1. 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.

2. fMRI data: 120 scans, 3 scans each of hot, rest, warm, rest, hot, rest, … T = (hot – warm effect) / S.d. ~ t 110 if no effect

3. FMRISTAT: fits a linear model for fMRI time series with AR(p) errors Linear model: ? ? Y t = (stimulus t * HRF) b + drift t c + error t AR(p) errors: ? ? ? error t = a 1 error t-1 + … + a p error t-p + s WN t unknown parameters

4. 050100150200250300350 0 1 2 Alternating hot and warm stimuli separated by rest (9 seconds each). hot warm hot warm 050 -0.2 0 0.2 0.4 Hemodynamic response function: difference of two gamma densities 050100150200250300350 0 1 2 Responses = stimuli * HRF, sampled every 3 seconds Time, seconds DESIGN example: pain perception

5. First step: estimate the autocorrelation AR(1) model: error t = a 1 error t-1 + s WN t Fit the linear model using least squares error t = Y t – fitted Y t â 1 = Correlation ( error t, error t-1 ) Estimating error t ’s changes their correlation structure slightly, so â 1 is slightly biased: Raw autocorrelation Smoothed 12.4mm Bias corrected â 1 ~ -0.05 ~ 0 ?

6. -0.5 0 0.5 1 Hot - warm effect, % 0 0.05 0.1 0.15 0.2 0.25 Sd of effect, % -6 -4 -2 0 2 4 6 T = effect / sd, 100 df Pre-whiten: Y t * = Y t – â 1 Y t-1, then fit using least squares: Second step: refit the linear model T > 4.93 (P < 0.05, corrected)

7. 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 050100150 0 2 4 6 8 10 12 14 Df Threshold Corrected for whole brain search One voxel

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

9. Effect of variability in sample acor on dbn of T: first idea Why not write linear model with e.g. AR(1) errors Y t = x t ’β + η t, η t = a 1 η t-1 + ε t where ε t iid ~N(0,σ 2 ), as Y t = a 1 Y t-1 + x t ’β + x t-1 ’(a 1 β) + ε t Least-squares estimates are ~max like, so Non-linear l.s.: df eff ~ n-(#a)-(#β) …. ???? or Linear l.s.: df eff ~ n-(#a)-(#β)-(#a)×(#β) …. ???? Doesn’t work (see later) because: –design matrix is random? –~max like only for large samples i.e. df = ∞?

10. Better idea: Harville et al. (1974), …, Kenward, Roger (1997) … SAS PROC MIXED … Linear model at a single voxel: Y ~ N n (Xβ, V(θ)), θ = (σ 2, a 1, …, a p ) 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

11. S depends on θ, and from ReML theory we know ~mean, ~variance of θ. Use linear approx to S 2 (θ) to find ~mean, ~variance of S 2 df eff is surrogate for variability of S 2 : df eff := 2 E(S 2 ) 2 /Var(S 2 ) Satterthwaite: S 2 ~ cons×χ 2 dfeff, T ~ t dfeff Continued …

12. Expression for df eff df eff 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) -1 c, so E = x’Y τ j = lag j acor of x, df residual = least-squares df 1/df eff = 1/df residual + 2(τ 1 2 + … + τ p 2 )/df residual

13. Effect of smoothing acor Assume ε ~ white noise smoothed by Gaussian filter, width FWHM data, GRF(FWHM data ) Autocors ~ GRF(FWHM data /√2) Smoothing acors in D dimensions by FWHM acor reduces variance by f = (2 FWHM acor 2 /FWHM data 2 + 1) D/2 Define df acor := f df residual 1/df eff = 1/df residual + 2(τ 1 2 + … + τ p 2 )/df acor

15. 0102030 0 50 100 FWHM acor 0102030 0 50 100 FWHM acor Summary Applications: Hot stimulus Hot-warm stimulus Target = 100 df Residual df = 110 Target = 100 df Residual df = 110 FWHM = 10.3mmFWHM = 12.4mm df acor = df residual ( 2 + 1 ) 1 1 2 acor(contrast of data) 2 df eff df residual df acor FWHM acor 2 3/2 FWHM data 2 = + Variability in acor lowers df Df depends on contrast Smoothing acor brings df back up: Contrast of data, acor = 0.79 Contrast of data, acor = 0.61 FWHM data = 8.79 df eff

16. 0 0.2 0.4 Autocorrelation a 1 No smoothing Effective df = 110 0 0.2 0.4 12.4mm FWHM smoothing Effective df = 1249 -5 0 5 T statistic for hot-warm Effective df = 49 -5 0 5 Effective df = 100 P = 0.05, corrected Threshold = 5.25 Threshold = 4.93 Application: Hot – warm stimulus

17. Refinements Could get a rough estimate of acor first, then use this to get better estimate of df eff, but this is time consuming Acor varies spatially, so df eff varies spatially, but we don’t have any random field theory for P-values Could use spatially varying filter to achieve ~constant df eff, but again this is time consuming All the theory built on asymptotic and/or questionable assumptions, so maybe can’t take it too far …