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Linear Hierarchical Models . Corinne Iola Giorgia Silani. SPM for Dummies. Outline. Fixed Effects versus Random Effects Analysis: how linear hierarchical models work Single-subject Multi-subjects Population studies. RFX: an example of hierarchical model. - PowerPoint PPT Presentation
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Linear Hierarchical Models
Corinne Iola Giorgia Silani
SPM for Dummies
Outline Fixed Effects versus Random Effects
Analysis: how linear hierarchical models work
Single-subject Multi-subjects Population studies
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)
Hierarchical form
1st level y = X(1) (1) + (1)
2nd level (1) = X(2) (2) + (2)
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
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
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[ Avg[Var()]
However, for balanced designs (N~12 and same n scans per subject). Avg[MEANEM Avg[Var()] = VAREM
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)
Single-subject FFX
Subj1= -1 1 0 0 0 0 0 0 0 0
t = ___1
with -in
^
Multi-subject FFX
Group= -1 1 -1 1 -1 1 -1 1 -1 1
t = ___i
i
with -in
^
RFX analysis
Subj1= -1 1 0 0 0 0 0 0 0 0Subj2= 0 0 -1 1 0 0 0 0 0 0
Subj5= 0 0 0 0 0 0 0 0 -1 1
t = ________i
i
with -in
^
ib
etween
^
@2nd level
Differences between RFX and FFX
1st Level 2nd Level
^
1^
^
2^
^
11^
^
12^
Data Design Matrix Contrast Images
)ˆ(ˆˆ
craV
ct
Random Effects Analysis : an fMRI study
SPM(t)
One-samplet-test @2nd level
Two populationsContrast images
Estimatedpopulation means
Two-samplet-test @2nd level
Example: Multi-session study of auditory processing
SS results EM results
Friston et al. (2003) Mixed effects and fMRI studies, Submitted.