1. Linear Hierarchical Models
Corinne Iola
Giorgia Silani
SPM for Dummies

2. Outline
Fixed Effects versus Random Effects Analysis: how linear hierarchical models work
Single-subject
Multi-subjects
Population studies

3. RFX: an example of 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
e(n) = N(m,2) error we assume there is a Gaussian distribution with a mean (m)
and variation (2)

4. Hierarchical form 1st level y = X(1) (1) +  (1)
2nd level (1) = X(2) (2)  (2)

5. Random Effects Analysis: why?
Interested in individual differences, but also
…interested in what is common
As experimentalists we know…
each subjects’ response 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 : why?
with-in subject variability – Fixed effects analysis (FFX) or 1st level analysis
Used to report case studies
Not possible to make formal inferences at population level
with-in and between subject variability – Random Effect analysis (RFX) or 2nd level analysis
possible to make formal inferences at population level

7. How do we perform a RFX?
RFX (Parameter and Hyperparameters (Variance components)) can be estimated using summary statistics or EM (ReML) algorithm
The gold standard approach to parameter and hyperparameter is the EM (expectation maximization)….(but takes more time…) EM
estimates population mean effect as MEANEM
the variance of this estimate as VAREM
For N subjects, n scans per subject and equal within-subject variance we have
VAREM = Var-between/N + Var-within/Nn
Summary statistics
Avg[a]
Avg[Var(a)]
However, for balanced designs (N~12 and same n scans per subject).
Avg[a] = MEANEM
Avg[Var(a)] = VAREM

8. Random Effects Analysis
Multi - subject PET study
Assumption - that the 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 (i.e. 1st level - multisubject analysis)
If we take both within and between subjects we get random effects analysis (2nd level analysis)

9. Single-subject FFX t = ___ Subj1= -1 1 0 0 0 0 0 0 0 0 1 s21 with -in^
Subj1=

10. Multi-subject FFX t = ___ Group= -1 1 -1 1 -1 1 -1 1 -1 1
<s2i> with -in ^
Group=

11. } RFX analysis t = ________ @2nd level <s2i> + with -in^
<s2i> between ^ }
Subj1=
Subj2=
@2nd level
Subj5=

12. Differences between RFX and FFX

13. Random Effects Analysis : an fMRI study
1st Level nd Level
Data Design Matrix Contrast Images 1 ^
SPM(t)
1 ^ 2 ^
2 ^
11 ^
11 ^  ^
One-sample
level
12 ^
12 ^

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

15. Example: Multi-session study of auditory processing
SS results
EM results
Friston et al. (2003) Mixed effects and fMRI studies, Submitted.