume of
of interest.
interest.
•• Depending
Depending on
on nature
nature of
of signal
signal we
we can
can trade-off
trade-off
anatomical
anatomical specificity
specificity for
for signal
signal sensitivity
sensitivity with
with the
the use
use
of
of cluster-level
cluster-level inference.
inference.
Nonparametric
Nonparametric
Permutation
Permutation Test
Test
•• Nonparametric
Nonparametric methods
methods
–– Use
Use data
data to
to find
find
distribution
distribution of
of statistic
statistic
under
under null
null hypothesis
hypothesis
–– Any
Any statistic!
statistic!
•• False
False Discovery
Discovery rate
rate (FDR)
(FDR)
•• SPM
SPM results
results
•• «
« Take
Take home
home »
» message
message
Permutation
Permutation Test
Test :: T
Toy
oy Example
Example
•• Data
Data from
from V1
V1 voxel
voxel in
in visual
visual stim.
stim. experiment
experiment
•• Parametric
Parametric methods
methods
–– Assume
Assume distribution
distribution of
of
statistic
statistic under
under null
null
hypothesis
hypothesis
•• Bonferroni
Random Field
correction/
Bonferroni correction/Random
correction/Random
Field Theory
Theory
•• Non-parametric approach
Non
Non-parametric
approach
A:
A: Active,
Active, flashing
flashing checkerboard
checkerboard B:
B: Baseline,
Baseline, fixation
fixation
66 blocks,
Just
blocks, ABABAB
ABABAB
Just consider
consider block
block averages...
averages...
5%
Parametric Null Distribution
A
B
A
B
A
B
103.00
90.48
99.93
87.83
99.76
96.06
•• Null
Null hypothesis
hypothesis H
Hoo
–– No
No experimental
experimental effect,
effect, AA &
& BB labels
labels arbitrary
arbitrary
•• Statistic
Statistic
5%
Nonparametric Null Distribution
–– Mean
Mean difference
difference
Permutation
Permutation Test
Test :: Toy
Toy Example
Example
•• Under
Under H
Hoo
–– Consider
Consider all
all equivalent
equivalent relabelings
relabelings
Permutation
Permutation Test
Test :: Toy
Toy Example
Example
•• Under
Under H
Hoo
–– Consider
Consider all
all equivalent
equivalent relabelings
relabelings
–– Compute
Compute all
all possible
possible statistic
statistic values
values
AAABBB
ABABAB
BAAABB
BABBAA
AAABBB 4.82
ABABAB 9.45
BAAABB -1.48
BABBAA -6.86
AABABB
ABABBA
BAABAB
BBAAAB
AABABB -3.25
ABABBA 6.97
BAABAB 1.10
BBAAAB 3.15
AABBAB
ABBAAB
BAABBA
BBAABA
AABBAB -0.67
ABBAAB 1.38
BAABBA -1.38
BBAABA 0.67
AABBBA
ABBABA
BABAAB
BBABAA
AABBBA -3.15
ABBABA -1.10
BABAAB -6.97
BBABAA 3.25
ABAABB
ABBBAA
BABABA
BBBAAA
ABAABB 6.86
ABBBAA 1.48
BABABA -9.45
BBBAAA -4.82
Permutation
Permutation Test
Test :: Toy
Toy Example
Example
•• Under
Under H
Hoo
–– Consider
Consider all
all equivalent
equivalent relabelings
relabelings
–– Compute
all
possible
Compute all possible statistic
statistic values
values
–– Find
Find 95%ile
95%ile of
of permutation
permutation distribution
distribution
AAABBB 4.82
ABABAB 9.45
BAAABB -1.48
BABBAA -6.86
AABABB -3.25
ABABBA 6.97
BAABAB 1.10
BBAAAB 3.15
AABBAB -0.67
ABBAAB 1.38
BAABBA -1.38
BBAABA 0.67
AABBBA -3.15
ABBABA -1.10
BABAAB -6.97
BBABAA 3.25
ABAABB 6.86
ABBBAA 1.48
BABABA -9.45
BBBAAA -4.82
Permutation
Permutation Test
Test :: Toy
Toy Example
Example
•• Under
Under H
Hoo
–– Consider
Consider all
all equivalent
equivalent relabelings
