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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 l.jpg

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 l.jpg

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 images l.jpg

Assessing StatisticImages…


Assessing statistic images4 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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


Multiple comparisons l.jpg

Multiple comparisons…


Hypothesis testing l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

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


Random field theory l.jpg

Random field theory…


Spm approach random fields l.jpg

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 l.jpg

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 l.jpg

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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 Minkowskifunctional 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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    Review:Levels of inference & power

    Set level…

    Cluster level…

    Voxel level…


    Random field theory limitations l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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


    False discovery rate l.jpg

    False Discovery Rate…


    Mcp solutions measuring false positives36 l.jpg

    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 l.jpg

    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 l.jpg

    False Discovery RateIllustration:

    Noise

    Signal

    Signal+Noise


    Slide39 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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 l.jpg

    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=0.001Cluster-wise FDR: 40 voxel cluster, PFDR 0.07Peak-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 l.jpg

    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 l.jpg

    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.


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