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Hierarchical Models and Variance Components 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 Summary statistic approach (t-tests @ 2 nd level) §General Framework Multiple variance components and Hierarchical models §Multiple variance components F-tests and conjunctions @2 nd level Modelling fMRI serial correlation @1 st level §Hierarchical models for Bayesian Inference SPMs versus PPMs

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Outline §Random Effects Analysis Summary statistic approach (t-tests @ 2 nd level) §General Framework Multiple variance components and Hierarchical models §Multiple variance components F-tests and conjunctions @2 nd level Modelling fMRI serial correlation @1 st level §Hierarchical models for Bayesian Inference SPMs versus PPMs

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

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Validity of approach §Gold Standard approach is EM – see later – estimates population mean effect as MEAN EM the variance of this estimate as VAR EM §For N subjects, n scans per subject and equal within-subject variance we have VAR EM = Var-between/N + Var-within/Nn §In this case, the SS approach gives the same results, on average: Avg[ MEAN EM Avg[Var( )] =VAR EM §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 resultsEM results Friston et al. (2003) Mixed effects and fMRI studies, Submitted.

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Two populations Contrast images Estimated population means Two-sample t-test @2 nd level

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Outline §Random Effects Analysis Summary statistic approach (t-tests @ 2 nd level) §General Framework Multiple variance components and Hierarchical models §Multiple variance components F-tests and conjunctions @2 nd level Modelling fMRI serial correlation @1 st level §Hierarchical models for Bayesian Inference SPMs versus PPMs

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y = X + N 1 N L L 1 N 1 2 Basic Assumptions §Identity §Independence The General Linear Model N N Error covariance We assume sphericity

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

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

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General Framework Hierarchical Models Multiple variance components 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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E-Step yCXC XCXC T yy T y 1 1 1 M-Step y Xyr for i and j { }{ }{}{ 11 11111 CQCQtrJ XCQCXC rCQCrCQ g ijij i T y i T ii } kk QCC gJ 1 Friston, K. et al. (2002), Neuroimage EM algorithm Estimation

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

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Outline §Random Effects Analysis Summary statistic approach (t-tests @ 2 nd level) §General Framework Multiple variance components and Hierarchical models §Multiple variance components F-tests and conjunctions @2 nd level Modelling fMRI serial correlation @1 st level §Hierarchical models for Bayesian Inference SPMs versus PPMs

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

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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 Non-Sphericity

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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 Non-Sphericity

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

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

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

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

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

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

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

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Outline §Random Effects Analysis Summary statistic approach (t-tests @ 2 nd level) §General Framework Multiple variance components and Hierarchical models §Multiple variance components F-tests and conjunctions @2 nd level Modelling fMRI serial correlation @1 st 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 Posterior Similar expressions exist for posterior distributions in multivariate models AIM: Make inferences based on posterior distribution But how do we compute the variance components or hyperparameters ?

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E-Step yCXC XCXC T yy T y 1 1 1 M-Step y Xyr for i and j { }{ }{}{ 11 11111 CQCQtrJ XCQCXC rCQCrCQ g ijij i T y i T ii } kk QCC gJ 1 Friston, K. et al. (2002), Neuroimage EM algorithm Estimation

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Estimating mean and variance Maximum Likelihood (ML), maximises p(Y|, ) Expectation-Maximisation (EM), maximises for vague prior on

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

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Estimating mean and variance at multiple voxels For a prior on over voxels with prior mean 0 and prior precision 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 WLS Parameters, REML Hyperparameters PEB Parameters and Hyperparameters No Priors Shrinkage priors

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

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LikelihoodPrior Posterior SPMs PPMs Bayesian Inference: Posterior Probability Maps

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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 AdvantagesDisadvantages One can infer a cause DID NOT elicit a response SPMs conflate effect-size and effect-variability P-values dont change with search volume For reasonable thresholds have intrinsically high specificity Use of shrinkage priors over voxels is computationally demanding Utility of Bayesian approach is yet to be established

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Summary §Random Effects Analysis Summary statistic approach (t-tests @ 2 nd level) §Multiple variance components F-tests and conjunctions @2 nd level Modelling fMRI serial correlation @1 st level §Hierarchical models for Bayesian Inference SPMs versus PPMs §General Framework Multiple variance components and Hierarchical models

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