1 / 14

Covariance, autocorrelation & non-sphericity

Learn how to model the covariance matrix, estimate the best model, and use it to run more accurate statistical tests. Correct for autocorrelation and non-sphericity to improve your results. Suitable for beginners.

murawski
Download Presentation

Covariance, autocorrelation & non-sphericity

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related