Bayesian Model Selection for Group Studies

Wellcome Trust Centre for Neuroimaging, Institute of Neurology, University College London, London, UK.
NeuroImage (Impact Factor: 6.36). 04/2009; 46(4):1004-17. DOI: 10.1016/j.neuroimage.2009.03.025
Source: PubMed


Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. BMS has recently found widespread application in neuroimaging, particularly in the context of dynamic causal modelling (DCM). However, so far, combining BMS results from several subjects has relied on simple (fixed effects) metrics, e.g. the group Bayes factor (GBF), that do not account for group heterogeneity or outliers. In this paper, we compare the GBF with two random effects methods for BMS at the between-subject or group level. These methods provide inference on model-space using a classical and Bayesian perspective respectively. First, a classical (frequentist) approach uses the log model evidence as a subject-specific summary statistic. This enables one to use analysis of variance to test for differences in log-evidences over models, relative to inter-subject differences. We then consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new variational Bayes method rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution which describes the probabilities for all models considered. These probabilities then define a multinomial distribution over model space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject as well as the exceedance probability of one model being more likely than any other model. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects, is more informative and appropriate than both the GBF and frequentist tests of the log-evidences. In particular, we found that the hierarchical Bayesian approach is considerably more robust than either of the other approaches in the presence of outliers. We expect that this new random effects method will prove useful for a wide range of group studies, not only in the context of DCM, but also for other modelling endeavours, e.g. comparing different source reconstruction methods for EEG/MEG or selecting among competing computational models of learning and decision-making.

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Available from: Karl J Friston, Oct 04, 2015
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    • "This can be done formally using a hierarchical model for the group that treats the subject-level model as a random variable, which allows the distribution of subject-level models to be estimated from the data (citations to justify this as a standard approach). This distribution can be approximated from the within-subject model probabilities that we already have by using a variational Bayesian method proposed by Stephan et al. (2009). In short, with this method, the expected multinomial distribution over subject-level models is estimated as the expected value of a Dirichlet distribution that is conditioned on the observed data. "
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    • "not through intrinsic hierarchical connections) provided the most accurate and parsimonious account of the at-risk functional MRI data. Taking the ratio of the posterior probability of models in both the control and at-risk groups yields an odds ratio of non-linear versus bilinear models of 2.6, corresponding to a robust effect (Stephan et al., 2009a). "
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    • "Model comparison was implemented using random-effects BMS in SPM8, using the DCM tool (DCM10 version) to compute exceedance and posterior probabilities (i.e. likelihood of a model given the data) at group level (Stephan et al., 2009). It should be noted that exceedance probabilities are conditional to the size of model space, and all exceedance probabilities sum to 1 over all models tested. "
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