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D ETECTING GLOBAL TREATMENT EFFECTS
ACROSS ROI
♦
George Minas♣ , Fabio Rigat , Tom Nichols♥ ,
John Aston♠ , Nigel Stallardz
♣
♥
Dept of Statistics & WCAS, University of Warwick
♦ Novartis Vaccines and Diagnostics
Neuroimaging Statistics, Dept of Statistics & WMG, University of Warwick
♠ Dept of Statistics, University of Warwick
z Health Sciences Research Institute, WMS, University of Warwick
FP7 Neurophysics Workshop: Pharmacological fMRI
University of Warwick, 23-24 Jan 2012
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O UTLINE
1
Introduction
1
2
3
2
Proposal: Optimal LC tests
1
2
3
Problem set-up
Optimal solution
Proposal Assessment
1
2
4
pharmacological MRI
ROI analysis
Detecting Global Treatment Effects
Simulations
Real data example
Discussion
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PHARMACOLOGICAL F MRI
BOLD F MRI The imaging contrast arising as a consequence of the
local changes in blood oxygenation accompanying
neuronal activation
C LINICAL T RIAL A controlled experiment to compare the effects of
different medical interventions on human subjects
PHARMACOLOGICAL F MRI
The use of fMRI in clinical trials
A PPLICATIONS Schizophrenia, Depression, Drug addiction,
Dementia, Parkinson’s, Pain, Epilepsy, Stroke, ...
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MRI: P RESENT AND P ROMISE
Honey and Bullmore (2004), JMRI SI: Clinical Potential of Brain
Mapping Using MRI (2006), Wong et al. (2009), Schwartz et al.
(2011)
new technique, principally used in licensed compounds, Wise
and Tracey (2006)
but, major pharma corps “are embracing this technology via
academic collaborations or by establishing it in-house”, Wise
and Tracey (2006)
Targets (among many): Use in early proof-of-concept studies for
novel therapies
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M ASS - UNIVARIATE AND ROI ANALYSIS
Wise and Tracey (2006):
To answer a question such as, “Where in the brain does my drug change
stimulus-related activity”
Hence, mass-univariate analysis at voxel-by-voxel resolution.
Wise and Tracey (2006):
To test a mechanism-based hypothesis, or one in which a specific drug target
is postulated, it is preferable to answer a more focused question concerning
the drug effect, for example, “Considering three brain areas A, B, and C, in
which of these does my drug reduce stimulus-related activity?”
Or, “Does my drug reduce stimulus-related activity across these regions”
Hence, ROI analysis.
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ROI ANALYSIS : POTENTIAL ADVANTAGES
Easier to explore the data
→ allowance for making specific mechanistic or spatial
hypotheses
Suitable for testing regional hypotheses
→ more strict hypotheses → (usually) more suitable for CT’s
Drastic reduction of multiple comparisons
→ statistical power increased
but strong spatial prior hypotheses are required
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H OW TO DO ROI ANALYSIS : I N ONE SLIDE
1
Define anatomical or functional ROI
2
Perform voxel-by-voxel mass-univariate analysis
3
Extract estimates βb of the treatment effect in each voxel
4
b across ROI
Average β’s
I
A measure of brain response per ROI/subject
Here, we use ROI data to see whether there is
a significant Global Treatment Effect (GTE) across ROI.
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T ESTING FOR GLOBAL TREATMENT EFFECTS ( GTE )
We have the response measure
Y = (Y1 , Y2 , ..., YK ) with E(Y) = µ
and we are willing to test the null hypothesis
H0 : µ = (µ1 , ..., µK )0 = (0, 0, ..., 0)0
I
Detection of GTE ≡ Rejection of H0
Targets:
I
control false positive rate: P( reject H0 | H0 true ) = α
I
maximise power function P( reject H0 | H0 false )
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B ONFERRONI - TYPE METHODS
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B ONFERRONI - TYPE METHODS
Test the local null hypotheses
H0k : µk = 0, k = 1, ..., K(e.g. ROI)
while Pr (at least one false positive) ≤ α.
Test:
reject H0k ⇔ pk ≤ αk
(pk : local p-value)
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B ONFERRONI - TYPE METHODS
Test the local null hypotheses
H0k : µk = 0, k = 1, ..., K(e.g. ROI)
while Pr (at least one false positive) ≤ α.