relabelings
–– Compute
all
possible
Compute all possible statistic
statistic values
values
–– Find
Find 95%ile
95%ile of
of permutation
permutation distribution
distribution
-8
-4
0
4
8
Permutation
Permutation Test
Test :: Toy
Toy Example
Example
•• Under
Under H
Hoo
–– Consider
Consider all
all equivalent
equivalent relabelings
relabelings
–– Compute
Compute all
all possible
possible statistic
statistic values
values
–– Find
Find 95%ile
95%ile of
of permutation
permutation distribution
distribution
Controlling
Controlling FWER:
FWER: Permutation
Permutation Test
Test
•• Parametric
Parametric methods
methods
–– Assume
Assume distribution
distribution of
of
max
max statistic
statistic under
under null
null
hypothesis
hypothesis
•• Nonparametric
Nonparametric methods
methods
AAABBB 4.82
ABABAB 9.45
BAAABB -1.48
BABBAA -6.86
AABABB -3.25
ABABBA 6.97
BAABAB 1.10
BBAAAB 3.15
AABBAB -0.67
ABBAAB 1.38
BAABBA -1.38
BBAABA 0.67
AABBBA -3.15
ABBABA -1.10
BABAAB -6.97
BBABAA 3.25
ABAABB 6.86
ABBBAA 1.48
BABABA -9.45
BBBAAA -4.82
Permutation
Permutation Test
Test &
& Exchangeability
Exchangeability
•• Exchangeability
Exchangeability is
is fundamental
fundamental
–– Def:
Def: Distribution
Distribution of
of the
the data
data unperturbed
unperturbed by
by
permutation
permutation
–– Under
Under H
H00,, exchangeability
exchangeability justifies
justifies permuting
permuting data
data
–– Allows
Allows us
us to
to build
build permutation
permutation distribution
distribution
•• Subjects
Subjects are
are exchangeable
exchangeable
–– Under
Under Ho,
Ho, each
each subject’s
subject’s A/B
A/B labels
labels can
can be
be flipped
flipped
•• Are
Are fMRI
fMRI scans
scans exchangeable
exchangeable under
under H
Hoo??
–– If
If no
no signal,
signal, can
can we
we permute
permute over
over time?
time?
5%
Parametric Null Max Distribution
–– Use
Use data
data to
to find
find
distribution
distribution of
of max
max statistic
statistic
under
null
hypothesis
under null hypothesis
–– Again,
Again, any
any max
max statistic!
statistic!
5%
Nonparametric Null Max Distribution
Permutation
Permutation Test
Test &
& Exchangeability
Exchangeability
•• fMRI
fMRI scans
scans are
are not
not exchangeable
exchangeable
–– Permuting
Permuting disrupts
disrupts order,
order, temporal
temporal autocorrelation
autocorrelation
•• Intrasubject
subject fMRI
Intra
Intrasubject
fMRI permutation
permutation test
test
–– Must
Must decorrelate
decorrelate data,
data, model
model before
before permuting
permuting
–– What
What is
is correlation
correlation structure?
structure?
•• Usually
Usually must
must use
use parametric
parametric model
model of
of correlation
correlation
–– E.g.
E.g. Use
Use wavelets
wavelets to
to decorrelate
decorrelate
•• Bullmore
-78
12:61
Bullmore et
et al
al 2001,
2001, HBM
HBM 12:6112:61-78
•• Intersubject
subject fMRI
Inter
Intersubject
fMRI permutation
permutation test
test
–– Create
Create difference
difference image
image for
for each
each subject
subject
–– For
For each
each permutation,
permutation, flip
flip sign
sign of
of some
some subjects
subjects
Permutation
Permutation Test
Test :: Example
Example
Permutation
Permutation Test
Test :: Example
Example
•• Permute!
