Rowe, MCW
A Complex way to Compute fMRI Activations
Daniel B. Rowe, Ph.D.
[email protected]
(Joint with BR Logan, MCW Biostat)
Department of Biophysics
Graduate School of Biomedical Sciences
8701 Watertown Plank Rd.
Milwaukee, WI 53226
Rowe, MCW
Outline
☞ Introduction
☞ Images
☞ Complex/Magnitude Time Course Model
☞ Simulation Example
☞ Real fMRI Example
☞ Discussion
Rowe, MCW
Introduction
In MRI/fMRI, (non)Fourier “image reconstruction” results in complex
valued proton spin densities that make up our voxel time course
observations.
The complex part of the proton spin density is a result of phase
imperfections due to magnetic field inhomogeneities.
Nearly all fMRI studies obtain a statistical measure of functional
“activation” based on magnitude image time courses.
Rowe, MCW
Introduction
However, it is the real and imaginary parts of the original signal that
are measured with normally distributed error, and not the magnitude.
A more accurate model should use the correct distributional
specification.
A model is presented that uses the original complex form of the data
and not the magnitude.
The result is the correct distribution and twice as many quantities
to estimate the model parameters which results in improved power
for low SNR.
Focus on a single axial slice.
Rowe, MCW
Complex Single Time Images
(a) real image
(b) imaginary image
Rowe, MCW
Magnitude/Phase Single Time Images
(c) magnitude image
(d) phase image
Rowe, MCW
Complex Time Course Images
In fMRI we observe a series of complex images over time.
Rowe, MCW
Magnitude Time Course Images
And not a series of real magnitude images. Phase not used.
Rowe, MCW
Complex Time Course Model
In a voxel, the complex valued quantity measured over time is
ρmt = (ρRt + ηRt) + i(ρIt + ηIt),
t = 1, ..., n
ρmt = complex voxel measurement at time t
ρRt = true real part of voxel measurement at time t
ηRt = noise real part voxel measurement at time t
ρIt = imaginary part voxel measurement at time t
ηIt = noise imaginary part voxel measurement at time t
(ηRt, ηIt)0 ∼ N (0, Σ), Σ = σ 2I2.
The distributional specification is on the real and imaginary parts
of the image and not on the magnitude.
Rowe, MCW
Magnitude Time Course Model
The typical method to compute activations is to use the magnitude
h
|ρmt| = (ρRt + nRt)2 + (ρIt + nIt)2
= yt
i1
2
,
t = 1, ..., n
that is Rician distributed and approximately normal for a large
signal to noise ratio (relatively small error variance).
The phase which may contain information is not used.
φt = tan−1
ρIt + nIt
ρRt + nRt
.
Rowe, MCW
Magnitude Time Course Model: Assumptions
The magnitude, does not have a normal distribution.
Ricean Distribution
Rayleigh Distribution
0.6
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0.5
0.5
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p(y)
t
p(r /σ)
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rt/σ
p(yt) =
(yt2+ρ2t ) −
yt
ρt ·yt
2σ 2 Io
e
σ2
σ2
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0
1
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y
p(yt) =
yt2
yt − 2σ 2
e
σ2
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Rowe, MCW
Magnitude Time Course Model: Assumptions
Data justification. Histogram of no signal, noise only outside voxels.
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Outer brain histogram
x 10
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3.5
N(y)
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y
Outside histogram
Magnitude Sequence
Rowe, MCW
Magnitude Time Course Model: Assumptions
Data justification. Histogram of no signal, noise only outside voxels.
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Outer brain histogram
x 10
Rayleigh Distribution
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0.5
3.5
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p(y)
N(y)
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Outside histogram
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Rayleigh PDF’s
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Rowe, MCW
Complex Time Course Model
A complex phase model was introduced that includes a phase
imperfection θ in which at time t, the measured proton spin density
is given by
ρmt = (ρt cos θ + ηRt) + i(ρt sin θ + ηIt)
ρRt = ρt cos θ
ρIt = ρt sin θ
(ηRt, ηIt)0 ∼ N (0, Σ), Σ = σ 2I2.
This model which includes a phase error is an accurate representation
of the true physical process that generates the data.
