Classical Inference on SPMs

Justin Chumbley

http://www.fil.ion.ucl.ac.uk/~jchumb/

SPM Course

Oct 23, 2008

realignment &motion

correctionsmoothing

normalisation

General Linear Modelmodel fittingstatistic image

Corrected thresholds & p-values

image data parameterestimates

designmatrix

anatomicalreference

kernel

StatisticalParametric Map

Random Field Theory

Frequentist ‘exceedence probabilities’: p(H>h)

1. (if h is fixed before)– a long-run property of

the decision-rule, i.e. all data-realisations

– E[ I(H>h) ]

2. ‘p-value’ (if h is observed data)

– a property of the this specific observation

3. just a parameter of a distribution

– (like dn on numbers in the set).

'' h

h

p-val

Null Distribution of H

h

Null Distribution of H

h

• More exceedence probabilities…

Bin(x|20, 0.3)Poi (x|20, 0.3)

Spatially independent noise

Independent Gaussian null Bernoulli Process

h

Null Distribution of T

h

N voxels

How many errors?

Errors accumulate

• AverageAverage number of errors is number of errors is• t = Number of errorst = Number of errors

(independence)(independence)

– Set h to ensure Bernoulli process rarely reaches height criterion anywhere in the field.

N

th

tNh

thh

tp

t

Ntp

1

);(

)1();(

Gives similar h to Bonferonni

hN

Independent Voxels Spatially Correlated Voxels

This is the WRONG model:1. Noise (Binomial/Bonferonni too conservative under spatially

dependent data)• There are geometric features in the noise:

2. Signal (under alternative distribution)• signal changes smoothly: neighbouring voxels should have

similar signal• signal is everywhere/nowhere (due to smoothing, K-space,

distributed neuronal responses)

WRONG APPROACH

Space

Repeatable

Space

Repeatable

Unrepeatable

Space

Repeatable

Unrepeatable

Observation

Binary decisions on signal geometry: How?!

• Set a joint threshold (H>h,S>s) to define a

set of regions with this geometric property.

One positive region

One departure from null/flat signal-geometry.

But how to calculate the number t of false-positive regions under the null!

Topological inference

• As in temporal analysis…– Assume a model for spatial dependence

• A Continuous Gaussian field vs Discrete 1st order Markov– estimate spatial dependence (under null)

• Use the component residual fields– Set a joint threshold (H>h,S>s) to define a class of regions with

some geometric property.

hs

Space

unrepeatable

Topological inference

• As in temporal analysis…– Assume a model for spatial dependence

• A Continuous Gaussian field vs Discrete 1st order Markov– estimate spatial dependence (under null)

• Use the component residual fields– Set a joint threshold (H>h,S>s) to define a class of regions with

some geometric property. Count regions whose topology surpasses threshold:

Space

h

s

R1R0

Topological inference

• As in temporal analysis…– Assume a model for spatial dependence

• A Continuous Gaussian field vs Discrete 1st order Markov– estimate spatial dependence (under null)

• Use the component residual fields– Set a joint threshold (H>h,S>s) to define a class of regions with

some geometric property. Count regions whose topology surpasses threshold:

Calibrate class definition, , to control false-positive class members.

What is the average number of false-positives?

hs

Topological inference• For ‘high’ h, assuming that errors are a Gaussian Field.

E(topological-false-positives per brain) =

sh

sSPhHP

sShHP

)()(

)(

!

)(),;(

t

etp

shtsh

sh

1

),;(t

shtp

hs

Topological attributes

)( hHP

Topological measure– threshold an image at h

– excursion set h

h) = # blobs - # holes

- At high h, h) = # blobs

P(h) > 0 )

)( hHP

)( hHP

• General form for expected Euler characteristic• 2, F, & t fields• restricted search regions

αh = Rd () d (h)

Unified Theory

Rd (): RESEL count; depends on

the search region – how big, how

smooth, what shape ?

d (h): EC density; depends on

type of field (eg. Gaussian, t) and thethreshold, h.

Au

Worsley et al. (1996), HBM

• General form for expected Euler characteristic• 2, F, & t fields• restricted search regions

αh = Rd () d (h)

Unified Theory

Rd (): RESEL count

R0() = () Euler characteristic of

R1() = resel diameter

R2() = resel surface area

R3() = resel volume

d (h): d-dimensional EC density –

E.g. Gaussian RF:

0(h) = 1- (u)

1(h) = (4 ln2)1/2 exp(-u2/2) / (2)

2(h) = (4 ln2) exp(-u2/2) / (2)3/2

3(h) = (4 ln2)3/2 (u2 -1) exp(-u2/2) / (2)2

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

Au

Worsley et al. (1996), HBM

5mm FWHM

10mm FWHM

15mm FWHM

Topological attributes

• Expected Cluster Size– E(S) = E(N)/E(L)– S cluster size– N suprathreshold

volume– L number of clusters

)( sSP

5mm FWHM

10mm FWHM

15mm FWHM

(2mm2 pixels)

Topological attributesunder independence

)()()( sSPhHPsShHP

shsSPhHPsShHP )()()(

3 related exceedence probabilities:

• Set-level

),( shfixed

ct

tsh

t

ecTp

sh

!

)()(

Summary: Topological F W E

• Brain images have spatially organised signal and noise. • Take this into account when compressing our 4-d data.• SPM infers the presence of departures from flat signal

geometry• inversely related (for fixed )• Exploit this for tall-thin/short-broad within one framework.

– ‘Peak’ level is optimised for tall-narrow departures– ‘Cluster’ level is for short-broad departures. – ‘Set’ level tells us there is an unusually large number

of regions.

)( shf

FDR

• Controls E( false-positives/total-positives )

• Doesn’t specify the subject of inference.

• On voxels?

• Preferably on Topological features.

THE END

Useful References

• http://www.fil.ion.ucl.ac.uk/spm/doc/biblio/Keyword/RFT.html

• http://www.math.mcgill.ca/keith/unified/unified.pdf

• http://www.sph.umich.edu/~nichols/Docs/FWEfNI.pdf