A Hierarchical Bayesian Mixture Model Approach for Analysis of Resting-State Functional Brain Connectivity: An Alternative to Thresholding

Abstract

This article proposes a Bayesian hierarchical mixture model to analyze functional brain connectivity where mixture components represent "positively connected" and "non-connected" brain regions. Such an approach provides a data-informed separation of reliable and spurious connections in contrast to arbitrary thresholding of a connectivity matrix. The hierarchical structure of the model allows simultaneous inferences for the entire population as well as for each individual subject. A new connectivity measure, the posterior probability of a given pair of brain regions of a specific subject to be connected given the observed correlation of regions' activity, can be computed from the model fit. The posterior probability reflects the connectivity of a pair of regions relative to the overall connectivity pattern of an individual, which is overlooked in traditional correlation analyses. This article demonstrates that using the posterior probability might diminish the effect of spurious connections on inferences, which is present when a correlation is used as a connectivity measure. In addition, simulation analyses reveal that the sparsification of the connectivity matrix using the posterior probabilities might outperform the absolute thresholding based on correlations. Therefore, we suggest that posterior probability might be a beneficial measure of connectivity compared with the correlation. The applicability of the introduced method is exemplified by a study of functional resting-state brain connectivity in older adults.

Keywords: brain aging; fMRI; functional connectivity; hierarchical modeling; lognormal distribution; resting state.

FIG. 1.

Heat map of pairwise Fisher-transformed correlations for one subject (left, axes represent brain regions, nodes), histogram of the pairwise Fisher-transformed correlations together with the model fit for the subject (center), heat map of posterior probabilities of being connected for one subject and all node pairs (right). Color images are available online.

FIG. 2.

95% Credible intervals for the relationship of covariates to the strength of connections (α, solid line) and proportion of connections (δ, dashed line). Sex equals 1 for males and 0 for females. Bl, block design; EM, episodic memory; FD, framewise displacement; Fl, word fluency; PS, processing speed.

FIG. 3.

Histogram of estimated proportion of connections per subject (in %).

FIG. 4.

(A–C) Histograms of Fisher-transformed correlations for three different individuals in the Betula project on a probability density function scale with densities of fitted normal, lognormal, and mixture distributions. (D) Scatter plot of posterior probabilities of being connected (y-axis) versus observed correlations (x-axis), for three selected individuals. Color images are available online.

FIG. 5.

The brain graph with edges corresponding to the node pairs with significant association between age and the posterior probability of being in the connected component.

FIG. 6.

Left: node pairs, grouped by Power, , network, where the correlations are significantly associated with age (Bonferroni corrected for 36,585 comparisons). Right: node pairs, grouped by network, where the posterior probabilities are significantly associated with age (Bonferroni corrected for 36,585 comparisons). Audi, Auditory; Cere, Cerebellum; Cing, Cingulate; Defa, Default mode network; Dors, Dorsal Attention; Fron, Fronto-parietal; Memo, Memory; Sali, Salience; Sens, Sensorimotor; Subc, Subcortical; Unce, Uncertain; Vent, Ventral attention; Visu, Visual.

FIG. 7.

Examples of node pairs that have a significant relationship of correlation between the blood oxygenation-level-dependent signal of their nodes and age but not between posterior probability and age. Top row: scatterplot of age versus posterior probability, bottom row: scatterplots of age versus Fisher-transformed correlation; observations that have zero (red diamonds) and positive (blue circles) posterior probability of being connected. Left: node pair 30–177 from Power parcellation, with mostly negative or small positive correlations; right: node pair 145–146 with positive posterior probabilities for 197 out of 198 subjects. Color images are available online.

FIG. 8.

A comparison of performance of absolute thresholding at 10⁻⁶ based on correlation (abs), absolute thresholding at 10⁻⁶ based on posterior probability (abs, pp), proposed mixture modeling and thresholding based on pseudo-false discovery rate (mixture), proportional threshold based on top 5% correlations (prop 5%) and on top 5% posterior probabilities (prop 5%, pp), and proportional threshold based on top 10% correlations (prop 10%) and on top 10% posterior probabilities (prop 10%, pp).