Covariance, autocorrelation & non-sphericity

1. Covariance, autocorrelation & non-sphericity Methods for Dummies

2. Ground to Cover • Quick Recap (IID etc) • Modelling the covariance matrix • Estimating the best model • Using the model to run more accurate stats 2/10

3. Y=βX+ε ε3 ε2 at same voxel error terms correlated across time ε1 Quick Recap • Considering temporal correlation • IID violated by short range serial effects (e.g. breathing) • Error terms correlated • t & F test too liberal (df & variance of parameters) 3/10

4. So What To Do? Model It • Capture the form of the Cov(єk) in a GLM & use this to correct the stats ID matrix auto-correlation white noise Cov(єk) = λ1Q1 + λ2Q2 + λ3Q3 + … correlation matrix hyperparameters to estimate basis set elements 4/10

5. The Same Thing, With Words • Auto-regressive [order 1] plus white noise model (AR[1]+wn) • Describes error at a voxel & relates to temporal neighbours • Requires 3 hyper-parameters & gives Cov(єk) • Estimated Cov(єk) with ReML as two components: • variance (local) • correlation matrix (global) 5/10

6. σ1 ε1 ε2 σ2 σ3 ε3 ε4 σ4 x s11 1 0.5 s12 0.2 s13 s14 0 s21 0.5 1 s22 0.5 s23 s24 0.2 Correlation matrix (V) Variance (σk2) Error term (ε) Covariance matrix Cov(εk) s31 0.2 s32 0.5 s33 1 s34 0.5 0 s41 s42 0.2 0.5 s43 s44 1 Local A Voxel by Voxel Account σ1 Global (good estimate) 6/10

7. We Have Cov(εk) - What Next? • Use it for t & F-tests: Components of model appear in new t statistic • Normally use t-statistic with DF to get p-value • But there’s a problem: V isn’t spherical so denominator isn’t a t-distribution 7/10

8. c σ2 c ρσ2 c ρσ2 c ρσ2 c ρσ2 σ2 c c ρσ2 ρσ2 c ρσ2 c c ρσ2 σ2 c c ρσ2 c ρσ2 ρσ2 c ρσ2 c c σ2 So Correct the Number of DF • Box’s measure (ε) measures Cov(εk) departure from spherical ε 1 1/(k-1) Number of measures Spherical Completely unspherical • Use εto correct DF using Satterthwaite approximation (Greenhouse-Geisser) 8/10

9. You’ve Never Had It So Good Temporal smoothing swamps autocorrelation & assume IID. Too liberal Temporal smoothing. Assume simple auto-correlation. Satterthwaite DF correction based on model Cov(εk). Less liberal. 7 6 5 AR(1) plus white noise. ReML etc. Best solution so far. 4 T value 3 2 1 0 SPM99 I SPM99 II SPM2 9/10

10. It’s Easy • Model Cov(εk): AR(1)+wn • Guess hyper-parameters with ReML • Use those parameters to perform better t-tests 10/10

11. Correct for DF Part II • Estimate Box’s εfrom modelled Cov(εk) • Use εto correct DF using Satterthwaite approximation • equivalent to Greenhouse-Geisser correction 11/10

12. A Psychology Example • Repeated measures of RT across subjects • RTs at level 2&3 might be more correlated than at 1&2 12/10

13. Other types of non-sphericity • 1st level • Temporal autocorrelation • Spatial (smoothness) • Unbalanced designs • 2nd level • Correlated repeated measures • Unequal variances between groups 13/10

14. Putting the ‘Re’ into ReML • Correlation matrix estimated with restricted basis set correlation matrix hyperparameters to estimate basis set elements 14/10