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Linear Hierarchical Models

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Linear Hierarchical Models

Corinne Iola

Giorgia Silani

SPM for Dummies

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

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)

- 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

- 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

- 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

- 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)

1

^

s21

with -in

Single-subject FFX

t = ___

Subj1= -1 1 0 0 0 0 0 0 0 0

<i>

^

<s2i> with -in

Multi-subject FFX

t = ___

Group= -1 1 -1 1 -1 1 -1 1 -1 1

<i>

^

^

<s2i> + with -in

<s2i> between

RFX analysis

t = ________

}

Subj1= -1 1 0 0 0 0 0 0 0 0

Subj2= 0 0 -1 1 0 0 0 0 0 0

@2nd level

Subj5= 0 0 0 0 0 0 0 0 -1 1

Differences between RFX and FFX

^

^

^

^

^

11

12

1

2

^

^

^

^

2

12

1

11

Random Effects Analysis : an fMRI study

1st Level 2nd Level

DataDesign MatrixContrast Images

SPM(t)

One-sample

t-test @2nd level

Estimated

population

means

Contrast images

Two-sample

t-test @2nd level

SS results

EM results

Friston et al. (2003) Mixed effects and fMRI studies, Submitted.