Rowe, MCW
Complex Time Course Model: Assumptions
Data justification. Phase and correlation of no signal noise only
outside voxels.
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0
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−2
−3
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Constant Phase
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R-I Uncorrelated
Rowe, MCW
Complex Time Course Model: Assumptions
Data justification. Phase and correlation of no signal noise only
outside voxels.
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Constant Phase
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−3
R-I Uncorrelated
Rowe, MCW
Complex Time Course Model: Assumptions
Data justification. Histogram of no signal noise only outside voxels.
Outside histogram
Complex Sequence
Rowe, MCW
Magnitude Time Course Model
The magnitude model from the complex phase model
h
i1
2
|ρmt| = (ρt cos θ + nRt)2 + (ρt sin θ + nIt)2
h
i1
2
yt = ρ2t + (n2Rt + n2It) + 2ρt(nRt cos θ + nIt sin θ)
"
#1
(nRt cos θ + nIt sin θ) (n2Rt + n2It) 2
= ρt 1 + 2
+
ρt
ρ2t
≈ ρt + t, t = 1, ..., n
t = nRt cos θ + nIt sin θ ∼ N (0, σ 2).
√
1 + u ≈ 1 + u/2, |u| 1.
Rowe, MCW
Time Course Model
In fMRI, we take repeated measurements over time while a subject
is performing a task.
We know the timing of the task, tap fingers for 16 seconds, rest
for 16 seconds, and repeat several times.
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In each voxel, compute a measure of association between observed
time course and a preassigned reference function that characterizes
the experimental paradigm (Bandettini et al., 1993; Cox et al., 1995).
Rowe, MCW
Magnitude Time Course Model
Linear multiple regression model individually for each voxel
ρt = x0tβ = β0 + β1x1t + · · · + βq xqt.
yt = x0tβ + t,
t = 1, ..., n
Also written as
y =
X
β
+ n×1
n × (q + 1) (q + 1) × 1
n×1
and ∼ N (0, σ 2Φ), Φ is the temporal correlation matrix
usually Φ = In.
Rowe, MCW
y
y1
..
y8
y9
..
y16
.. =
..
y241
..
y248
y249
..
y256
en
1
..
1
1
..
1
..
..
1
..
1
1
..
1
cn
1
..
8
9
..
16
..
..
241
..
248
249
..
256
rn
1
..
1
-1
..
-1
.. ∗
..
1
..
1
-1
..
-1
β
β0
+
β1
β2
1
..
8
9
..
16
..
..
241
..
248
249
..
256
Rowe, MCW
Magnitude Time Course Model
The magnitude activation likelihood is given by
p(y|β, σ 2, X) =
0(y−Xβ)
(y−Xβ)
n 2 −n −
−
2σ 2
(2π) 2 (σ ) 2 e
.
We want to see if the observed time course has a component related
to the reference function.
H0 : Cβ = γ vs H1 : Cβ 6= γ
i.e. Is the coefficient for the reference function zero.
C = (0, ..., 0, 1), β 0 = (β0, β1, · · · , βq ), γ = 0
Rowe, MCW
Magnitude Time Course Model
By maximizing the likelihood under the unconstrained alternative
β̂ = (X 0X)−1X 0y,
0 1
y − X β̂
σ̂ 2 =
y − X β̂ .
n
By maximizing the likelihood under the constrained null hypotheses
β̃ = Ψβ̂
+ (X 0X)−1C 0[C(X 0X)−1C 0]−1γ,
0 1
y − X β̃
σ̃ 2 =
y − X β̃
n
Ψ = Iq+1 − (X 0X)−1C 0[C(X 0X)−1C 0]−1C .
Rowe, MCW
Magnitude Model
By maximizing the likelihood under constrained null and
unconstrained alternative hypotheses the GLR statistic is
λ =
p(y|β̃, σ̃ 2, X, H0)
p(y|β̂, σ̂ 2, X, H1)
− n
2
= σ̃ 2/σ̂ 2
0[C(X 0X)−1C 0]−1 (C β̂ − γ)
2
(C
β̂
−
γ)
n−q−1
(λ− n − 1) =
r
rnσ̂ 2/(n − q − 1)
This magnitude model uses n quantities to estimate the q + 2
parameters being the q + 1 regression coefficients, the 1 variance.
