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**Hierarchical Models and**

Variance Components Will Penny Wellcome Department of Imaging Neuroscience, University College London, UK SPM Course, London, May 2003

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**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

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**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

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**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 ^

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**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. ^ ^

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**Example: Multi-session study of auditory processing**

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

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**Two populations Estimated population means Contrast images Two-sample**

level

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**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

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**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’

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**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’

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**Non-Sphericity Errors are independent but not identical**

Errors are not independent and not identical Error Covariance

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**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.

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**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

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**Algorithm Equivalence**

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

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**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

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**Non-Sphericity Errors are independent but not identical**

Errors are not independent and not identical Error Covariance

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**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

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**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

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**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?

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**Independent but Non-Identical Error**

1st Level Controls Blinds 2nd Level Controls and Blinds Conjunction between the 2 groups

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**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 ?

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**= = = ? ? ? Non-Independent and Non-Identical Error**

1st Leve visual sound hand motion ? = ? = ? = 2nd Level F-test

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**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?

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**Non-Independent and Non-Identical Error**

1st Level Sentence > Symbols No-Priming>Priming Orthogonal contrasts 2nd Level Conjunction of 2 contrasts Left Anterior Temporal

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**Example IV Modelling serial correlation in fMRI time series**

Model errors for each subject as AR(1) + white noise.

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**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

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Bayes Rule

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**Example 2:Univariate model**

Likelihood and Prior Posterior Relative Precision Weighting

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**Example 2:Univariate model**

Likelihood and Prior AIM: Make inferences based on posterior distribution Similar expressions exist for posterior distributions in multivariate models Posterior But how do we compute the variance components or ‘hyperparameters’ ?

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**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

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**Estimating mean and variance**

Maximum Likelihood (ML), maximises p(Y|m,b) Expectation-Maximisation (EM), maximises for ‘vague’ prior on m

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**Estimating mean and variance**

For a prior on m with prior mean 0 and prior precision a Expectation-Maximisation (EM) gives where Larger a more shrinkage

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**Estimating mean and variance at multiple voxels**

For a prior on m over voxels with prior mean 0 and prior precision a Expectation-Maximisation (EM) gives at voxel i=1..V, scan n=1..N where Prior precision can be estimated from data. If mean activation over all voxels is 0 then these EM estimates are more accurate than ML

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**The Interface PEB WLS Parameters Parameters, and REML Hyperparameters**

No Priors Shrinkage priors

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**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

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**Bayesian Inference: Posterior Probability Maps**

PPMs Posterior Likelihood Prior SPMs

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**SPMs and PPMs PPMs: Show activations of a given size**

SPMs: show voxels with non-zero activations

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**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 P-values don’t change with search volume For reasonable thresholds have intrinsically high specificity

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**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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