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

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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.