Test:
I
reject H0k ⇔ pk ≤ αk
(pk : local p-value)
Rejection of a local H0k ⇒ Rejection of global H0
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B ONFERRONI - TYPE METHODS
Test the local null hypotheses
H0k : µk = 0, k = 1, ..., K(e.g. ROI)
while Pr (at least one false positive) ≤ α.
Test:
I
reject H0k ⇔ pk ≤ αk
(pk : local p-value)
Rejection of a local H0k ⇒ Rejection of global H0
C LASSICAL B ONFERRONI αk = α/K
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B ONFERRONI - TYPE METHODS
Test the local null hypotheses
H0k : µk = 0, k = 1, ..., K(e.g. ROI)
while Pr (at least one false positive) ≤ α.
Test:
I
reject H0k ⇔ pk ≤ αk
(pk : local p-value)
Rejection of a local H0k ⇒ Rejection of global H0
C LASSICAL B ONFERRONI αk = α/K
W ESTFALL ET AL . (2007) Compute αk > 0:
K
P
αk = α and
k=1
arg max E(# rejections | D), D: prior information
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B ONFERRONI - TYPE METHODS
Test the local null hypotheses
H0k : µk = 0, k = 1, ..., K(e.g. ROI)
while Pr (at least one false positive) ≤ α.
Test:
I
reject H0k ⇔ pk ≤ αk
(pk : local p-value)
Rejection of a local H0k ⇒ Rejection of global H0
C LASSICAL B ONFERRONI αk = α/K
W ESTFALL ET AL . (2007) Compute αk > 0:
K
P
αk = α and
k=1
arg max E(# rejections | D), D: prior information
+: Simple
−: Overconservative for high correlations, big K
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M ULTIVARIATE TESTING PROCEDURES (MTP)
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M ULTIVARIATE TESTING PROCEDURES (MTP)
Combine the evidence for treatment effects arising from the local
outcomes
Provide a single global statement for the global treatment effect
Assumption:
iid
Yi ∼ NK (µ, Σ) , i = 1, ..., N
Global null hypothesis
H0 : µ = 0
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M ULTIVARIATE TESTING PROCEDURES (MTP)
Combine the evidence for treatment effects arising from the local
outcomes
Provide a single global statement for the global treatment effect
Assumption:
iid
Yi ∼ NK (µ, Σ) , i = 1, ..., N
Global null hypothesis
H0 : µ = 0
+: Incorporate correlations
−: Less simple than classical Bonferroni
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MTP: H OTELLING ’ S T2 TEST
Classical multivariate testing procedure
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MTP: H OTELLING ’ S T2 TEST
Classical multivariate testing procedure
Test statistic:
T2 = Ny0 S−1
y y
y, Sy : sample mean, var-covar matrix of Y
Test:
reject H0 ⇔
T2 N − K
> F(K,N−K),α
N−1 K
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MTP: H OTELLING ’ S T2 TEST
Classical multivariate testing procedure
Test statistic:
T2 = Ny0 S−1
y y
y, Sy : sample mean, var-covar matrix of Y
Test:
reject H0 ⇔
T2 N − K
> F(K,N−K),α
N−1 K
+: Scale invariant, uniformly most powerful
−: Requires N > K, lacks power if N ' K
⇒ inappropriate for trials where K: moderate or large
and N: small or moderate (e.g. typical fMRI studies)
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MTP: L INEAR C OMBINATION T ESTS
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MTP: L INEAR C OMBINATION T ESTS
Define the linear combination
0
Lw = w Y =
K
X
k=1
wk Yk (w 6= 0)
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MTP: L INEAR C OMBINATION T ESTS
Define the linear combination
0
Lw = w Y =
K
X
wk Yk (w 6= 0)
k=1
Test statistic:
Lw
Lw
√ ,
√
Σ unknown :
σLw / N
sLw / N
: sample mean, variance and sample variance of Lw
Σ known :
Lw , σL2w , s2Lw
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MTP: L INEAR C OMBINATION T ESTS
Define the linear combination
0
Lw = w Y =
K
X
wk Yk (w 6= 0)
k=1
Test statistic:
Lw
Lw
√ ,
√
Σ unknown :
σLw / N
sLw / N
: sample mean, variance and sample variance of Lw
Σ known :
Lw , σL2w , s2Lw
Critical issue: The selection of the weighting vector w.
