corinne introduction/overview & examples (behavioral) giorgia functional brain imaging examples,...

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

Models

Hierarchical

Linear

MFD

Giorgia & Corinne

Linear Hierarchical Models: Definitions

mixed effects analysis

random effects analysis

fixed effects analysis

hierarchical linear models

general linear models

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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

Linear Hierarchical Models: Approaches

Linear Hierarchical Models

DCM

Bayesian Theorem

Population inferences

fMRI as priors for EEG source location

Expectation Maximisation Summary statistics

fMRI, EEG, PET, behavioral Data, observational surveys

PPM

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







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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

Linear Hierarchical Models: General

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

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 generalized to 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!


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

Linear Hierarchical Models: SPSS

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 the fixed 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


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

Linear Hierarchical Models: fMRI

Linear Hierarchical Models: fMRI


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


» 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)

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.

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


» 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)


Linear Hierarchical Models: Population Inferences

1st level = within-subjects

2nd level = between-subjects

Likelihood

Prior

)2()2()2()1(

)1()1()1(

X

XyContrast images

One-samplet-test at 2nd level

Linear Hierarchical Models: Bayesian

1st level = within-voxel

2nd level = between-voxels

Likelihood

Prior

)2()2()2()1(

)1()1()1(

X

Xy


Linear Hierarchical Models

DCM

Bayesian Theorem


PPM

fMRI, EEG, PET, behavioral Data, observational surveys

EM Summary statistics


DCM

Bayesian Theorem


PPM

EM Summary statisticsfMRI, EEG, PET, behavioral Data, observational surveys

Statistical Analysis Model

1st level GLM

More flexible to describe relation between dependent variabls and set of independent variables

2st level HLM

Linear Hierarchical Models(mixed Model)

General Linear Model

More flexible with nested-structure data (i.e. correlated data) Model the mean, variance and covariance

Applied inUse for







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