Permute!
•• fMRI
fMRI Study
Study of
of Working
Working Memory
Memory
–– 12
12 subjects,
subjects, block
block design
design
–– Item
Item Recognition
Recognition
Active
Marshuetz
Marshuetz et
et al
al (2000)
(2000)
•• Active:View
Active:View five
five letters,
letters, 2s
2s pause,
pause,
view
view probe
probe letter,
letter, respond
respond
•• Baseline:
Baseline: View
View XXXXX,
XXXXX, 2s
2s pause,
pause,
view
view YY or
or N,
N, respond
respond
•• Second
Second Level
Level RFX
RFX
–– Difference
Difference image,
image, A-B
A-B constructed
constructed
for
for each
each subject
subject
–– One
One sample,
sample, smoothed
smoothed variance
variance tt test
test
...
...
D
UBKDA
12 =
–– 2
212
= 4,096
4,096 ways
ways to
to flip
flip 12
12 A/B
A/B labels
labels
–– For
For each,
each, note
note maximum
maximum of
of tt image
image
..
yes
Baseline
...
...
N
XXXXX
no
Permutation Distribution
Maximum t
Maximum Intensity Projection
Thresholded t
Does
Does this
this Generalize?
Generalize?
RFT
RFT vs
vs Bonf.
Bonf. vs
vs Perm.
Perm.
uPerm = 7.67
58 sig. vox.
t11 Statistic, Nonparametric Threshold
uRF = 9.87
uBonf = 9.80
5 sig. vox.
t11 Statistic, RF & Bonf. Threshold
• Compare with Bonferroni
α = 0.05/110,776
• Compare with parametric RFT
110,776 2×2×2mm voxels
5.1×5.8×6.9mm FWHM
smoothness
462.9 RESELs
Test Level vs. t11 Threshold
Verbal Fluency
Location Switching
Task Switching
Faces: Main Effect
Faces: Interaction
Item Recognition
Visual Motion
Emotional Pictures
Pain: Warning
Pain: Anticipation
df
4
9
9
11
11
11
11
12
22
22
t Threshold
(0.05 Corrected)
RF
Bonf Perm
4701.32 42.59 10.14
11.17
9.07
5.83
10.79 10.35
5.10
10.43
9.07
7.92
10.70
9.07
8.26
9.87
9.80
7.67
11.07
8.92
8.40
8.48
8.41
7.15
5.93
6.05
4.99
5.87
6.05
5.05
RFT
RFT vs
vs Bonf.
Bonf. vs
vs Perm.
Perm.
Verbal Fluency
Location Switching
Task Switching
Faces: Main Effect
Faces: Interaction
Item Recognition
Visual Motion
Emotional Pictures
Pain: Warning
Pain: Anticipation
df
4
9
9
11
11
11
11
12
22
22
No. Significant Voxels
(0.05 Corrected)
t
RF Bonf Perm
0
0
0
0
0
158
4
6
2241
127
371
917
0
0
0
5
5
58
626 1260
1480
0
0
0
127
116
221
74
55
182
Performance
Performance Summary
Summary
•• Bonferroni
Bonferroni
–– Not
Not adaptive
adaptive to
to smoothness
smoothness
–– Not
Not so
so conservative
conservative for
for low
low
smoothness
smoothness
•• Random
Random Field
Field
–– Adaptive
Adaptive
–– Conservative
Conservative for
for low
low smoothness
smoothness &
& df
df
•• Permutation
Permutation
–– Adaptive
Adaptive (Exact)
(Exact)
“Old”
“Old” Conclusions
Conclusions
Contents
Contents
•• tt random
random field
field results
results conservative
conservative
for
for
•• Recap
Recap &
& Introduction
Introduction
–– Low
Low df
df &
& smoothness
smoothness
–– 9
df
&
≤12
9 df & ≤12 voxel
voxel FWHM;
FWHM; 19
19 df
df &
&<
< 10
10
voxel
voxel FWHM
FWHM
•• Bonferroni
Bonferroni surprisingly
surprisingly satisfactory
satisfactory
for
for low
low smoothness
smoothness
•• Nonparametric
Nonparametric methods
methods perform
perform
well
well overall
overall