It affects:
I
the false positive rate
I
the power of the test
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MTP: T ESTS BASED ON LINEAR COMBINATIONS 2
Some solutions:
O’Brien (1984): wOLS = 1K , wGLS = Σ−1 1K , 1K = (1, 1, ..., 1)0
Powerful if the effect has same size and sign across outcomes
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MTP: T ESTS BASED ON LINEAR COMBINATIONS 2
Some solutions:
O’Brien (1984): wOLS = 1K , wGLS = Σ−1 1K , 1K = (1, 1, ..., 1)0
Powerful if the effect has same size and sign across outcomes
Lauter et al. (1996): w uniquely determined by Y0 Y, Y = (Yi )Ni=1
Powerful if the effect has certain factorial structures
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MTP: T ESTS BASED ON LINEAR COMBINATIONS 2
Some solutions:
O’Brien (1984): wOLS = 1K , wGLS = Σ−1 1K , 1K = (1, 1, ..., 1)0
Powerful if the effect has same size and sign across outcomes
Lauter et al. (1996): w uniquely determined by Y0 Y, Y = (Yi )Ni=1
Powerful if the effect has certain factorial structures
+: Scale invariant, not constrained to have K < N
−: Require specific structures of E(Y) and/or Var(Y) to be powerful.
I
Proposal (coming next): Attempts to tackle the last problem by
using pilot data and prior information to select w optimally
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F ORMULATION OF THE PROBLEM
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F ORMULATION OF THE PROBLEM
Response:
iid
Yi ∼ NK (µ, Σ) , i = 1, 2, ..., ny
(1)
Hypotheses:
H0 : µ = 0 (no effect)
versus
H1 : µ 6= 0
(2)
Global Measure: the linear combination
Lw,i = w0 Yi , i = 1, 2, ..., ny (w 6= 0)
(3)
Test statistic:
Σ known : Zw =
Lw
Lw
√ , Σ unknown : Tw =
√ (4)
σLw / N
sLw / N
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F ORMULATION OF THE PROBLEM 2
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F ORMULATION OF THE PROBLEM 2
Hypothesis Test:
Σ known : reject H0 ⇔ |Zw | > zα/2
Σ unknown : reject H0 ⇔ |Tw | > tny −1,α/2
(5)
(6)
Power:
Σ known : βz = P |Zw | > zα/2
(7)
(8)
Σ unknown : βt = P |Tw | > tny −1,α/2
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O PTIMAL WEIGHTING VECTOR
Target: Select w so that the power for any µ 6= 0 is maximised
T HEOREM 1
Under (1) and when µ 6= 0, the weighting vector maximising the
power functions β (w, µ, ny ) and βt (w, µ, Σ, ny ) is
ω + = Σ−1 µ.
I
ω + optimal with respect to power.
But,...
I
The weighting vector ω + depends on µ (which is unknown!)
I
We wish to collect information for µ (and Σ) to select w
(9)
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R ESULTS OF T HEOREM 1
C OROLLARY 1
Under (1), βt+ = βt (ω + , µ, Σ, ny ) is larger or equal than the power
of the Hotelling’s T 2 test βT 2 (µ, Σ, ny ), for any value of µ, Σ
(ny > K).
In fact, βT 2 (µ, Σ, ny ) βt (ω + , µ, Σ, ny ) especially for ny ' K
1
2
3
4
µ
(0.5, 0.5, 0.5, 0, 0)
(0.5, 0.5, 0.5, 0.5, 0.5)
ny
8
10
17
20
βt+ (ny )
0.99
0.99
0.72
0.79
βT 2 (ny )
0.11
0.61
0.30
0.38
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LC t TESTS VS H OTELLING ’ S T 2 TEST
F IGURE : βt? and βT 2 vs sample size n, K = 15.
For n < 30, βt dominates βT 2
I
There is scope for developing linear combination tests.
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P RIOR INFORMATION - P ILOT DATA
Prior for µ:
(µ|Σ, D0 ) ∼ NK (m0 , Σ/n0 ) ,
m0 : prior estimate for µ
Prior for Σ:
n0 : # of prior observations
(Σ|D0 ) ∼ IW K×K ν0 , S0−1
iid
Xi ∼ NK (µ, Σ) , i = 1, 2, ..., nx
I
(11)
ν0 : degrees of freedom
S0 : scale matrix
Pilot data:
(10)
(12)
We use the information set D1 = {x, D0 } to select optimally w
Optimality? Max predictive power given D1
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S ELECTING THE WEIGHTING VECTOR
I
Predictive power: P(reject H0 | available information)
Σ known : Bz = P |Zw | > zα/2 | D1
Σ unknown : Bt = P |Tw | > tny −1,α/2 | D1
(13)
(14)
T HEOREM 2
Under (1), (10), (11) and (12) the weighting vector maximising Bz is
w?z = Σ−1 m1 .
