Hierarchical Models
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 [3]. 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. Model fitting and statistical inference for hierarchical models can be implemented using a Parametric Empirical Bayes (PEB) algorithm described in Chapter 17 and in [4]. The algorithm is sufficiently general to accomodate multiple levels in the hierarchy and allows for the error covariances to take on arbitrary form. This generality is particularly appealing as it renders the method applicable to a wide variety of modelling scenarios. Because of this generality, however, and the complexity of scenarios in which the method is applied, readers wishing to learn about PEB for the first time are advised to read this chapter first. We provide an introduction to hierarchical models and focus on some relatively simple examples. Each model and PEB algorithm we present is a special case of that described in [4]. Whilst there are a number of tutorials on hierarchical modelling [9],[2] what we describe here has been tailored for functional imaging applications. We also note that a tutorial on hierarchical models is, to our minds, also a tutorial on Bayesian inference as higher levels act as priors for parameters in lower levels. Readers are therefore encouraged to also consult background texts on Bayesian inference, such as [5]. We restrict our attention to two-level models and show, in section 2, how one computes the posterior distributions over the firstand second-level parameters. These are derived, initially, for completely general design and error covariance matrices. We then consider two special cases (i) models with equal error variances and (ii) separable models. In section 3 of the chapter we show how the parameters and covariance components can be estimated using PEB. In section 4 we show how a two-level hierarchical model can be used for Random-Effects Analysis. For equal subject error variances at the first level and the same first-level design matrices (ie. balanced designs) we show that