Spatial smoothing of autocorrelations to control the degrees of freedom in

fMRI analysis

Keith Worsley

Department of Mathematics and Statistics, McGill University,

McConnell Brain Imaging Centre, Montreal Neurological Institute.

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1000First scan of fMRI data

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T statistic for hot - warm effect

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870880890 hot

restwarm

Highly significant effect, T=6.59

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820hotrestwarm

No significant effect, T=-0.74

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790800810

Drift

Time, seconds

fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …

T = (hot – warm effect) / S.d. ~ t110 if no effect

FMRISTAT: fits a linear model for fMRI time series with AR(p) errors

• Linear model: ? ? Yt = (stimulust * HRF) b + driftt c + errort

• AR(p) errors: ? ? ? errort = a1 errort-1 + … + ap errort-p + s WNt

unknown parameters

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2Alternating hot and warm stimuli separated by rest (9 seconds each).

hot

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Hemodynamic response function: difference of two gamma densities

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2Responses = stimuli * HRF, sampled every 3 seconds

Time, seconds

DESIGN example: pain perception

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First step: estimate the autocorrelationAR(1) model: errort = a1 errort-1 + s WNt

• Fit the linear model using least squares

• errort = Yt – fitted Yt

• â1 = Correlation ( errort , errort-1)

• Estimating errort’s changes their correlation structure slightly, so â1 is slightly biased:

Raw autocorrelation Smoothed 12.4mm Bias corrected â1

~ -0.05 ~ 0~ -0.05 ~ 0

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1 Hot - warm effect, %

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0.25Sd of effect, %

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6 T = effect / sd, 100 df

Pre-whiten: Yt* = Yt – â1 Yt-1, then fit using least squares:

Second step: refit the linear model

T > 4.93 (P < 0.05, corrected)

Why bother to smooth the acor?

• Sample variability in estimated acor adds variability to sd

• Lowers effective

df of T statistic

• Increases

threshold

• Less power

• Particularly after

correction for search 0 50 100 150

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Df

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Corrected for whole brain search

One voxel

Gautama et al. (2005): Smooth autocorrelations, choose amount of smoothing to optimally predict autocorrelations using e.g. cross-validation, model selection.

Effect of variability in sample acor on dbn of T: first idea

• Why not write linear model with e.g. AR(1) errors

Yt = xt’β + ηt, ηt = a1ηt-1 + εt

where εt iid ~N(0,σ2), as

Yt = a1Yt-1 + xt’β + xt-1’(a1β) + εt

• Least-squares estimates are ~max like, so

• Non-linear l.s.: dfeff ~ n-(#a)-(#β) …. ???? or

• Linear l.s.: dfeff ~ n-(#a)-(#β)-(#a)×(#β) …. ????

• Doesn’t work (see later) because: – design matrix is random?– ~max like only for large samples i.e. df = ∞?

Better idea: Harville et al. (1974), …, Kenward, Roger (1997) … SAS PROC MIXED …

• Linear model at a single voxel:

Y ~ Nn(Xβ, V(θ)), θ = (σ2, a1, …, ap)

• Fit by ReML, interested in effect

E = c’β, S = Sd(E)

• T = E / S

• E depends on β, S depends on θ

• β, θ ~independent so variability in θ only affects S

• S depends on θ, and from ReML theory we know ~mean, ~variance of θ.

• Use linear approx to S2(θ) to find ~mean, ~variance of S2

• dfeff is surrogate for variability of S2:

dfeff := 2 E(S2)2/Var(S2)

• Satterthwaite: S2 ~ cons×χ2dfeff , T ~ tdfeff

Continued …

Expression for dfeff

• dfeff depends on contrast(!) and θ,

– Could plug in θ, but don’t know θ in advance– Explicit expression if acors = 0– Hope it is a good approx for when acors ≠ 0

• Contrast in obs: x = X(X’X)-1c, so E = x’Y

• τj = lag j acor of x, dfresidual = least-squares df

• 1/dfeff = 1/dfresidual + 2(τ12 + … + τp

2)/dfresidual

Effect of smoothing acor

• Assume ε ~ white noise smoothed by Gaussian filter, width FWHMdata, GRF(FWHMdata)

• Autocors ~ GRF(FWHMdata/√2)

• Smoothing acors in D dimensions by FWHMacor reduces variance by

f = (2 FWHMacor2/FWHMdata

2 + 1)D/2

• Define dfacor := f dfresidual

• 1/dfeff = 1/dfresidual + 2(τ12 + … + τp

2)/dfacor

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Sim, a1=

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Hot, 1=0.61

FWHMf ilter

/FWHMdata

Eff

ectiv

e df

Residual df = 114

Theory,a

1=0

x=

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Sim, a1=

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Hot + Warm, 1=0.5

FWHMf ilter

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Eff

ectiv

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Residual df = 114

Theory,a

1=0

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Sim, a1=

00.10.20.30.4

Hot - Warm, 1=0.79

FWHMf ilter

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Eff

ectiv

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Residual df = 114

Theory,a

1=0

x=

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Sim, a1=

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Cubic drift, 1=0.94

FWHMf ilter

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Eff

ectiv

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Residual df = 114

Theory,a

1=0

x=

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FWHMacor

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FWHMacor

Summary

Applications: Hot stimulus Hot-warm stimulus

Target = 100 df

Residual df = 110

Target = 100 df

Residual df = 110

FWHM = 10.3mm FWHM = 12.4mm

dfacor = dfresidual(2 + 1) 1 1 2 acor(contrast of data)2

dfeff dfresidual dfacor

FWHMacor2 3/2

FWHMdata2

= +

• Variability in acor lowers df• Df depends on contrast • Smoothing acor brings df back up:

Contrast of data, acor = 0.79Contrast of data, acor = 0.61

FWHMdata = 8.79

dfeff dfeff

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Autocorrelation a1

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smoo

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Effective df = 110

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FW

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sm

ooth

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Effective df = 1249

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T statistic for hot-warm

Effective df = 49

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Effective df = 100

P = 0.05, corrected

Threshold = 5.25

Threshold = 4.93

Application: Hot – warm stimulus

Refinements

• Could get a rough estimate of acor first, then use this to get better estimate of dfeff, but this is time consuming

• Acor varies spatially, so dfeff varies spatially, but we don’t have any random field theory for P-values

• Could use spatially varying filter to achieve ~constant dfeff, but again this is time consuming

• All the theory built on asymptotic and/or questionable assumptions, so maybe can’t take it too far …