(15)
For large ν1 = ν0 + nx , Bt (w, D1 ) is maximised by w?t = S1−1 m1 .
m1 =
n0 m0 + nx x
n0 nx
, S1 = S0 + (nx − 1)Sx +
(x − m0 )(x − m0 )0
n0 + nx
n0 + nx
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z? AND t? TESTING PROCEDURES
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z? AND t? TESTING PROCEDURES
0. P LANNING STAGE Elicit priors for µ and Σ (if unknown)
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z? AND t? TESTING PROCEDURES
0. P LANNING STAGE Elicit priors for µ and Σ (if unknown)
x
1. P ILOT STUDY Collect pilot data x = (xi )ni=1
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z? AND t? TESTING PROCEDURES
0. P LANNING STAGE Elicit priors for µ and Σ (if unknown)
x
1. P ILOT STUDY Collect pilot data x = (xi )ni=1
2. I NTERIM ANALYSIS Using Theorem 2, compute the optimal
weighting vector w?z or w?t (if Σ unknown)
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z? AND t? TESTING PROCEDURES
0. P LANNING STAGE Elicit priors for µ and Σ (if unknown)
x
1. P ILOT STUDY Collect pilot data x = (xi )ni=1
2. I NTERIM ANALYSIS Using Theorem 2, compute the optimal
weighting vector w?z or w?t (if Σ unknown)
3. M AIN STUDY Collect responses Yj , for j = 1, ..., ny
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z? AND t? TESTING PROCEDURES
0. P LANNING STAGE Elicit priors for µ and Σ (if unknown)
x
1. P ILOT STUDY Collect pilot data x = (xi )ni=1
2. I NTERIM ANALYSIS Using Theorem 2, compute the optimal
weighting vector w?z or w?t (if Σ unknown)
3. M AIN STUDY Collect responses Yj , for j = 1, ..., ny
4. F INAL ANALYSIS Perform the z? - or t? -test (if Σ unknown)
using Zw? or Tw?t , respectively
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P ROPERTIES OF z? - AND t? - TESTS
control false positive rate
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P ROPERTIES OF z? - AND t? - TESTS
control false positive rate
the distribution of Zw? , Tw?
I
do not depend on K (unlike Hotelling’s T2 )
I
invariant to scale transformations Y → cY
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P ROPERTIES OF z? - AND t? - TESTS
control false positive rate
the distribution of Zw? , Tw?
I
do not depend on K (unlike Hotelling’s T2 )
I
invariant to scale transformations Y → cY
the weighting vectors w?z and w?t
I
optimal given the available information at interim
I
invariant to scale transformation Y → cY
I
components with strong effects receive larger weights
I
intuitive relation to the expected treatment effect
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S IMULATIONS ALGORITHM
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S IMULATIONS ALGORITHM
Set the inputs:
I
Sample sizes: nx , ny and the FPR: α
I
Prior hyperparameters: m0 , n0 , S0 , ν0
I
Parameters: µ, Σ
For r = 1, ..., R(= 15000):
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S IMULATIONS ALGORITHM
Set the inputs:
I
Sample sizes: nx , ny and the FPR: α
I
Prior hyperparameters: m0 , n0 , S0 , ν0
I
Parameters: µ, Σ
For r = 1, ..., R(= 15000):
1
x
Generate pilot data xr = (xri )ni=1
from N(µ, Σ)
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S IMULATIONS ALGORITHM
Set the inputs:
I
Sample sizes: nx , ny and the FPR: α
I
Prior hyperparameters: m0 , n0 , S0 , ν0
I
Parameters: µ, Σ
For r = 1, ..., R(= 15000):
1
x
Generate pilot data xr = (xri )ni=1
from N(µ, Σ)
2
Compute w?z (xr ) (or w?t (xr ))
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S IMULATIONS ALGORITHM
Set the inputs:
I
Sample sizes: nx , ny and the FPR: α
I
Prior hyperparameters: m0 , n0 , S0 , ν0
I
Parameters: µ, Σ
For r = 1, ..., R(= 15000):
1
x
Generate pilot data xr = (xri )ni=1
from N(µ, Σ)
2
Compute w?z (xr ) (or w?t (xr ))
3
Compute βz? (w?z (xr ))(or βt? (w?t (xr )))
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S IMULATIONS ALGORITHM
Set the inputs:
I
Sample sizes: nx , ny and the FPR: α
I
Prior hyperparameters: m0 , n0 , S0 , ν0
I
Parameters: µ, Σ
For r = 1, ..., R(= 15000):
1
x
Generate pilot data xr = (xri )ni=1
from N(µ, Σ)
2
Compute w?z (xr ) (or w?t (xr ))
3
Compute βz? (w?z (xr ))(or βt? (w?t (xr )))
We examine the produced distributions of βz? (X) (or βt? (X))
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T OTAL SAMPLE SIZE
F IGURE : Simulated percentiles of βt? (X) vs nt .
