Introduction to Linear Hierarchical Models: Definitions and Examples

1. Linear MFD Giorgia & Corinne Hierarchical Models Corinne Introduction/Overview & Examples (behavioral) Giorgia functional Brain Imaging Examples, Fixed Effects Analysis vs. Random Effects Analysis

2. Linear Hierarchical Models: Definitions ? general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

3. Linear Hierarchical Models: Definitions • Introduction • Penny, W. & Friston K.J. (2003) Human Brain Function • Hierarchical models are central to many current analyses of functional imaging data including random effects analysis, models using fMRI as priors for EEG source localization and spatiotemporal Bayesian modelling of imaging data • These hierarchical models posit linear relations between variables with error terms that are Gaussian • The General Linear Model (GLM), which to date has been so central to the analysis of functional imaging data, is a special case of these hierarchical models consisting of just a single layer • Central to many analysis of functional imaging data (e.g., random effects analysis) • Linear relations between variables • GLM = special HLM of just a single layer

4. Linear Hierarchical Models: Approaches DCM PPM fMRI as priors for EEG source location Linear Hierarchical Models Bayesian Theorem fMRI, EEG, PET, behavioral Data, observational surveys Population inferences Expectation Maximisation Summary statistics

5. Linear Hierarchical Model Figure 1: Two-level hierarchical model. The data y are explained as deriving from an effect w and a zero-mean Gaussian random variation with covariance C. The effects w in turn are random effrects deriving from a superordinate effect u and zero-mean Gaussian random variation with covariance P. The goal of Bayesian inference is to make inferences about u and w from the posterior distribution p(u¦y) and p(w¦y) Penny, W. & Friston K.J. (2003) Human Brain Function

6. Linear Hierarchical Models: Definitions ? general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

7. Linear Hierarchical Models: Definitions ? general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

8. Linear Hierarchical Models: Definitions ? general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

9. Linear Hierarchical Models: Definitions ? general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

10. Linear Hierarchical Models: Definitions ? general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

11. Linear Hierarchical Models: General • Definition: • Hierarchical Linear Models (HLM) incorporate data from multiple levels in an attempt to determine the impact of individual and grouping factors upon some individual level outcome • Example • Student achievment as a function of student level characteristics (e.g., IQ, study habits), classroom level factors (e.g., instructions style, textbook), school level factors (e.g., wealth), and so on. • HLMs or multilevel models can incorporate such factors in a manner better than ordinary least squares since HLMs take into account error structures at each level

12. Linear Hierarchical Models: Screening Example • Example: • Dummies knowledge increase as a function of • Talk in Methods for Dummies • Research group (topic) • Previous knowledge level • GLM • 4 types of tests • Dummies ID • Research group ID • Does the talk have a significant effect on knowledge increase? • Suspecting research group having a significant effect • talk_MFD\testdummies_rest.sav

13. Linear Hierarchical Models: Screening Example • Results and Inferences of GLM: • There is a lot of variability in the „research group ID“, so it is an important effect to model; • Hence the model results cannot be generalizedto other research groups as „research group ID“ is a fixed effect. • Aside from the first research group which generated considerably higher knowledge increase than the others, the research group effects seem to be randomly scattered (Gaussian Distribution) • Therefore, using another procedure, you might be able to specify „reserach group ID“as a random effect • To use ordinary least squares with these data as „independent“ (exogenous) variables isn‘t right because of the correlations among Dummies in the same research field. • No inferences about other research groups dummies!

14. Linear Hierarchical Models: SPSS • GLM: • Results: • first talk was more effective than the others • considerable variation in knowledge increase by factor research group • How to do generalization of these results to other Dummies?! • GLM Repeated Measures procedure does not allow for random effects • GLM Univariate procedure does not allow for within-subjects effects • Linear Mixed Model procedure: • specify research groups ID as a random effect • Gives greater control over the specification of the covariance matrix for the within-subjects factor • Models the mean of a response variable and its covariance structure • talk_MFD\testdummies_mntestres.sav

