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.
0
500
1000First scan of fMRI data
-5
0
5
T statistic for hot - warm effect
0 100 200 300
870880890 hot
restwarm
Highly significant effect, T=6.59
0 100 200 300
800
820hotrestwarm
No significant effect, T=-0.74
0 100 200 300
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
0 50 100 150 200 250 300 350-1
0
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2Alternating hot and warm stimuli separated by rest (9 seconds each).
hot
warm
hot
warm
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0.4
Hemodynamic response function: difference of two gamma densities
0 50 100 150 200 250 300 350-1
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2Responses = stimuli * HRF, sampled every 3 seconds
Time, seconds
DESIGN example: pain perception
-0.1
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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
?
-1
-0.5
0
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1 Hot - warm effect, %
0
0.05
0.1
0.15
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0.25Sd of effect, %
-6
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-2
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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
0
2
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Df
Th
resh
old
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
0 1 2 3 40
20
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Sim, a1=
00.10.20.30.4
Hot, 1=0.61
FWHMf ilter
/FWHMdata
Eff
ectiv
e df
Residual df = 114
Theory,a
1=0
x=
0 1 2 3 40
20
40
60
80
100
120
Sim, a1=
00.10.20.30.4
Hot + Warm, 1=0.5
FWHMf ilter
/FWHMdata
Eff
ectiv
e df
Residual df = 114
Theory,a
1=0
x=
0 1 2 3 40
20
40
60
80
100
120
Sim, a1=
00.10.20.30.4
Hot - Warm, 1=0.79
FWHMf ilter
/FWHMdata
Eff
ectiv
e df
Residual df = 114
Theory,a
1=0
x=
0 1 2 3 40
20
40
60
80
100
120
Sim, a1=
00.10.20.30.4
Cubic drift, 1=0.94
FWHMf ilter
/FWHMdata
Eff
ectiv
e df
Residual df = 114
Theory,a
1=0
x=
0 10 20 30
0
50
100
FWHMacor
0 10 20 300
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100
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
0
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Autocorrelation a1
No
smoo
thin
g
Effective df = 110
0
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12.4
mm
FW
HM
sm
ooth
ing
Effective df = 1249
-5
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T statistic for hot-warm
Effective df = 49
-5
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5
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 …
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