Download presentation
Published byDeborah Shields Modified over 9 years ago
1
Wellcome Dept. of Imaging Neuroscience University College London
Hierarchical Models Stefan Kiebel Wellcome Dept. of Imaging Neuroscience University College London
2
Overview Why hierarchical models? The model Estimation
Expectation-Maximization Summary Statistics approach Variance components Examples
3
Overview Why hierarchical models? Why hierarchical models? The model
Estimation Expectation-Maximization Summary Statistics approach Variance components Examples
4
Data Time fMRI, single subject EEG/MEG, single subject
fMRI, multi-subject ERP/ERF, multi-subject Hierarchical model for all imaging data!
5
Reminder: voxel by voxel
model specification parameter estimation Time hypothesis statistic Time Intensity single voxel time series SPM
6
Overview Why hierarchical models? The model The model Estimation
Expectation-Maximization Summary Statistics approach Variance components Examples
7
General Linear Model = +
Model is specified by Design matrix X Assumptions about e N: number of scans p: number of regressors
8
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.
9
Example: Two level model
= + = + Second level First level
10
Algorithmic Equivalence
Parametric Empirical Bayes (PEB) Hierarchical model EM = PEB = ReML Single-level model Restricted Maximum Likelihood (ReML)
11
Overview Why hierarchical models? The model Estimation Estimation
Expectation-Maximization Summary Statistics approach Variance components Examples
12
Estimation EM-algorithm E-step M-step Assume, at voxel j:
Friston et al. 2002, Neuroimage
13
And in practice? Most 2-level models are just too big to compute.
And even if, it takes a long time! Moreover, sometimes we‘re only interested in one specific effect and don‘t want to model all the data. Is there a fast approximation?
14
Summary Statistics approach
First level Second level Data Design Matrix Contrast Images SPM(t) One-sample 2nd level
15
Validity of approach The summary stats approach is exact under i.i.d. assumptions at the 2nd level, if for each session: Within-session covariance the same First-level design the same One contrast per session All other cases: Summary stats approach seems to be robust against typical violations.
16
Mixed-effects Summary statistics EM approach Step 1 Step 2
Friston et al. (2004) Mixed effects and fMRI studies, Neuroimage
17
Overview Why hierarchical models? The model Estimation
Expectation-Maximization Summary Statistics approach Variance components Variance components Examples
18
Sphericity Scans ‚sphericity‘ means: i.e. Scans
19
2nd level: Non-sphericity
Error covariance Errors are independent but not identical Errors are not independent and not identical
20
1st level fMRI: serial correlations
with autoregressive process of order 1 (AR-1) autocovariance- function COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!
21
Restricted Maximum Likelihood
observed ReML estimated COMPUTATIONAL EFFICIENCY To test a hypothesis a lot of images (on the order of 104) are accessed. Acceptable for interpolated channel data, but probably not for source reconstructed 3D-data. For fMRI, this procedure is ok, because there are not really many hypotheses (HRF up or HRF down + differences/interactions) For EEG, there are many more hypotheses to test, because there are many potential different hypotheses Some hypotheses cannot be tested with conventional model: any contrasts with lots of different regressors at 2nd level!!
22
Estimation in SPM5 ReML (pooled estimate) Ordinary least-squares
Maximum Likelihood optional in SPM5 one pass through data statistic using effective degrees of freedom 2 passes (first pass for selection of voxels) more accurate estimate of V
23
2nd level: Non-sphericity
Errors can be non-independent and non-identical when… Several parameters per subject, e.g. repeated measures design Complete characterization of the hemodynamic response, e.g. taking more than 1 HRF parameter to 2nd level Example data set (R Henson) on:
24
Overview Why hierarchical models? The model Estimation
Expectation-Maximization Summary Statistics approach Variance components Examples Examples
25
Example I Stimuli: Subjects: Scanning:
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 U. Noppeney et al.
26
Population differences
1st level: Controls Blinds 2nd level:
27
Auditory Presentation (SOA = 4 secs) of words
Example II Stimuli: Auditory Presentation (SOA = 4 secs) of words Motion Sound Visual Action “jump” “click” “pink” “turn” Subjects: (i) 12 control subjects fMRI, 250 scans per subject, block design Scanning: What regions are affected by the semantic content of the words? Question: U. Noppeney et al.
28
Repeated measures Anova
1st level: Motion Sound Visual Action ? = ? = ? = 2nd level:
29
Example 3: 2-level ERP model
Single subject Multiple subjects Subject 1 Trial type 1 . . . . . . Subject j Trial type i . . . . . . Subject m Trial type n
30
Test for difference in N170
ERP-specific contrasts: Average between 150 and 190 ms Example: difference between trial types 2nd level contrast -1 1 . . . = = + Identity matrix second level first level
31
Some practical points RFX: If not using multi-dimensional contrasts at 2nd level (F-tests), use a series of 1-sample t-tests at the 2nd level. Use mixed-effects model only, if seriously in doubt about validity of summary statistics approach. If using variance components at 2nd level: Always check your variance components (explore design).
32
Conclusion 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. 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.
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.