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Klaas Enno Stephan Laboratory for Social and Neural Systems Research

Institute for Empirical Research in Economics

University of Zurich

Functional Imaging Laboratory (FIL)

Wellcome Trust Centre for Neuroimaging

University College London

With many thanks for slides & images to:

FIL Methods group,

particularly Will Penny

Methods & models for fMRI data analysis26 November 2008

Statistical parametric map (SPM)

Design matrix

Image time-series

Kernel

Realignment

Smoothing

General linear model

Gaussian

field theory

Statistical

inference

Normalisation

p <0.05

Template

Parameter estimates

Why hierachical models?

fMRI, single subject

EEG/MEG, single subject

time

fMRI, multi-subject

ERP/ERF, multi-subject

Hierarchical models for all imaging data!

specification

parameter

estimation

hypothesis

statistic

Reminder: voxel-wise time series analysis!

Time

Time

BOLD signal

single voxel

time series

SPM

X

+

y

=

- Model is specified by
- Design matrix X
- Assumptions about e

N: number of scans

p: number of regressors

The design matrix embodies all available knowledge about experimentally controlled factors and potential confounds.

GLM assumes Gaussian “spherical” (i.i.d.) errors

Examples for non-sphericity:

sphericity = iid:error covariance is scalar multiple of identity matrix:

Cov(e) = 2I

non-identity

non-independence

Multiple covariance components at 1st level

enhanced noise model

error covariance components Q

and hyperparameters

V

Q2

Q1

1

+ 2

=

Estimation of hyperparameters with ReML (restricted maximum likelihood).

Group level inference: fixed effects (FFX)

- assumes that parameters are “fixed properties of the population”
- all variability is only intra-subject variability, e.g. due to measurement errors
- Laird & Ware (1982): the probability distribution of the data has the same form for each individual and the same parameters
- In SPM: simply concatenate the data and the design matrices lots of power (proportional to number of scans), but results are only valid for the group studied, can’t be generalized to the population

Group level inference: random effects (RFX)

- assumes that model parameters are probabilistically distributed in the population
- variance is due to inter-subject variability
- Laird & Ware (1982): the probability distribution of the data has the same form for each individual, but the parameters vary across individuals
- In SPM: hierarchical model much less power (proportional to number of subjects), but results can (in principle) be generalized to the population

Linear hierarchical model

Multiple variance components at each level

Hierarchical model

At each level, distribution of parameters is given by level above.

What we don’t know: distribution of parameters and variance parameters.

Non-hierarchical model

Estimating 2nd level effects

Variance components at 2nd level

between-level non-sphericity

Additionally: within-level non-sphericity at both levels!

Friston et al. 2005, NeuroImage

Algorithmic equivalence

Parametric

Empirical

Bayes (PEB)

Hierarchical

model

EM = PEB = ReML

Single-level

model

Restricted

Maximum

Likelihood

(ReML)

Practical problems

Most 2-level models are just too big to compute.

And even if, it takes a long time!

Moreover, sometimes we are only interested in one specific effect and do not want to model all the data.

Is there a fast approximation?

Summary statistics approach

First level

Second level

DataDesign MatrixContrast Images

SPM(t)

One-sample

t-test @ 2nd level

Validity of the summary statistics approach

The summary stats approach is exact if for each session/subject:

Within-session covariance the same

First-level design the same

One contrast per session

All other cases: Summary stats approach seems to be fairly robust against typical violations.

2nd level: non-sphericity

Error

covariance

- Errors are independent
- but not identical:
- e.g. different groups (patients, controls)

- Errors are not independent
- and not identical:
- e.g. repeated measures for each subject (like multiple basis functions)

Example 1: non-indentical & independent errors

Auditory Presentation (SOA = 4 secs) of

(i) words and (ii) words spoken backwards

Stimuli:

e.g.

“Book”

and

“Koob”

(i) 12 control subjects

(ii) 11 blind subjects

Subjects:

Scanning:

fMRI, 250 scans per subject, block design

Noppeney et al.

Example 2: non-indentical & non-independent errors

Stimuli:

Auditory Presentation (SOA = 4 secs) of words

Subjects:

(i) 12 control subjects

1. Words referred to body motion. Subjects decided if the body movement was slow.

2. Words referred to auditory features. Subjects decided if the sound was usually loud

3. Words referred to visual features. Subjects decided if the visual form was curved.

4. Words referred to hand actions. Subjects decided if the hand action involved a tool.

fMRI, 250 scans per

subject, block design

Scanning:

What regions are affected

by the semantic content of

the words?

Question:

Noppeney et al.

Practical conclusions

- Linear hierarchical models are general enough for typical multi-subject imaging data (PET, fMRI, EEG/MEG).
- Summary statistics are robust approximation to mixed-effects analysis.
- Use mixed-effects model only, if seriously in doubt about validity of summary statistics approach.
- RFX: If not using multi-dimensional contrasts at 2nd level (F-tests), use a series of 1-sample t-tests at the 2nd level.
- To minimize number of variance components to be estimated at 2nd level, compute relevant contrasts at 1st level and use simple test at 2nd level.

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