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Covariance, autocorrelation & non-sphericity. Methods for Dummies. Ground to Cover. Quick Recap (IID etc) Modelling the covariance matrix Estimating the best model Using the model to run more accurate stats. Y= β X+ ε. ε 3. ε 2. at same voxel error terms correlated across time. ε 1.

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Presentation Transcript
ground to cover
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

quick recap

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

so what to do model it
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

the same thing with words
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

a voxel by voxel account

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

we have cov k what next
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

so correct the number of df

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

you ve never had it so good
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

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

10/10

correct for df part ii
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

a psychology example
A Psychology Example
  • Repeated measures of RT across subjects
  • RTs at level 2&3 might be more correlated than at 1&2

12/10

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

13/10

putting the re into reml
Putting the ‘Re’ into ReML
  • Correlation matrix estimated with restricted basis set

correlation matrix

hyperparameters to estimate

basis set elements

14/10