nT % ⇒ βz? (X), βt? (X)%
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S AMPLE ALLOCATION
F IGURE : Simulated percentiles of βt? (X) versus f = nx /nT , nT = 14.
Higher βz? (X), βt? (X) for balanced allocations (f ∈ (0.3, 0.5))
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P OWER COMPARISONS
Power of OLS, SS, PC and t? -test (median) for nx = 5, ny = 15,
e’s, βT 2 (nT ) = 0.38.
various m0 , n0 and µ = 0.90 × √ 0µe −1 . For all µ
µ
eΣ
1
2
3
4
5
6
7
8
9
10
µ
e
µ
e, K = 11
(1, ..., 1)0
βtOLS (nT )
βtSS (nT )
βtPC (nT )
0.97
0.96
0.96
(5, 1, ..., 1)0
0.16
0.14
0.14
(1/4, ..., 1/4)0
(1/4, 0, ..., 0)0
(6, 6, 4, 4, 2, 2, 1, ..., 1)0
0.28
0.26
0.27
(−6, 6, 4, 4, 2, 2, 1, ..., 1)0
0.07
0.07
0.07
(1/4, ..., 1/4)0
(0.1, 0.1, 0.05, 0.05, 0.05, 0.05, 0, ..., 0)0
(1/4, ..., 1/4)0
(−0.1, 0.05, 0.05, 0.05, 0.05, 0.05, 0, ..., 0)0
m0
(1/4, ..., 1/4)0
I
For small nT , the prior estimates of the treatment effect are
highly influential on βt?
I
For relatively precise prior estimates, βt? (ny ) is substantially
greater than βtOLS (nT ), βtSS (nT ), βtPC (nT )
n0
1
5
0
1
1
5
1
1
1
1
?
βt,0.50
(ny )
0.60
0.74
0.33
0.37
0.58
0.68
0.42
0.58
0.33
0.59
PH
MRI - ROI ANALYSIS
T ESTING FOR GTE
P ROPOSAL
P OWER A NALYSIS
D ISCUSSION
R EAL E XAMPLE - MRI STUDY: DATA
11 subjects participated in a GSK study informing drug
development using fMRI recordings
11 ROI were defined
We suppose that nx = 3, ny = 8
TABLE : Sample means (1,2), variances (3,4) and correlations (5-15) of x and y
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
ROI
xk
yk
sx,k
sy,k
AC
A
C
DLPFC
GP
I
OFC
P
SA
T
VS
AC
0.21
-0.14
0.54
0.22
1.00
0.40
0.78
0.78
0.73
0.85
0.48
0.71
0.32
0.86
0.63
A
0.23
-0.05
0.36
0.32
0.97
1.00
0.20
0.34
0.75
0.64
0.56
0.57
0.79
0.33
0.63
C
0.05
-0.15
0.18
0.15
0.99
0.99
1.00
0.96
0.66
0.71
0.37
0.72
0.19
0.81
0.62
DLPFC
-0.04
-0.13
0.36
0.19
0.99
0.95
0.98
1.00
0.68
0.71
0.52
0.73
0.36
0.84
0.69
GP
-0.02
-0.18
0.49
0.32
0.90
0.96
0.93
0.85
1.00
0.93
0.32
0.94
0.60
0.70
0.91
I
0.22
-0.13
0.30
0.24
0.97
0.99
0.99
0.94
0.97
1.00
0.30
0.82
0.49
0.70
0.78
OFC
-0.19
-0.12
0.62
0.27
0.94
0.99
0.96
0.89
0.99
0.99
1.00
0.24
0.32
0.58
0.30
P
0.20
-0.15
0.59
0.35
0.99
0.98
0.99
0.99
0.90
0.98
0.94
1.00
0.51
0.77
0.94
SA
0.06
-0.12
0.09
0.39
0.70
0.54
0.63
0.77
0.32
0.53
0.41
0.68
1.00
0.19
0.62
T
0.15
-0.25
0.47
0.23
0.99
0.98
0.99
0.99
0.91
0.98
0.95
0.99
0.67
1.00
0.73
VS
0.16
-0.22
0.44
0.26
0.79
0.90
0.84
0.73
0.98
0.90
0.95
0.81
0.13
0.81
1.00
PH
MRI - ROI ANALYSIS
T ESTING FOR GTE
P ROPOSAL
P OWER A NALYSIS
R EAL E XAMPLE - F MRI STUDY: RESULTS
TABLE : P-values of t? -test for various hyperparameters and nx = 3, ny = 8
compared with the p-values of T 2 , OLS, SS and PC tests.
