1. 'Linear Hierarchical Models'
Definition
Computation
Applications
……the girlie way

2. Hierarchical models provide a “framework” for both
Definition
Hierarchical models provide a “framework” for both
Classical inference (at first & second level) &
Parametric Empirical Bayes (PEB) at first level (constrained by higher levels).
Extension of the general linear model but with more variance components.
Computation
Both Classical and Bayesian approaches rest on
estimating the covariance components
Covariance components refer to the multiple variance components in the data
(including within & between subject variance)
Covariance components estimation is based on:
Maximum Likelihood estimator (ML) or the Expectation Maximization (EM) algorithm
= ML plus expectation step.

3. y = response variable;
X = design matrix/explanatory variables;
= parameters (betas) e = error
1st Level: Activation over scans
(within session & subject)
2nd Level: Activation over sessions
(within subjects)
3rd Level: Subject specific effects
(over subjects)
Parameters (q ): Quantities that determine the expected response (the effect)
Can be estimated from mean without knowing its variance
Hyperparameters: Variance of these quantities (within & between subject)
(classically estimated using residual sum of squares)

4. covariance components
Classical approach:
how large are parameters with respect to standard error
inference usually made at highest level
(using between subject variability).
Bayesian inference:
probability of the parameters given the data
prior covariance = error covariance at level above
i.e. inference at lower levels (constrained by higher levels)
e.g. Estimate of single subject can be constrained by knowledge of population using
between subject variability from 2nd level as priors for parameters at first level.
SPM uses activation over voxels (within subject) as the priors
E.g. Within subject
1st Level: Activation over scans
2nd Level: Activation over voxels
Both Classical and Bayesian approaches rest on estimating the
covariance components

5. Covariance component estimation
= estimating hyperparameters (within & between subject contributions to error)
(hierarchically organised & linearly mixed variance components)
Flattened to:
Variance components are estimated with:
Expectation Maximization (EM) Algorithm

6. Expectation Maximization (EM) Algorithm
EM estimates both parameters and hyperparameters from the data
(using generic iterative / re-estimation procedure)
E. step: finds expectation & covariance of the parameters
holding hyperparameters fixed.
M. step: updates maximum likelihood estimate of hyperparameters
holding parameters fixed.
EM used when hyperparameters of the likelihood and prior densities are not known.
these hyperparameters become variance components that can be
estimated using restricted maximum likelihood (ReML)
ReML involves M step but not E step
finds unknown variance components without explicit reference to the parameters.
……over to Raj