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Inference on SPMs: Random Field Theory & Alternatives. Thomas Nichols, Ph.D. Department of Statistics & Warwick Manufacturing Group University of Warwick FIL SPM Course 12 May, 2011. image data. parameter estimates. design matrix. kernel. Thresholding & Random Field Theory.

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inference on spms random field theory alternatives
Inference on SPMs:Random Field Theory & Alternatives

Thomas Nichols, Ph.D.

Department of Statistics &

Warwick Manufacturing Group

University of Warwick

FIL SPM Course

12 May, 2011

slide2

image data

parameter

estimates

designmatrix

kernel

Thresholding &Random Field Theory

  • General Linear Model
    • model fitting
    • statistic image

realignment &motioncorrection

smoothing

normalisation

StatisticalParametric Map

anatomicalreference

Corrected thresholds & p-values

assessing statistic images4

t > 0.5

t > 3.5

t > 5.5

Assessing Statistic Images

Where’s the signal?

High Threshold

Med. Threshold

Low Threshold

Good SpecificityPoor Power(risk of false negatives)

Poor Specificity(risk of false positives)Good Power

  • ...but why threshold?!
blue sky inference what we d like
Blue-sky inference:What we’d like
  • Don’t threshold, model the signal!
    • Signal location?
      • Estimates and CI’s on(x,y,z) location
    • Signal magnitude?
      • CI’s on % change
    • Spatial extent?
      • Estimates and CI’s on activation volume
      • Robust to choice of cluster definition
  • ...but this requires an explicit spatial model

space

blue sky inference what we need
Blue-sky inference:What we need
  • Need an explicit spatial model
  • No routine spatial modeling methods exist
    • High-dimensional mixture modeling problem
    • Activations don’t look like Gaussian blobs
    • Need realistic shapes, sparse representation
      • Some work by Hartvig et al., Penny et al.
real life inference what we get
Real-life inference:What we get
  • Signal location
    • Local maximum – no inference
    • Center-of-mass – no inference
      • Sensitive to blob-defining-threshold
  • Signal magnitude
    • Local maximum intensity – P-values (& CI’s)
  • Spatial extent
    • Cluster volume – P-value, no CI’s
      • Sensitive to blob-defining-threshold
voxel level inference
Voxel-level Inference
  • Retain voxels above -level threshold u
  • Gives best spatial specificity
    • The null hyp. at a single voxel can be rejected

u

space

Significant Voxels

No significant Voxels

cluster level inference
Cluster-level Inference
  • Two step-process
    • Define clusters by arbitrary threshold uclus
    • Retain clusters larger than -level threshold k

uclus

space

Cluster not significant

Cluster significant

k

k

cluster level inference10
Cluster-level Inference
  • Typically better sensitivity
  • Worse spatial specificity
    • The null hyp. of entire cluster is rejected
    • Only means that one or more of voxels in cluster active

uclus

space

Cluster not significant

Cluster significant

k

k

set level inference
Set-level Inference
  • Count number of blobs c
    • Minimum blob size k
  • Worst spatial specificity
    • Only can reject global null hypothesis

uclus

space

k

k

Here c = 1; only 1 cluster larger than k

hypothesis testing

u

Null Distribution of T

t

P-val

Null Distribution of T

Hypothesis Testing
  • Null Hypothesis H0
  • Test statistic T
    • t observed realization of T
  •  level
    • Acceptable false positive rate
    • Level  = P( T>u | H0)
    • Threshold u controls false positive rate at level 
  • P-value
    • Assessment of t assuming H0
    • P( T > t | H0 )
      • Prob. of obtaining stat. as largeor larger in a new experiment
    • P(Data|Null) not P(Null|Data)
multiple comparisons problem

