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.
fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, …
First scan of fMRI data
Highly significant effect, T=6.59
1000
hot
rest
warm
890
880
870
500
0
100
200
300
No significant effect, T=-0.74
820
hot
rest
warm
0
800
T statistic for hot - warm effect
5
0
-5
T = (hot – warm effect) / S.d.
~ t110 if no effect
0
100
0
100
200
Drift
300
810
800
790
200
Time, seconds
300
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:
unknown parameters
?
?
?
errort = a1 errort-1 + … + ap errort-p + s WNt
DESIGN example: pain perception
Alternating hot and warm stimuli separated by rest (9 seconds each).
2
1
0
-1
0
50
100
150
200
250
300
350
Hemodynamic response function: difference of two gamma densities
0.4
0.2
0
-0.2
0
50
Responses = stimuli * HRF, sampled every 3 seconds
2
1
0
-1
0
50
100
150
200
Time, seconds
250
300
350
First step: estimate the autocorrelation
?
AR(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
~ -0.05
Bias corrected â1
~0
0.3
0.2
0.1
0
-0.1
Second step: refit the linear model
Pre-whiten: Yt* = Yt – â1 Yt-1, then fit using least squares:
Hot - warm effect, %
Sd of effect, %
1
0.25
0.2
0.5
0.15
0
0.1
-0.5
0.05
-1
0
T = effect / sd, 100 df
6
4
2
0
-2
-4
-6
T > 4.93
(P < 0.05,
corrected)
Why bother to smooth the acor?
Threshold
• Sample variability in estimated acor adds
variability to sd 14
• Lowers effective 12
10
df of T statistic
8
Corrected for whole
• Increases
brain search
6
threshold
4
One voxel
• Less power
2
• Particularly after 0 0
50
100
Df
correction for search
150
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
Continued …
• 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
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 + … + τp2)/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/FWHMdata2 + 1)D/2
• Define dfacor := f dfresidual
• 1/dfeff = 1/dfresidual + 2(τ12 + … + τp2)/dfacor
Hot, 1=0.61
Hot + Warm, 1=0.5
120 Residual df = 114
Sim, a1=
0.4
0.3
0.2
0.1
0
80
60
40
x=
20
0
100
Theory,
a1=0
Effective df
Effective df
100
120 Residual df = 114
0
Sim, a1=
0.4
0.3
0.2
0.1
0
80
60
40
x=
20
1
2
3
FWHMfilter/FWHMdata
0
4
0
1
2
3
FWHMfilter/FWHMdata
Hot - Warm, 1=0.79
80
60
40
100
Theory,
a1=0
x=
20
0
120 Residual df = 114
Effective df
Effective df
Sim, a1=
0.4
0.3
0.2
0.1
0
0
4
Cubic drift, 1=0.94
120 Residual df = 114
100
Theory,
a1=0
Sim, a1=
0
0.1
0.2
0.3
0.4
80
60
40
x=
20
1
2
3
FWHMfilter/FWHMdata
4
0
Theory,
a1=0
0
1
2
3
FWHMfilter/FWHMdata
4
Summary
• Variability in
2
3/2
FWHM
acor
acor lowers df
dfacor = dfresidual 2 FWHM 2 + 1
• Df depends
data
1
1
2 acor(contrast of data)2
on contrast
=
+
• Smoothing acor
dfeff dfresidual
dfacor
brings df back up:
Applications: Hot stimulus FWHMdata = 8.79 Hot-warm stimulus
(
Residual df = 110
100
Target = 100 df
50
Contrast of data, acor = 0.61
dfeff
0
0
10
20
30
FWHM = 10.3mm
FWHMacor
)
Residual df = 110
100
Target = 100 df
50
Contrast of data, acor = 0.79
dfeff
0
0
10
20
30
FWHM = 12.4mm FWHMacor
Application: Hot – warm stimulus
Autocorrelation a1
0.2
0
0
Effective df = 110
-5
Threshold = 5.25
Effective df = 49
5
0.4
0.2
0
0
Effective df = 1249
P = 0.05, corrected
5
0.4
No smoothing
12.4mm FWHM smoothing
T statistic for hot-warm
-5
Effective df = 100
Threshold = 4.93
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 …