Transcript Document

Linear
Hierarchical
Models
Corinne
Giorgia
MFD
Giorgia &
Corinne
Introduction/Overview & Examples (behavioral)
functional Brain Imaging Examples,
Fixed Effects Analysis vs. Random Effects Analysis
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
?
fixed effects analysis
hierarchical linear models
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
Linear Hierarchical Models: Approaches
DCM
PPM
Linear Hierarchical Models
fMRI as priors for
EEG source location
Bayesian Theorem
fMRI, EEG, PET, behavioral Data, observational surveys
Population inferences
Expectation Maximisation
Summary statistics
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
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
?
fixed effects analysis
hierarchical linear models
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
?
fixed effects analysis
hierarchical linear models
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
?
fixed effects analysis
hierarchical linear models
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
?
fixed effects analysis
hierarchical linear models
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
?
fixed effects analysis
hierarchical linear models
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
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
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!
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
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 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
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
Linear Hierarchical Models: fMRI
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
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)
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.
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
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)
Linear Hierarchical Models: Population Inferences
1st level = within-subjects
Contrast images
One-sample
t-test at 2nd level
y  X (1) (1)   (1)
(1)
( 2)
( 2) ( 2)
  X  
2nd level = between-subjects
Likelihood
Prior
Linear Hierarchical Models: Bayesian
1st level = within-voxel
y  X (1) (1)   (1)
(1)
( 2)
( 2) ( 2)
  X  
2nd level = between-voxels
Likelihood
Prior
Linear Hierarchical Models: Approaches
DCM
PPM
Linear Hierarchical Models
Bayesian Theorem
fMRI, EEG, PET, behavioral Data, observational surveys
Population inferences
EM
Summary statistics
Linear Hierarchical Models: Approaches
Use for
Statistical Analysis Model
More flexible to
describe relation
between dependent
variabls and set of
independent variables
More flexible with
nested-structure data
(i.e. correlated data)
Model the mean,
variance and
covariance
Applied in
General Linear Model
Linear Hierarchical Models
(mixed Model)
1st level GLM
DCM
PPM
Bayesian Theorem
Population inferences
2st level HLM
EM Summary statistic
fMRI, EEG, PET, behavioral Data, observational surveys
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
!
fixed effects analysis
hierarchical linear models
Linear Hierarchical Models: Definitions
general linear models
mixed effects analysis
random effects analysis
!
fixed effects analysis
-> SPM: Giorgia
-> more detail SPM course ….
hierarchical linear models