multiple testing
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Multiple testing
Justin ChumbleyLaboratory for Social and Neural Systems ResearchInstitute for Empirical Research in EconomicsUniversity of Zurich
With many thanks for slides & images to:FIL Methods group
Overview of SPM
Realignment Smoothing
Normalisation
General linear model
Statistical parametric map (SPM)Image time-series
Parameter estimates
Design matrix
Template
Kernel
Gaussian field theory
p <0.05
Statisticalinference
Inference at a single voxel
t =
contrast ofestimated
parameters
varianceestimate
t
Inference at a single voxel
H0 , H1: zero/non-zero activation
t =
contrast ofestimated
parameters
varianceestimate
t
Inference at a single voxel
Decision:H0 , H1: zero/non-zero activation
t =
contrast ofestimated
parameters
varianceestimate
t
h
Inference at a single voxel
Decision:H0 , H1: zero/non-zero activation
t =
contrast ofestimated
parameters
varianceestimate
t
h
h
Multiple tests
t =
contrast ofestimated
parameters
varianceestimate
t
h
ht
h
h
h
t
h
h
What is the problem?
Multiple tests
t =
contrast ofestimated
parameters
varianceestimate
t
h
ht
h
h
h
t
h
h
( 1 )
( )
hp or more FP FWERFPE FDR
All positives
Multiple tests
t =
contrast ofestimated
parameters
varianceestimate
t
h
ht
h
h
h
t
h
h hh
FWERN
h h
Bonferonni
FWER N
Convention: Choose h to limit assuming family-wise H0
hFWER
Smooth errors: the facts• intrinsic smoothness
– MRI signals are aquired in k-space (Fourier space); after projection on anatomical space, signals have continuous support
– diffusion of vasodilatory molecules has extended spatial support
• extrinsic smoothness– resampling during preprocessing– matched filter theorem
deliberate additional smoothing to increase SNR
Model errors as Gaussian random fields
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Surprising qualitative features in this landscape?
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Height Spatial extentLocationTotal number
• General form for expected Euler characteristic• 2, F, & t fields• restricted search regions
<Q ; h> = S Rd (W) rd (h)
Number of regions, Q?
Rd (W): RESEL count; depends on the search region – how big, how smooth, what shape ?
rd (h): EC density; depends on type of field (eg. Gaussian, t) and thethreshold, h.
Au
W
Worsley et al. (1996), HBM
• General form for expected Euler characteristic• 2, F, & t fields• restricted search regions
<Q ; h> = S Rd (W) rd (h)
Number of regions, Q?
Rd (W): RESEL count
R0(W) = (W) Euler characteristic ofWR1(W) = resel diameterR2(W) = resel surface areaR3(W) = resel volume
rd (h): d-dimensional EC density –E.g. Gaussian RF:
r0(h) = 1- (u)
r1(h) = (4 ln2)1/2 exp(-u2/2) / (2p)
r2(h) = (4 ln2) exp(-u2/2) / (2p)3/2
r3(h) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2p)2
r4(h) = (4 ln2)2 (u3 -3u) exp(-u2/2) / (2p)5/2
Au
W
Worsley et al. (1996), HBM
Space
height, h
Size of a region, S?)|( hHsp ii
1i 2i
1s 2s
Location of events?
Space
‘Poisson clumping heuristic’high maxima locations follow a spatial Poisson process.
Number of events: Poisson distributed, under high thresholds (Adler & Hasofer, 1981).
Location of events?
Space
‘Poisson clumping heuristic’high maxima locations follow a spatial Poisson process - Thinned
Number of events: Poisson distributed, under high thresholds (Adler & Hasofer, 1981).
‘F W E’ decisions1. Induce a set of regions,
2. Ensure
Each element of A departure from flat signal
95.0)0|(| , shAp
},...,,{ ,,,, 2211 NN shshshsh AAAA
))(;Poi(~|| , sSphQA ish
F W E
• Advantage– Adaptive to spatial noise (unlike voxel-wise
approaches, voxBonf/voxFDR).
• Disadvantage– Stingy
FDR
• Consider A … – not a set of positive decisions – a set of candidates, with unknown noise/ signal
partition• Decide on a partition?
– BH algorithm 1. Arranges candidates according to ‘improbability’ 2. Determines a partition
},{ hs
hn AAA
},{ hs
hn AA
FDR
• Select desired limit on FDR• Order p-values,
p(1) p(2) ... p(V)
• Let r be largest i such that
i.e. decides on a partition of A controlling
JRSS-B (1995)57:289-300
|)(|||)( AcAip i
sr Appp ˆ)()2()1( ,...,,
p(i)
i/|A|
(i/|A|) p-va
lue
0 1
01
i/|A|= proportion of all selected regions
||/||
||||
ˆ
ˆ, AAAA
EFDR ns
sn
Conclusions
• There is a multiple testing problem (From either ‘voxel’ or ‘blob’ perspective)
• ‘Corrections’ necessary to counterbalance this• FWE
– Random Field Theory• Inference about blobs (peaks, clusters)• Excellent for large sample sizes (e.g. single-subject analyses or large
group analyses)• Afford littles power for group studies with small sample size
consider non-parametric methods (not discussed in this talk)
• FDR• represents false positive risk over whole set of regions
• Height, spatial extent, total number
Further reading• Friston KJ, Frith CD, Liddle PF, Frackowiak RS. Comparing
functional (PET) images: the assessment of significant change. J Cereb Blood Flow Metab. 1991 Jul;11(4):690-9.
• Genovese CR, Lazar NA, Nichols T. Thresholding of statistical maps in functional neuroimaging using the false discovery rate. Neuroimage. 2002 Apr;15(4):870-8.
• Worsley KJ Marrett S Neelin P Vandal AC Friston KJ Evans AC. A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping 1996;4:58-73.
Thank you
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