Bayesian model selection maps for group studies

Abstract

This technical note describes the construction of posterior probability maps (PPMs) for Bayesian model selection (BMS) at the group level. This technique allows neuroimagers to make inferences about regionally specific effects using imaging data from a group of subjects. These effects are characterised using Bayesian model comparisons that are analogous to the F-tests used in statistical parametric mapping, with the advantage that the models to be compared do not need to be nested. Additionally, an arbitrary number of models can be compared together. This note describes the integration of the Bayesian mapping approach with a random effects analysis model for BMS using group data. We illustrate the method using fMRI data from a group of subjects performing a target detection task.

Fig. 1

Graphical models underlying (A) fixed and (B) random effects inference on model space at the group level. FFX assigns a model, drawn using r, to be used by all members of the group, while for RFX, a (potentially different) model is assigned to each member of the group. Mult(m;1, r) corresponds to Mult(m; N, r), when the number of observations N is equal to 1. See the main text for a detailed explanation of the two different inference approaches.

Fig. 2

Schematic representation of the method for constructing Bayesian model selection (BMS) maps for group studies. (1) The first step involves estimating log-evidence maps for each subject and model. (2) The RFX approach for BMS described in the text is then applied in a voxel-wise manner to the log-evidence data. (3) The BMS maps (posterior probability map, PPM; exceedance probability map, EPM) for each model are then constructed by plotting the posterior and exceedance probabilities at each voxel (〈r_ki〉 and φ_ki, respectively), using a threshold, γ, to visualise the resulting image. See the main text for a detailed explanation of the different steps involved in this procedure.

Fig. 3

Group-level PPMs for the ‘Validity’ model from (A) fixed and (B) random effects analysis. The maps therefore show brain regions encoding cue validity. These maps were thresholded to show regions where the posterior model probability of the ‘Validity’ model is greater than γ = 0.75. The FFX approach does not account for between-subject variability and, consequently, can appear over-confident.

Fig. 4

Group-level PPMs (z = 59 mm, Talairach coordinates) for the ‘Validity’ model from (A) fixed and (B) random effects analysis. The maps were thresholded to show regions where the posterior probability of the ‘Validity’ model is greater than γ = 0.75. The position of the crossbars (Talairach coordinates: [− 21, − 73, 59] mm) indicates a cluster that is only visible for the FFX maps, suggesting that this approach may be over-confident.

Fig. 5

Posterior model probabilities obtained by comparing the ‘Validity’ and ‘Null’ model (models 1 and 2, respectively) at an example voxel, [− 21, − 73, 59] mm (Talairach coordinates), using a (A) fixed and (B) random effects analysis. For the RFX analysis, we include the exceedance probabilities at the same voxel. As can be seen, the RFX analysis produces lower posterior probabilities for model 1 than does the FFX approach.

Fig. 6

(A) Group-level exceedance probability map (EPM) (log-odds scale) for the ‘Validity’ model. The map was thresholded to show regions where the exceedance probability for the ‘Validity’ model is greater than γ = 0.95. (B) Posterior distribution and exceedance probability for the same model at an example voxel, [− 21, − 73, 59] mm (Talairach coordinates).

Fig. 7

Log-model evidence differences between the ‘Null’ and ‘Validity’ models (model 2 and model 1, respectively) at voxel [− 29, 0, 49] mm (Talairach coordinates), for the 12 subjects analysed. The data clearly show that one subject (bottom row) is an outlier.

Fig. 8

Posterior model probabilities obtained by comparing the ‘Validity’ and ‘Null’ model (models 1 and 2, respectively) at voxel [− 29, 0, 49] mm (Talairach coordinates), using a (A) fixed and (B) random effects analysis. For the RFX analysis, we include the exceedance probabilities at the same voxel. The voxel chosen here belongs to a brain region where FFX and RFX analyses yield different results due to the presence of an outlier (see Fig. 7).

Fig. 9

Group-level PPMs (slice z = 49 mm, Talairach coordinates) for the ‘Validity’ model from (A) fixed and (B) random effects analysis. The maps were thresholded to show regions where the posterior model probability of the ‘Validity’ model is greater than γ = 0.75. The crossbars indicate a cluster of voxels where one of the subjects is clearly an outlier (Fig. 7).

Fig. 10

Group-level PPM for the ‘Ideal Observer’ model from random effects analysis. The map is thresholded to show regions where the posterior model probability of the ‘Ideal Observer’ model is greater than γ = 0.6.