15. Linear Hierarchical Models: Screening Example • Results and Inferences of linear mixed model procedure: • Fixed effects • Test significance values (that is, less than 0.05) indicate that the effect research group contributes to the model talk_MFD\FIXED EFFECTS.HTM • Estimates of thefixed model effects their significance talk_MFD\ESTIMATES OF FIXED EFFECTS.HTM • The individual effects of the first two talks are significantly different from the third • The estimates of the effects suggest that the first talk is the best, for it is associated with higher sales than either of the other promotions • Random effects • Estimates of Covariance Parametes • Random Effect Covariance Structure

16. Linear Hierarchical Models: fMRI • Mixed model procedure = linear hierarchical model ? • Mixed model: • Fixed effects (1st level, GLM) • Random effects (2nd level, LHM) • Why so important in fMRI ? • Study design: • Single subjects study vs. studies invonlving many subjects • Individual differences vs. population effects • Trial to trial (within subjects) and subject to subject (between subjecs) • 2 levels of variance

17. Linear Hierarchical Models: fMRI

18. Linear Hierarchical Model Figure 1: Two-level hierarchical model. The data y are explained as deriving from an effect w and a zero-mean Gaussian random variation with covariance C. The effects w in turn are random effrects deriving from a superordinate effect u and zero-mean Gaussian random variation with covariance P. The goal of Bayesian inference is to make inferences about u and w from the posterior distribution p(u¦y) and p(u¦y) Penny, W. & Friston K.J. (2003) Human Brain Function

19. Linear Hierarchical Model • Y = X(1)(1) + e(1)(1st level) – within subject : • (1) = X(2)(2) + e(2)(2nd level) – between subject • Y = scans from all subjects • X(n) = design matrix at nth level • (n)= parameters - basically the s of the GLM • e(n) = N(m,2) error we assume there is a Gaussian distribution with a mean (m) and variation (2)

20. Linear Hierarchical Models: Considerations • Assumptions • Dependent variable assumed to be linearly related to the fixed factors, random factors, and covariates. • The fixed effects model the mean of the dependent variable. • The random effects model the covariance structure of the dependent variable.

21. Linear Hierarchical Models: Considerations • Assumptions • Related procedures • Examine the data before running an analysis • If you do not suspect there to be correlated or non-constant variability, you can alternatively use the GLM Univariate or GLM Repeated Measures procedure. • Correlated or non-constant variablility in fMRI?! • Limitations of SPM

22. Linear Hierarchical Models: Considerations • Solving the problems • By using the probability that a voxel had activated, or indeed its activation was greater than some threshold • likelihood of getting the data, given no activation • Classical approach • Probablility distribution of the activation given the data • posterior probability used in Bayesian inference • Posterior distribution needs • Likelihood, afforded by assumption about the distribution of errors • Prior probability of activation (as values or estimated from the data, provided we have observed multiple instances of the effect we are insterested in)

23. Linear Hierarchical Models: Population Inferences 1st level = within-subjects Likelihood Contrast images Prior One-sample t-test at 2nd level 2nd level = between-subjects

24. Linear Hierarchical Models: Bayesian 1st level = within-voxel Likelihood Prior 2nd level = between-voxels

25. Linear Hierarchical Models: Approaches DCM PPM Linear Hierarchical Models Bayesian Theorem fMRI, EEG, PET, behavioral Data, observational surveys Population inferences EM Summary statistics

26. Linear Hierarchical Models: Approaches Use for Statistical Analysis Model Applied in More flexible to describe relation between dependent variabls and set of independent variables General Linear Model DCM PPM More flexible with nested-structure data (i.e. correlated data) Model the mean, variance and covariance Linear Hierarchical Models (mixed Model) Bayesian Theorem 1st level GLM Population inferences 2st level HLM Summary statistics EM fMRI, EEG, PET, behavioral Data, observational surveys

27. Linear Hierarchical Models: Definitions ! general linear models random effects analysis fixed effects analysis mixed effects analysis hierarchical linear models

28. Linear Hierarchical Models: Definitions ! general linear models random effects analysis fixed effects analysis mixed effects analysis -> SPM: Giorgia -> more detail SPM course …. hierarchical linear models