t > 2.5

t > 4.5

t > 0.5

t > 1.5

t > 3.5

t > 5.5

t > 6.5

Multiple Comparisons Problem
  • Which of 100,000 voxels are sig.?
    • =0.05  5,000 false positive voxels
  • Which of (random number, say) 100 clusters significant?
    • =0.05  5 false positives clusters
mcp solutions measuring false positives
MCP Solutions:Measuring False Positives
  • Familywise Error Rate (FWER)
    • Familywise Error
      • Existence of one or more false positives
    • FWER is probability of familywise error
  • False Discovery Rate (FDR)
    • FDR = E(V/R)
    • R voxels declared active, V falsely so
      • Realized false discovery rate: V/R
mcp solutions measuring false positives16
MCP Solutions:Measuring False Positives
  • Familywise Error Rate (FWER)
    • Familywise Error
      • Existence of one or more false positives
    • FWER is probability of familywise error
  • False Discovery Rate (FDR)
    • FDR = E(V/R)
    • R voxels declared active, V falsely so
      • Realized false discovery rate: V/R
fwe mcp solutions bonferroni
FWE MCP Solutions: Bonferroni
  • For a statistic image T...
    • Tiith voxel of statistic image T
  • ...use  = 0/V
    • 0 FWER level (e.g. 0.05)
    • V number of voxels
    • u -level statistic threshold, P(Ti u) = 
  • By Bonferroni inequality...

FWER = P(FWE) = P( i {Tiu} | H0)i P( Tiu| H0 )

= i = i0 /V = 0

Conservative under correlation

Independent: V tests

Some dep.: ? tests

Total dep.: 1 test

spm approach random fields
Consider statistic image as lattice representation of a continuous random field

Use results from continuous random field theory

SPM approach:Random fields…

lattice represtntation

fwer mcp solutions random field theory
FWER MCP Solutions:Random Field Theory
  • Euler Characteristic u
    • Topological Measure
      • #blobs - #holes
    • At high thresholds,just counts blobs
    • FWER = P(Max voxel u | Ho) = P(One or more blobs | Ho) P(u  1 | Ho) E(u| Ho)

Threshold

Random Field

No holes

Never more than 1 blob

Suprathreshold Sets

rft details expected euler characteristic

Only very upper tail approximates1-Fmax(u)

RFT Details:Expected Euler Characteristic

E(u) () ||1/2 (u 2 -1) exp(-u 2/2) / (2)2

    • Search regionR3
    • (volume
    • ||1/2roughness
  • Assumptions
    • Multivariate Normal
    • Stationary*
    • ACF twice differentiable at 0
  • Stationary
    • Results valid w/out stationary
    • More accurate when stat. holds
random field theory smoothness parameterization

FWHM

Autocorrelation Function

Random Field TheorySmoothness Parameterization
  • E(u) depends on ||1/2
    •  roughness matrix:
  • Smoothness parameterized as Full Width at Half Maximum
    • FWHM of Gaussian kernel needed to smooth a whitenoise random field to roughness 
random field theory smoothness parameterization23

1

2

3

4

5

6

7

8

9

10

1

2

3

4

Random Field TheorySmoothness Parameterization
  • RESELS
    • Resolution Elements
    • 1 RESEL = FWHMx FWHMy FWHMz
    • RESEL Count R
      • R = () || = (4log2)3/2 () / ( FWHMx FWHMy FWHMz )
      • Volume of search region in units of smoothness
      • Eg: 10 voxels, 2.5 FWHM 4 RESELS
  • Beware RESEL misinterpretation
    • RESEL are not “number of independent ‘things’ in the image”
      • See Nichols & Hayasaka, 2003, Stat. Meth. in Med. Res.