Rowe, MCW
Magnitude Model
For example with q = 2, X = (en, cn, rn), H0 : β2 = 0 can be
evaluated with C = (0, 0, 1), γ = 0, and any of the test statistics
t =
F =
χ2 =
β̂2
1
2
[W33nσ̂ /(n−2−1)] 2
β̂22
W33nσ̂ 2/(n−2−1)
n log σ̃ 2/σ̂ 2
∼ tn−2−1
∼ F1,n−2−1
·
∼
Where W33 is (3, 3) element of W = (X 0X)−1.
χ21
Rowe, MCW
Complex Time Course Model
The previous complex model
ρmt = (ρt cos θ + ηRt) + i(ρt sin θ + ηIt)
can be written as
yRt
yIt
=
ρt cos θ
ρt sin θ
+
where again (ηRt, ηIt)0 ∼ N (0, Σ), Σ = σ 2I2.
ηRt
ηIt
Rowe, MCW
Complex Time Course Model
A non-linear multiple regression model can be introduced in which
ρt = x0tβ = β0 + β1x1t + · · · + βq xqt
as in the magnitude model then written in terms of matrices as
y
2n × 1
=
X 0
β cos θ
+
η
0 X
β sin θ
2n × 2(q + 1) 2(q + 1) × 1
2n × 1
0 , y 0 )0 and η = (η 0 , η 0 )0 ∼ N (0, Σ ⊗ Φ)
where y = (yR
I
Rt It
Σ = σ 2I2, Φ = In.
Rowe, MCW
yR
yR1
..
yR8
yR9
..
yR16
vec ..
..
yR241
..
yR248
yR249
..
yR256
yI
en
yI1
1
..
..
yI8
1
yI9
1
..
..
yI16
1
.. =I ⊗ ..
2
..
..
yI241
1
..
..
yI248
1
yI249
1
..
..
yI256
1
cn
1
..
8
9
..
16
..
..
241
..
248
249
..
256
rn
ηR
ηI
1
ηR1 ηI1
..
..
..
1
ηR8 ηI8
-1
ηR9 ηI9
..
..
..
β
phase
-1
ηR16 ηI16
β
.. ∗ cos θ ⊗ 0 +vec ..
..
β1
.. sin θ
..
..
β2
1
ηR241 ηI241
..
..
..
1
ηR248 ηI248
-1
ηR249 ηI249
..
..
..
-1
ηR256 ηI256
Rowe, MCW
Complex Time Course Model
The non-linear multiple regression model
X 0
β cos θ
y
=
+
η
0 X
β sin θ
2n × 1
2n × 2(q + 1) 2(q + 1) × 1
2n × 1
0 , y 0 )0, η = (η 0 , η 0 )0 ∼ N (0, Σ ⊗ Φ), Σ = σ 2I and Φ = I .
y = (yR
n
2
I
Rt It
The likelihood is
h
2n 2 − 2n −
2
−
p(y|β, θ, σ , X) = (2π) 2 (σ ) 2 e 2σ2
where
0 X 0
β cos θ
X 0
β cos θ
h = y−
y−
.
0 X
β sin θ
0 X
β sin θ
Rowe, MCW
Complex Time Course Model
Just as in the magnitude model, we want to see if the observed time
course has a component related to the reference function.
H0 : Cβ = γ vs H1 : Cβ 6= γ
i.e. Is the coefficient for the reference function zero.
C = (0, ..., 0, 1), β 0 = (β0, β1, · · · , βq ), γ = 0
Rowe, MCW
Complex Time Course Model
By maximizing the likelihood under the unconstrained alternative
"
0 (X 0X)β̂
2
β̂
1
I
R
θ̂ = tan−1
0 (X 0X)β̂ − β̂ 0 (X 0X)β̂
2
β̂R
R
I
I
β̂ = β̂R cos θ̂ + β̂I sin θ̂,
"
1
X 0
2
y−
σ̂ =
0 X
2n
β̂ cos θ̂
β̂ sin θ̂
!#0 "
#
y−
β̂R = (X 0X)−1X 0yR,
β̂I = (X 0X)−1X 0yI .
