Spatial smoothing of autocorrelations to control the degrees of freedom in fMRI analysis
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Transcript of Spatial smoothing of autocorrelations to control the degrees of freedom in fMRI analysis
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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T statistic for hot - warm effect
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870880890 hot
restwarm
Highly significant effect, T=6.59
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No significant effect, T=-0.74
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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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Hemodynamic response function: difference of two gamma densities
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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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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
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Residual df = 114
Theory,a
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FWHMf ilter
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Residual df = 114
Theory,a
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Hot - Warm, 1=0.79
FWHMf ilter
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Residual df = 114
Theory,a
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Sim, a1=
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Cubic drift, 1=0.94
FWHMf ilter
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Residual df = 114
Theory,a
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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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Effective df = 110
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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 …