•• Inference
Inference &
& multiple
multiple
comparison
comparison
•• Single/multiple
Single/multiple voxel
voxel inference
inference
•• Family
Family wise
wise error
error rate
rate (FWER)
(FWER)
•• False
Discovery
rate
(FDR)
False Discovery rate (FDR)
•• SPM
SPM results
results
•• «
« Take
Take home
home »
» message
message
False
False Discovery
Discovery Rate
Rate
Illustration:
Illustration:
False
False Discovery
Discovery Rate
Rate
ACTION
TRUTH
H True (o)
H False (x)
Reject
TN
FP
At u2
TP
FDR=1/12=8%
α=1/10=10%
FN
Noise
FDR=3/17=18%
α=3/10=30%
Signal
Eg. t-scores
from regions
that truly do and
do not activate
FDR = FP/(FP+TP)
α = FP/(FP+TN)
At u1
Don’t
Reject
Signal+Noise
oooooooxxxooxxxoxxxxxxxx
u1
u2
Control of Per Comparison Rate at 10%
11.3% 11.3% 12.5% 10.8% 11.5% 10.0% 10.7% 11.2% 10.2% 9.5%
Percentage of Null Pixels that are False Positives
••
••
••
Select
Select desired
desired limit
limit qq on
on E(FDR)
E(FDR)
≤
p (2) ≤≤ ...
Order
pvalues,
p
p
... ≤≤ pp((V)
Order p-values, p(1)
V)
(1) ≤ p(2)
Let
r
be
largest
i
such
that
Let r be largest i such that
p(i) ≤ i/V*q
1
Control of Familywise Error Rate at 10%
Benjamini
Benjamini &
& Hochberg
Hochberg Procedure
Procedure
•• Reject
Reject all
all hypotheses
hypotheses
corresponding
corresponding to
to
, ... , p r)..
pp(1)
(1), ... , p((r)
6.7% 10.4% 14.9% 9.3% 16.2% 13.8% 14.0% 10.5% 12.2% 8.7%
Percentage of Activated Pixels that are False Positives
0
Control of False Discovery Rate at 10%
p(i)
p-value
Occurrence of Familywise Error
FWE
0
JRSS-B (1995) 57:289-300
i/V × q/c(V)
i/V
1
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Varying
Varying Signal
Signal Extent
Extent
p=
Signal Intensity 3.0
Varying
Varying Signal
Signal Extent
Extent
z=
Signal Extent 1.0
p=
Noise Smoothness 3.0
Signal Intensity 3.0
z=
Signal Extent 2.0
Noise Smoothness 3.0
1
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Varying
Varying Signal
Signal Extent
Extent
p=
Signal Intensity 3.0
Varying
Varying Signal
Signal Extent
Extent
z=
Signal Extent 3.0
2
p = 0.000252
Noise Smoothness 3.0
Signal Intensity 3.0
3
z = 3.48
Signal Extent 5.0
Noise Smoothness 3.0
4
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Varying
Varying Signal
Signal Extent
Extent
p = 0.001628
Signal Intensity 3.0
Varying
Varying Signal
Signal Extent
Extent
z = 2.94
Signal Extent 9.5
p = 0.007157
Noise Smoothness 3.0
Signal Intensity 3.0
z = 2.45
Signal Extent16.5
Noise Smoothness 3.0
5
Benjamini
Benjamini &
& Hochberg:
Hochberg:
Benjamini
Benjamini &
& Hochberg:
Hochberg: Properties
Properties
Varying
Varying Signal
Signal Extent
Extent
p = 0.019274
6
z = 2.07
•• Adaptive
Adaptive
–– Larger
Larger the
the signal,
signal, the
the lower
lower the
the threshold
threshold
–– Larger
the
signal,
the
more
false
Larger the signal, the more false positives
positives
•• False
False positives
positives constant
constant as
as fraction
fraction of
of
rejected
rejected tests
tests
•• Not
’s sparse
imaging
Not aa problem
problem with
with imaging’
imaging’s
sparse
signals
signals
•• Smoothness