.

random field theory smoothness estimation
Random Field TheorySmoothness Estimation
  • Smoothness est’dfrom standardizedresiduals
    • Variance ofgradients
    • Yields resels pervoxel (RPV)
  • RPV image
    • Local roughness est.
    • Can transform in to local smoothness est.
      • FWHM Img = (RPV Img)-1/D
      • Dimension D, e.g. D=2 or 3
random field intuition
Random Field Intuition
  • Corrected P-value for voxel value t

Pc = P(max T > t) E(t) () ||1/2t2 exp(-t2/2)

  • Statistic value t increases
    • Pc decreases (but only for large t)
  • Search volume increases
    • Pc increases (more severe MCP)
  • Roughness increases (Smoothness decreases)
    • Pc increases (more severe MCP)
rft details unified formula
RFT Details:Unified Formula
  • General form for expected Euler characteristic
      • 2, F, & t fields • restricted search regions •D dimensions •

E[u(W)] = SdRd(W)rd (u)

Rd (W):d-dimensional Minkowski functional of W

– function of dimension, spaceWand smoothness:

R0(W) = (W) Euler characteristic of W

R1(W) = resel diameter

R2(W) = resel surface area

R3(W) = resel volume

rd (W):d-dimensional EC density of Z(x)

– function of dimension and threshold, specific for RF type:

E.g. Gaussian RF:

r0(u) = 1- (u)

r1(u) = (4 ln2)1/2 exp(-u2/2) / (2p)

r2(u) = (4 ln2) exp(-u2/2) / (2p)3/2

r3(u) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2p)2

r4(u) = (4 ln2)2 (u3 -3u) exp(-u2/2) / (2p)5/2

random field theory cluster size tests
Expected Cluster Size

E(S) = E(N)/E(L)

S cluster size

N suprathreshold volume({T > uclus})

L number of clusters

E(N) = () P( T > uclus )

E(L)  E(u)

Assuming no holes

5mm FWHM

10mm FWHM

15mm FWHM

Random Field TheoryCluster Size Tests
random field theory cluster size distribution
Random Field TheoryCluster Size Distribution
  • Gaussian Random Fields (Nosko, 1969)
    • D: Dimension of RF
  • t Random Fields (Cao, 1999)
    • B: Beta distn
    • U’s: 2’s
    • c chosen s.t.E(S) = E(N) / E(L)
random field theory cluster size corrected p values
Random Field TheoryCluster Size Corrected P-Values
  • Previous results give uncorrected P-value
  • Corrected P-value
    • Bonferroni
      • Correct for expected number of clusters
      • Corrected Pc = E(L) Puncorr
    • Poisson Clumping Heuristic (Adler, 1980)
      • Corrected Pc = 1 - exp( -E(L) Puncorr )
review levels of inference power
Review:Levels of inference & power

Set level…

Cluster level…

Voxel level…

random field theory limitations

Lattice ImageData

Continuous Random Field

Random Field Theory Limitations
  • Sufficient smoothness
    • FWHM smoothness 3-4× voxel size (Z)
    • More like ~10× for low-df T images
  • Smoothness estimation
    • Estimate is biased when images not sufficiently smooth
  • Multivariate normality
    • Virtually impossible to check
  • Several layers of approximations
  • Stationary required for cluster size results
real data

Active

...

...

yes

Baseline

...

...

D

UBKDA

N

XXXXX

no

Real Data
  • fMRI Study of Working Memory
    • 12 subjects, block design Marshuetz et al (2000)
    • Item Recognition
      • Active:View five letters, 2s pause, view probe letter, respond
      • Baseline: View XXXXX, 2s pause, view Y or N, respond
  • Second Level RFX
    • Difference image, A-B constructedfor each subject
    • One sample t test
real data rft result
Real Data:RFT Result
  • Threshold
    • S = 110,776
    • 2  2  2 voxels5.1  5.8  6.9 mmFWHM
    • u = 9.870
  • Result
    • 5 voxels above the threshold
    • 0.0063 minimumFWE-correctedp-value

-log10 p-value

real data snpm promotional
Nonparametric method more powerful than RFT for low DF

“Variance Smoothing” even more sensitive

FWE controlled all the while!

uRF = 9.87uBonf = 9.805 sig. vox.

t11Statistic, RF & Bonf. Threshold

378 sig. vox.

uPerm = 7.67 58 sig. vox.

