1. Linear Hierarchical Modelling
Hierarchical models are central to many of the current analyses for functional imaging data in SPM2
This includes:
Random effects analysis
Bayesian modelling
Dynamic Causal Modelling
Models using fMRI as priors for EEG source location

2. A Two Level Hierarchical Model
Y = X(1)(1) + e(1) (1st level) – within subject :
(1) = X(2)(2) + e(2) (2nd level) – between subject
Y = scans from all subjects
X(n) = design matrix at nth level
(n) = parameters - basically the s of the GLM
(but becomes  in hierarchical model – to confuse me!!!!!!!)
e(n) = N(m,2) error we assume there is a Gaussian distribution with a mean (m) and variation (2)

3. Squiggly line!! squiggly line!!
Hierarchical form
Squiggly line!! squiggly line!!
Where are the donuts??
1st level y = X(1) (1)  (1)
2nd level (1) = X(2) (2)  (2)

4. Hierarchical models
Hierarchical models a posit linear relations between variables with error terms which are Gaussian
In simple terms the GLM is a special case of hierarchical model with a single layer
Hierarchical models used at the second level can support Bayesian inference at the first level.
Thus, hierarchical models are essential for Bayesian statistics
Parameters - determine the expected response estimated from sample means (e.g the s from the GLM)
Hyperparameter – determines what causes the variation of the measured parameters (e.g for the t test - variation of observation error termed 2)

5. Random Effect Analysis
RFX is an example of how hierarchical modelling is used.
Used to find out what the stereotypical effect in the population is from the group of subjects that we have
We know that each subjects response’s varies from trial to trial (with-in subject variability)
Also, responses vary from subject to subject (between subject variability)
Both these are important when we make inference about the population

6. Random Effects Analysis - 2
12 subject study
Assumption - that the 12 subjects are drawn at random from the normal distributed population
If we only take into account the within subject variability we get the fixed effect analysis
If we take both within and between subjects we get random effects analysis

7. Data example

8. Differences between RFX and FFX

9. Random Effects Analysis
1st Level nd Level
Data Design Matrix Contrast Images a1 ^
SPM(t)
s21 ^ a2 ^
s22 ^
a11 ^
s211 ^ a ^
One-sample
level
a12 ^
s212 ^

10. Two populations Estimated population means Contrast images Two-sample
level

11. How do we estimate the parameters and hyperparameters?
Parameter and Hyperparameters (Variance components) can be estimated using summary statistics or EM (ReML)
The gold standard approach to parameter and hyperparameter is the EM (expectation maximization). EM
estimates population mean effect as MEANEM
the variance of this estimate as VAREM
Summary statistics
Avg[a]
Avg[Var(a)]
However, for balanced designs.
Avg[a] = MEANEM
Avg[Var(a)] = VAREM

12. Summary
In classical statistics - the parameters are be estimated (e.g. using the ordinary least squares)
Hyperparameter (covariance of the error) uses ReML
All the error terms of all the levels are grouped together and treated as a composite error for RFX
ReML – this is used to estimate unknown variance components without explicit reference to the parameters.
In classical inference one is usually interested only in the inference of the parameters at the highest level. (I.e. second level)

13. Thanks to Will Any questions? Ask the Specialist ?? SPM ?? ??RFX??
DONUTS