Smoothness OK
OK
–– Smoothing
Smoothing introduces
introduces positive
positive correlations
correlations
Signal Intensity 3.0
Signal Extent25.0
Noise Smoothness 3.0
7
Contents
Contents
Summary:
Summary: Levels
Levels of
of inference
inference &
& power
power
SPM intensity
•• Recap
Recap &
& Introduction
Introduction
•• Inference
Inference &
& multiple
multiple
comparison
comparison
•• Single/multiple
Single/multiple voxel
voxel inference
inference
•• Family
wise
error
rate
Family wise error rate (FWER)
(FWER)
h
u
Sensitivity
L1
SPM position
: significant at the set level
: significant at the cluster level
: significant at the voxel level
L1 > spatial extent threshold
L2 < spatial extent threshold
•• False
False Discovery
Discovery rate
rate (FDR)
(FDR)
•• SPM
SPM results
results
•• «
« Take
Take home
home »
» message
message
Levels
…
Levels of
of inference
inference…
voxelvoxel-level
P(c ≥ 1 | n ≥ 0, t ≥ 4.37) = 0.048 (corrected)
P(t
P(t ≥ 4.37) = 1 - Φ{4.37} < 0.001 (uncorrected)
omnibus
P(c≥
P(c≥7 | n ≥ 0, t ≥ 3.09) = 0.031
n=12
t=4.37
n=82
n=32
clustercluster-level
P(c ≥ 1 | n ≥ 82, t ≥ 3.09) = 0.029 (corrected)
P(n ≥ 82 | t ≥ 3.09) = 0.019 (uncorrected)
setset-level
P(c ≥ 3 | n ≥ 12, t ≥ 3.09) = 0.019
Parameters
u
k
S
FWHM
D
- 3.09
- 12 voxels
- 323 voxels
- 4.7 voxels
-3
Test based on
L2
SPM
SPM results...
results...
The intensity of a
voxel
The spatial extent above u
or the spatial extent and the
maximum peak height
Parameters
set by the user
• Low pass filter
• Low pass filter
• intensity threshold u
The number of clusters
above u with size greater
than n
• Low pass filter
• intensity thres. u
• spatial threshold n
The sum of square of the
SPM or a MANOVA
• Low pass filter
Regional
specificity
SPM
SPM results...
results...
SPM
SPM results...
results...
fMRI :
Activations
significant at
voxel and
cluster level
Contents
Contents
Conclusions:
Conclusions: FWER
FWER vs
vs FDR
FDR
•• Must
Must account
account for
for multiplicity
multiplicity
–– Otherwise
fishing expedition”
”
expedition
Otherwise have
have aa ““fishing
expedition”
•• Recap
Recap &
& Introduction
Introduction
•• Inference
Inference &
& multiple
multiple
comparison
comparison
•• «
« Take
Take home
home »
» message
message
•• FWER
FWER
–– Very
Very specific,
specific, less
less sensitive
sensitive
•• FDR
FDR
–– Less
Less specific,
specific, more
more sensitive
sensitive
–– Trouble
with
cluster
…
inference
Trouble with cluster inference…
inference…
More
!
Ya
More Power
Power to
to Ya!
Ya!
Statistical Power
• the probability of rejecting the null hypothesis when it is
actually false
• “if there’s an effect, how likely are you to find it”?
Effect size
• bigger effects, more power
• e.g., MT localizer (moving rings - stationary runs) -- 1 run is usually
enough
• looking for activation during imagined motion might require many more
runs
Sample size
• larger n, more power
• more subjects - longer runs - more runs
Signal:Noise Ratio
• better SNR, more power
• stronger, cleaner magnet - more focal coil - fewer artifacts - more
filtering