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Hierarchical Models and

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1 Hierarchical Models and
Variance Components Will Penny Wellcome Department of Imaging Neuroscience, University College London, UK Cyclotron Research Centre, University of Liege, April 2003

2 Outline Random Effects Analysis General Framework
Summary statistic approach 2nd level) General Framework Multiple variance components and Hierarchical models Multiple variance components F-tests and level Modelling fMRI serial level Hierarchical models for Bayesian Inference SPMs versus PPMs

3 Outline Random Effects Analysis General Framework
Summary statistic approach 2nd level) General Framework Multiple variance components and Hierarchical models Multiple variance components F-tests and level Modelling fMRI serial level Hierarchical models for Bayesian Inference SPMs versus PPMs

4 Random Effects Analysis:Summary-Statistic Approach
1st Level nd Level Data Design Matrix Contrast Images 1 ^ SPM(t) 1 ^ 2 ^ 2 ^ 11 ^ 11 ^ ^ One-sample level 12 ^ 12 ^

5 Validity of approach ^ ^ Gold Standard approach is EM – see later –
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 In this case, the SS approach gives the same results, on average: Avg[a] = MEANEM Avg[Var(a)] =VAREM In other cases, with N~12, and typical ratios of between-subject to within-subject variance found in fMRI, the SS approach will give very similar results to EM. ^ ^

6 Two populations Estimated population means Contrast images Two-sample
level

7 Outline Random Effects Analysis General Framework
Summary statistic approach 2nd level) General Framework Multiple variance components and Hierarchical models Multiple variance components F-tests and level Modelling fMRI serial level Hierarchical models for Bayesian Inference SPMs versus PPMs

8 The General Linear Model
y = X  + e N  N  L L  N  1 Error covariance N 2 Basic Assumptions Identity Independence N We assume ‘sphericity’

9 Multiple variance components
y = X  + e N  N  L L  N  1 Error covariance N Errors can now have different variances and there can be correlations N We allow for ‘nonsphericity’

10 Non-Sphericity Errors are independent but not identical
Errors are not independent and not identical Error Covariance

11 General Framework Multiple variance components Hierarchical Models
at each level With hierarchical models we can define priors and make Bayesian inferences. If we know the variance components we can compute the distributions over the parameters at each level.

12 Estimation EM algorithm Friston, K. et al. (2002), Neuroimage ( ) å y
E-Step ( ) y C X T 1 - = e q h M-Step r for i and j { } { Q tr J g i j ij k å + l Friston, K. et al. (2002), Neuroimage

13 Pooling assumption Decompose error covariance at each voxel, i, into
a voxel specific term, r(i), and voxel-wide terms. The hyperparameters are now estimated once for the whole brain.

14 Algorithm Equivalence
Parametric Empirical Bayes (PEB) Hierarchical model EM=PEB=ReML Restricted Maximimum Likelihood (ReML) Single-level model

15 Outline Random Effects Analysis General Framework
Summary statistic approach 2nd level) General Framework Multiple variance components and Hierarchical models Multiple variance components F-tests and level Modelling fMRI serial level Hierarchical models for Bayesian Inference SPMs versus PPMs

16 Non-Sphericity Errors are independent but not identical
Errors are not independent and not identical Error Covariance

17 Non-Sphericity Error can be Independent but Non-Identical when…
1) One parameter but from different groups e.g. patients and control groups 2) One parameter but design matrices differ across subjects e.g. subsequent memory effect

18 Non-Sphericity Error can be Non-Independent and Non-Identical when…
1) Several parameters per subject e.g. Repeated Measurement design 2) Conjunction over several parameters e.g. Common brain activity for different cognitive processes 3) Complete characterization of the hemodynamic response e.g. F-test combining HRF, temporal derivative and dispersion regressors

19 Example I U. Noppeney et al.
Stimuli: Auditory Presentation (SOA = 4 secs) of (i) words and (ii) words spoken backwards Subjects: (i) 12 control subjects (ii) 11 blind subjects jump touch koob “click” Scanning: fMRI, 250 scans per subject, block design Q. What are the regions that activate for real words relative to reverse words in both blind and control groups?

20 Independent but Non-Identical Error
1st Level Controls Blinds 2nd Level Controls and Blinds Conjunction between the 2 groups

21 Example 2 U. Noppeney et al.
Stimuli: Auditory Presentation (SOA = 4 secs) of words motion sound visual action jump touch “jump” “click” “pink” “turn” “click” Subjects: (i) 12 control subjects Scanning: fMRI, 250 scans per subject, block design Q. What regions are affected by the semantic content of the words ?

22 = = = ? ? ? Non-Independent and Non-Identical Error
1st Leve visual sound hand motion ? = ? = ? = 2nd Level F-test

23 Example III U. Noppeney et al.
Stimuli: (i) Sentences presented visually (ii) False fonts (symbols) Some of the sentences are syntactically primed Scanning: fMRI, 250 scans per subject, block design Q. Which brain regions of the “sentence reading system” are affected by Priming?

24 Non-Independent and Non-Identical Error
1st Level Sentence > Symbols No-Priming>Priming Orthogonal contrasts 2nd Level Conjunction of 2 contrasts Left Anterior Temporal

25 Example IV Modelling serial correlation in fMRI time series
Model errors for each subject as AR(1) + white noise.

26 Outline Random Effects Analysis General Framework
Summary statistic approach 2nd level) General Framework Multiple variance components and Hierarchical models Multiple variance components F-tests and level Modelling fMRI serial level Hierarchical models for Bayesian Inference SPMs versus PPMs

27 The Interface PEB OLS Parameters Parameters, and REML Hyperparameters
No Priors Shrinkage priors

28 Bayes Rule

29 Example 2:Univariate model
Likelihood and Prior Posterior Relative Precision Weighting

30 Bayesian Inference 1st level = within-voxel Likelihood Shrinkage Prior
In the absence of evidence to the contrary parameters will shrink to zero 2nd level = between-voxels

31 Bayesian Inference: Posterior Probability Maps
PPMs Posterior Likelihood Prior SPMs

32 SPMs and PPMs PPMs: Show activations of a given size
SPMs: show voxels with non-zero activations

33 PPMs Advantages Disadvantages Use of shrinkage One can infer a cause
priors over voxels is computationally demanding Utility of Bayesian approach is yet to be established One can infer a cause DID NOT elicit a response SPMs conflate effect-size and effect-variability No multiple comparisons problem (hence no smoothing) P-values don’t change with search volume

34 Summary Random Effects Analysis Multiple variance components
Summary statistic approach 2nd level) Multiple variance components F-tests and level Modelling fMRI serial level Hierarchical models for Bayesian Inference SPMs versus PPMs General Framework Multiple variance components and Hierarchical models


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