1
2
3
6
7
8
9
10
11
12
13
14
15
16
17
I
I
pOLS
0.33
pSS
0.31
pPC
0.34
S0
CS:s20 = 0.05, r0 = 0.6
m0
0.1 × 1K
(0.1, ..., 0.1, 0.2, 0.2)
0 = 0.9
same but r3,4
0.1 × 1K
(0.1, ..., 0.1, 0.2, 0.2)
0 = 0.2
same but r3,4
0.1 × 1K
(0.1, ..., 0.1, 0.2, 0.2)
n0
0
1
3
1
3
0
1
3
1
3
0
1
3
1
3
pt?
0.70
0.29
0.22
0.02*
0.01*
0.53
0.20
0.17
0.02*
0.01*
0.97
0.32
0.14
0.03*
0.01*
The prior estimates are highly influential to the p-values
Even for fairly poor prior estimates, t? succeeds substantially
lower p-values than the other tests.
D ISCUSSION
PH
MRI - ROI ANALYSIS
T ESTING FOR GTE
P ROPOSAL
P OWER A NALYSIS
D ISCUSSION
S UMMARY
ROI analysis is useful for pharmacological fMRI
Available testing procedures often not suitable for detecting
global treatment effects across ROI in pH MRI
We proposed a novel optimal procedure based on multivariate
assumptions
The proposed test statistic is based on linear combinations of the
response.
PH
MRI - ROI ANALYSIS
T ESTING FOR GTE
P ROPOSAL
P OWER A NALYSIS
D ISCUSSION
S UMMARY
The weighting vector of the linear combination is a key element
of the test.
The procedure is split into two stages.
First-stage: collect information to select the weighting vector.
Second stage: collect observations and do the test.
The proposed testing procedures:
I
I
I
I
I
scale invariant test statistics
control the false positive rate
well behaved power function
exploit all sources of available information
efficient if the collected information is not misleading
Development to adaptive design and analysis
PH
MRI - ROI ANALYSIS
T ESTING FOR GTE
P ROPOSAL
P OWER A NALYSIS
D ISCUSSION
R EFERENCES
T. W. Anderson. An introduction to multivariate statistical analysis,
2nd edition. John Wiley and Sons, 2003
G. Honey, E. Bullmore, Human pharmacological MRI, Trends.
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R. A. Poldrack, Region of interest analysis for fMRI, SCAN, 2007, 2,
67-70
G. Minas, F. Rigat, T.E. Nichols, J.A.D. Aston. A hybrid procedure for
detecting Global Treatment Effects in Multivariate Clinical Trials, Stat.
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PH
MRI - ROI ANALYSIS
T ESTING FOR GTE
P ROPOSAL
P OWER A NALYSIS
D ISCUSSION
R EFERENCES
G. D. Mitsis, G. D. Iannetti, T. S. Smart, I. Tracey, and R. G. Wise.
Regions of interest analysis in pharmacological fMRI: How do the
definition criteria influence the inferred result? Neuroimage,
40:121-132, 2007
P. C. O’Brien. Procedures for comparing samples with multiple
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D. J. Spiegelhalter, K. R. Abrams, and J. P. Myles. Bayesian
approaches to clinical trials and health-care evaluation. John Wiley and
Sons, 2004
P. Westfall, A. Krishen, and S. Young. Using prior information to
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