Smoothed Variance t Statistic,Nonparametric Threshold

t11Statistic, Nonparametric Threshold

Real Data:SnPM Promotional
mcp solutions measuring false positives36
MCP Solutions:Measuring False Positives
  • Familywise Error Rate (FWER)
    • Familywise Error
      • Existence of one or more false positives
    • FWER is probability of familywise error
  • False Discovery Rate (FDR)
    • FDR = E(V/R)
    • R voxels declared active, V falsely so
      • Realized false discovery rate: V/R
false discovery rate37
False Discovery Rate
  • For any threshold, all voxels can be cross-classified:
  • Realized FDR

rFDR = V0R/(V1R+V0R) = V0R/NR

    • If NR = 0, rFDR = 0
  • But only can observe NR, don’t know V1R & V0R
    • We control the expected rFDR

FDR = E(rFDR)

false discovery rate illustration
False Discovery RateIllustration:

Noise

Signal

Signal+Noise

slide39

11.3%

11.3%

12.5%

10.8%

11.5%

10.0%

10.7%

11.2%

10.2%

9.5%

6.7%

10.5%

12.2%

8.7%

10.4%

14.9%

9.3%

16.2%

13.8%

14.0%

Control of Per Comparison Rate at 10%

Percentage of Null Pixels that are False Positives

Control of Familywise Error Rate at 10%

FWE

Occurrence of Familywise Error

Control of False Discovery Rate at 10%

Percentage of Activated Pixels that are False Positives

benjamini hochberg procedure

p(i) i/V q

Benjamini & HochbergProcedure
  • Select desired limit q on FDR
  • Order p-values, p(1)p(2) ...  p(V)
  • Let r be largest i such that
  • Reject all hypotheses corresponding top(1), ... , p(r).

JRSS-B (1995)57:289-300

1

p(i)

p-value

i/V q

0

0

1

i/V

adaptiveness of benjamini hochberg fdr
Adaptiveness of Benjamini & Hochberg FDR

Ordered p-values p(i)

P-value threshold when no signal: /V

P-value thresholdwhen allsignal:

Fractional index i/V

real data fdr example

FWER Perm. Thresh. = 9.877 voxels

Real Data: FDR Example
  • Threshold
    • Indep/PosDepu = 3.83
    • Arb Covu = 13.15
  • Result
    • 3,073 voxels aboveIndep/PosDep u
    • <0.0001 minimumFDR-correctedp-value

FDR Threshold = 3.833,073 voxels

fdr changes
FDR Changes
  • Before SPM8
    • Only voxel-wise FDR
  • SPM8
    • Cluster-wise FDR
    • Peak-wise FDR
    • Voxel-wise available: edit spm_defaults.m to readdefaults.stats.topoFDR = 0;

Item Recognition data

Cluster-forming threshold P=0.001 Cluster-wise FDR: 40 voxel cluster, PFDR 0.07 Peak-wise FDR: t=4.84, PFDR 0.836

Cluster-forming threshold P=0.01

Cluster-wise FDR: 1250 - 4380 voxel clusters, PFDR <0.001

Cluster-wise FDR: 80 voxel cluster, PFDR 0.516

Peak-wise FDR: t=4.84, PFDR 0.027

conclusions
Conclusions
  • Must account for multiplicity
    • Otherwise have a fishing expedition
  • FWER
    • Very specific, not very sensitive
  • FDR
    • Voxel-wise: Less specific, more sensitive
    • Cluster-, Peak-wise: Similar to FWER
references
References
  • Most of this talk covered in these papers

TE Nichols & S Hayasaka, Controlling the Familywise Error Rate in Functional Neuroimaging: A Comparative Review. Statistical Methods in Medical Research, 12(5): 419-446, 2003.

TE Nichols & AP Holmes, Nonparametric Permutation Tests for Functional Neuroimaging: A Primer with Examples. Human Brain Mapping, 15:1-25, 2001.

CR Genovese, N Lazar & TE Nichols, Thresholding of Statistical Maps in Functional Neuroimaging Using the False Discovery Rate. NeuroImage, 15:870-878, 2002.