X 0
0 X
β̂ cos θ̂
β̂ sin θ̂
!#
Rowe, MCW
Complex Time Course Model
By maximizing the likelihood under the constrained null hypotheses
"
#
0 Ψ(X 0X)β̂
2
β̂
1
I
R
θ̃ = tan−1
0 Ψ(X 0X)β̂ − β̂ 0 Ψ(X 0X)β̂
2
β̂R
R
I
I
β̃ = Ψ[β̂R cos θ̃ + β̂I sin θ̃]
+ (X 0X)−1C 0[C(X 0X)−1C 0]−1γ,
0 1
X 0
β̃ cos θ̃
X 0
β̃ cos θ̃
2
σ̃ =
y−
y−
0 X
0
X
β̃ sin θ̃
β̃ sin θ̃
2n
Ψ = Iq+1 − (X 0X)−1C 0[C(X 0X)−1C 0]−1C .
Rowe, MCW
Complex Time Course Model
The way this is formulated, we have to worry about phase angles.
An alternative formulation is to let α1 = cos θ and α2 = sin θ
y =
X 0
0 X
α1β
α2β
+ η,
α12 + α22 = 1
0 , y 0 )0, η = (η 0 , η 0 )0 ∼ N (0, Σ ⊗ Φ), Σ = σ 2I and Φ = I .
y = (yR
n
2
I
Rt It
With this formulation, it can be seen that this is a reduced rank
regression model (Reinsel, 1998).
Rowe, MCW
Complex Time Course Model
By maximizing the likelihood under the unconstrained alternative
α̂1 = β̂ 0(X 0X)β̂R/[(β̂ 0(X 0X)β̂R)2 + (β̂ 0(X 0X)β̂I )2]1/2
α̂2 = β̂ 0(X 0X)β̂I /[(β̂ 0(X 0X)β̂R)2 + (β̂ 0(X 0X)β̂I )2]1/2
β̂ = α̂1β̂R + α̂2β̂I ,
"
1
X 0
2
y−
σ̂ =
0 X
2n
α̂1β̂
α̂2β̂
!#0 "
y−
β̂R = (X 0X)−1X 0yR,
β̂I = (X 0X)−1X 0yI .
X 0
0 X
α̂1β̂
α̂2β̂
!#
Rowe, MCW
Complex Time Course Model
By maximizing the likelihood under the constrained null hypotheses
α̃1 = β̃ 0(X 0X)β̂R/[(β̃ 0(X 0X)β̂R)2 + (β̃ 0(X 0X)β̂I )2]1/2
α̃2 = β̃ 0(X 0X)β̂I /[(β̃ 0(X 0X)β̂R)2 + (β̃ 0(X 0X)β̂I )2]1/2
β̃ = Ψ(α̃1β̂R + α̃2β̂I )
+ (X 0X)−1C 0[C(X 0X)−1C 0]−1γ,
0 1
X 0
α̃1β̃
X 0
α̃1β̃
2
y
−
y
−
σ̃ =
0 X
0 X
α̃2β̃
α̃2β̃
2n
Ψ = Iq+1 − (X 0X)−1C 0[C(X 0X)−1C 0]−1C .
Rowe, MCW
Complex Time Course Model
GLR statistic is
λ =
p(y|β̃, σ̃ 2, X, H0)
p(y|β̂, σ̂ 2, X, H1)
−n
= σ̃ 2/σ̂ 2
or
−2 log λ = 2n log σ̃ 2/σ̂ 2 .
This complex model uses all the 2n observations to estimate
the q + 3 parameters being the q + 1 regression coefficients,
the 1 variance, and the 1 phase error or trigonometric coefficient.
Rowe, MCW
Real fMRI Experiment
Imaging Parameters:
1.5T GE Signa
5 axial slices of 128x128
96 acq.-2.0833mm2
128 recon.-1.5625mm2
FOV =20cm
TR=1000ms
TE=47ms
FA=90◦
Task:
Bilateral finger tapping
Block design
16 off + 8×(16on+16off);
Rowe, MCW
Time Course Models
Compare the two models for testing H0 : β2 = 0.
