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

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

3. 3/10 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) ε3ε3 ε2ε2 ε1ε1 Y=βX+ε at same voxel error terms correlated across time

4. 4/10 So What To Do? Model It  Capture the form of the Cov(є k ) in a GLM & use this to correct the stats Cov(є k )=λ1Q1λ1Q1 λ2Q2λ2Q2 λ3Q3λ3Q3 +++… hyperparameters to estimate correlation matrix basis set elements white noise auto- correlation ID matrix

5. 5/10 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(є k )  Estimated Cov(є k ) with ReML as two components:  variance (local)  correlation matrix (global)

6. 6/10 A Voxel by Voxel Account Global (good estimate) σ1σ1 Covariance matrix Cov(εk) Correlation matrix (V) Variance ( σ k 2) s 21 s 22 s 23 s 24 s 11 s 12 s 13 s 14 s 41 s 42 s 43 s 44 s 31 s 32 s 33 s 34 0.51 0.2 10.50.20 0 0.51 0.20.51 x σ2σ2 σ1σ1 σ4σ4 σ3σ3 ε2ε2 ε1ε1 ε4ε4 ε3ε3 Error term (ε) Local

7. 7/10 We Have Cov(ε k ) - What Next?  Use it for t & F-tests:  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 Components of model appear in new t statistic

8. 8/10 So Correct the Number of DF  Box’s measure (ε) measures Cov(ε k ) departure from spherical ρσ 2 σ2σ2 σ2σ2 σ2σ2 σ2σ2 cccc cccc cccc cccc SphericalCompletely unspherical 11/(k-1) ε Number of measures  Use ε to correct DF using Satterthwaite approximation (Greenhouse-Geisser)

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

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

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

12. 12/10 A Psychology Example  Repeated measures of RT across subjects  RTs at level 2&3 might be more correlated than at 1&2 Treatment Level 123 Task 0 back recall 1 back recall 2 back recall No. Subjects 12

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

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