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Introduction

Functional magnetic resonance imaging studies are typically used to address questions about activation effects in populations of subjects. This generally involves a multi-subject and/or multi-session approach where data are analysed in such a way as to allow for hypothesis tests at the group level (15,28), e.g. in order to assess whether the observed effects are common and stable across or between groups of interest.

Calculating the level and probability of brain activation for a single subject is typically achieved using a linear model of the signal together with a Gaussian noise model for the residuals. This model is commonly referred to as the General Linear Model (GLM) and much attention to date has been focussed on ways of modelling and fitting the (time-series) signal and residual noise at the individual single-session level (25,27,4).

In order to be able to generate results that extend to the population, we also need to account for the fact that the individual subjects themselves are sampled from the population and thus are random quantities with associated variances. It is exactly this step that marks the transition from a simple fixed-effects model to a mixed-effects model1 and it is imperative to formulate a model at the group-level that allows for the explicit modelling of these additional variance terms (15,7).

We can formulate the problem of group statistics in neuro-imaging as being hierarchical (1,11). For example, the different levels of the hierachy could be separate GLMs for a session-level, subject-level and group-level. In this paper we attempt to deal with inference on these multi-level GLM hierarchies by utilising a fully Bayesian framework. Typically, the most important inference is at the top-level of the hierarchy, for example we may be looking for significance of a group mean. Whether we are looking to infer at the top-level with the within-session FMRI time-series data (11) or with summary statistic results from the level below (15,28), a fully Bayesian approach provides us with the means to assess the full uncertainty in the parameter of interest (contrasts of regression parameters) at the top-level; taking into account all of the unknown variance components (fixed and random) in the model.

Bayesian statistics provides the only generic tool for inferring model parameter probability distribution functions from data. It provides strict rules for the rational and consistent adjustment of belief (in the form of probability density functions) in the presence of new information (5), which are not available in the frequentist literature. The major consequences of this are twofold. First, we may make inference about the absolute value of the parameters of interest. i.e. we may ask questions of our parameters such as, ``What is the probability that $ \theta$ lies in the interval $ [\theta_0,\theta_1]$?'', a question unavailable to any frequentist technique. Frequentist statistics is typically limited to posing questions of the data under the ``Null hypothesis'' that the parameter value is zero. Inference in a frequentist framework is then limited to the simple acceptance or rejection of this null hypothesis without being able make any statement about the parameter values. Second, Bayesian statistics gives us a tool for inferring on any model we choose, and guarantees that uncertainty will be handled correctly. Only in certain special cases (not including the model presented here) is it possible to derive analytical forms for the null distributions required by frequentist statistics. In their absence, frequentist solutions rely on null distributions derived from the data (e.g. permutation tests), losing the statistical power gained from educated assumptions about, for example, the distribution of the noise.

These features of Bayesian analysis mean that we may make inference on physiological parameters of the haemodynamic response in the complex non-linear balloon model (9) or on spatial noise relationships in multivariate spatial auto-regressive models of FMRI data (24) or, in this paper, on higher level statistics in the presence of multiple variance components.

One important ingredient in a Bayesian approach is the choice of prior on the variance components and top-level regression parameters. Due to the typically small numbers of observations in neuro-imaging above the first-level (e.g. small numbers of subjects), this choice of prior is critical. To solve this problem we introduce to neuro-image modelling the approach of reference priors, which drives the choice of prior such that it is non-informative in an information-theoretic sense. For GLMs where a frequentist solution is available, reference analysis gives the same inference as a frequentist approach. Importantly, reference analysis allows us to perform inference when frequentist solutions are unavailable.

Using fully Bayesian reference analysis we propose two approaches to inferring at the top-level; these are a fast approximation to the marginal posterior, and a slower approach utilising Markov Chain Monte Carlo (MCMC) followed by a multivariate non-central t-distribution fit to the MCMC chains.

In (11) the hierarchical model is solved ``all-in-one'' using the within-session FMRI time-series data as input. However, in neuro-imaging, where the human and computational costs involved in data analysis are relatively high it is desirable to be able to make top level inferences using the results of separate lower-level analyses without the need to re-analyse any of the lower-level data; an approach commonly referred to as the summary statistics approach to FMRI analysis (15). Within such a summary statistic split-level approach, group parameters of interest can easily be refined as more data become available.

In (15), when inferring at the top level, this summary statistic split-level approach is shown to be equivalent to inferring all-in-one under certain conditions (e.g. the approach in (15) requires balanced designs). (1) show that top-level inference using the split-level summary statistics approach can be made equivalent to the all-in-one approach with no restrictions, if we pass up the correct summary statistics (in particular, the covariances from previous levels). Furthermore, (1) demonstrate that by taking into account lower-level covariances and heterogeneity a substantial increase in higher-level z-statistic is possible. However, (1) only show that this is the case when all variance components are known. Independently, in this paper, using the fully Bayesian approach, we show this equivalence for when the variance components (excluding autocorrelation) are unknown. The equivalence relies on the assumption that the summary statistics, which correspond to the marginal distributions of the GLM regressions parameters, can be represented as a multivariate non-central t-distributions. Between the first-level (within session) and the second-level this can be shown analytically. For summary statistics at higher-levels this is an assumption which we test empirically using artificial data.

In summary, there are three main contributions presented in this paper. Firstly, we introduce reference analysis to neuro-imaging. Secondly, we propose two inference techniques at the top-level for multi-level hierarchies (a fast approach and a slower more accurate approach). Thirdly, we demonstrate that we can infer on the top-level of multi-level hierarchies by inferring on the split levels separately and passing summary statistics between them.



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