(q = 2, X = (en, cn, rn), C = (0, 0, 1), γ = 0)
· 2
∼ χ1
·
∼ χ21
2 /σ̂ 2
χ2M = n log σ̃M
M
2 /σ̂ 2
χ2C = 2n log σ̃C
C
Both χ21 distributed for large samples!
Rowe, MCW
Real fMRI-Magnitude H1 Estimated
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Rowe, MCW
Real fMRI-Magnitude H0 Estimated
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Rowe, MCW
Real fMRI-Complex H1 Estimated
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=
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*
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*
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Rowe, MCW
Real fMRI-Complex H1 Estimated
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128
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Rowe, MCW
Real fMRI-Complex H0 Estimated
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128
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96 104 112 120 128
128
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Rowe, MCW
Real fMRI-Magnitude/Complex H1 Estimated β̂2
0.05
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56
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0.04
0.03
128
0.02
0.01
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−0.01
−0.02
−0.03
−0.04
128
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96 104 112 120 128
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These coefficients are not visually that different.
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−0.05
Rowe, MCW
Real fMRI-Magnitude/Complex −2 log(λ) Maps
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·
112
These voxel statistics are ∼ χ21!
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0
Rowe, MCW
Real fMRI-Magnitude/Complex −2 log(λ) Maps
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5% Unadjusted Threshold
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Rowe, MCW
Real fMRI-Magnitude/Complex −2 log(λ) Maps
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5% Bonferroni Threshold
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Rowe, MCW
Real fMRI-Magnitude/Complex −2 log(λ) Maps
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5% FDR Threshold
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Rowe, MCW
Simulation
In each voxel, simulate complex valued time courses like real data.
yt = [(β0 + β1x1t + β2x2t)α1 + nRt]
+i[(β0 + β1x1t + β2x2t)α1 + nIt]
2
From a real dataset, fitted complex model, took β̂C and σ̂C
from a “highly activated” voxel. α̂1’s and α̂2’s from whole image.
Created complex data where the coefficients in each voxel were the
first two elements of β̂C . Effect to noise ratio EN R = β2/σ̂C .
Created four 7 × 7 square ROI’s, EN R = 1, 1/2, 1/4, 1/8,
β2 = 0 outside ROI’s.
2 ). Varied SN R = β /σ.
Added normal noise N (0, σ̂C
0
Rowe, MCW
Activation: 5% Unadjusted
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M: SNR = 1
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M: SNR = 5
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C: SNR = 2.5
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Rowe, MCW
Activation: 5% Bonferroni
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M: SNR = 1
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M: SNR = 5
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Rowe, MCW
Activation: 5% FDR
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Rowe, MCW
Simulation
Repeated simulation 1000 times.
For each thresholding method, the power in, or relative frequency over the
1000 simulated images with which each voxel was detected as active, was
recorded.
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5% Unadjusted, 5% FDR, and 5% Bonferroni thresholds.
Rowe, MCW
Power: 5% Unadjusted
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100
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128
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Rowe, MCW
Power: 5% Bonferroni
100
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128
128
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Rowe, MCW
Power: 5% FDR
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Rowe, MCW
Power Differences: 5% Unadjusted
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Rowe, MCW
Power Differences: 5% Bonferroni
10
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SNR = 5
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−10
Rowe, MCW
Power Differences: 5% FDR
10
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2
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−2
−4
−6
−8
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SNR = 1
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SNR = 2.5
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−10
SNR = 5
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−10
Rowe, MCW
100
90
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30
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SNR
6
7
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9
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10
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100
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SNR
6
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9
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ENR=1/4,1/8
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ENR=1,1/2
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Power
Power
100
Power
Power
Power versus SNR: Complex (blue) and magnitude (red)
0
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6
7
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Rowe, MCW
Discussion
A complex data fMRI activation model was presented .
Complex and magnitude models activation compared on real data.
Complex and magnitude models power compared on simulated data.
For a given ENR the complex model power constant irrespective
of SNR while the magnitude model power decreases.
For smaller SNR’s, the complex activation model demonstrated
better power
The complex model more useful as SNR decreases with voxel size.