Abstract
This article seeks to investigate the impact of aging on functional connectivity across different cognitive control scenarios, particularly emphasizing the identification of brain regions significantly associated with early aging. By conceptualizing functional connectivity within each cognitive control scenario as a graph, with brain regions as nodes, the statistical challenge revolves around devising a regression framework to predict a binary scalar outcome (aging or normal) using multiple graph predictors. Popular regression methods utilizing multiplex graph predictors often face limitations in effectively harnessing information within and across graph layers, leading to potentially less accurate inference and predictive accuracy, especially for smaller sample sizes. To address this challenge, we propose the Bayesian Multiplex Graph Classifier (BMGC). Accounting for multiplex graph topology, our method models edge coefficients at each graph layer using bilinear interactions between the latent effects associated with the two nodes connected by the edge. This approach also employs a variable selection framework on node-specific latent effects from all graph layers to identify influential nodes linked to observed outcomes. Crucially, the proposed framework is computationally efficient and quantifies the uncertainty in node identification, coefficient estimation, and binary outcome prediction. BMGC outperforms alternative methods in terms of the aforementioned metrics in simulation studies. An additional BMGC validation was completed using an fMRI study of brain networks in adults. The proposed BMGC technique identified that sensory motor brain network obeys certain lateral symmetries, whereas the default mode network exhibits significant brain asymmetries associated with early aging.
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S.G. conceived the idea, formulated the model, calculated the priors for parameters and full conditionals for MCMC computation. S.G. took the lead in drafting the article. J.R.A. implemented the model and drafted parts of the simulation study and the real data analysis, generated and added simulation figures and tables. I.D.D. took the lead in drafting the real data analysis section, editing the draft, providing scientific data and valuable scientific insights, generated the real data result figures. All authors reviewed the manuscript.
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Appendix
Appendix
Below we present the full conditional posterior distributions for all parameters within the generalized linear model detailed in “Model and Prior Formulation” section. These distributions are utilized for implementing a Markov Chain Monte Carlo (MCMC) algorithm via Gibbs sampling. The samples derived from the MCMC algorithm yield samples from the complete posterior distributions for the model parameters jointly, facilitating posterior inference as elaborated in “Posterior Inference” section. Let \(\kappa _i=(y_i-0.5)/\omega _i\), and \({\varvec{\Omega }} = diag\left( \frac{1}{\omega _1},...,\frac{1}{\omega _n}\right)\). Assuming \({\varvec{X}}\) to be an \(n\times p\) matrix with its ith row as \({\varvec{x}} _i\), and \({\varvec{A}} ^{(\alpha )}\) to be an \(n\times V(V-1)/2\) dimensional matrix with its ith row given by \({\varvec{w}} _i^{(\alpha )}\). Consequently, the full conditional distributions for the model parameters are expressed as:
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\(\mu |-\sim N\left( \frac{{\textbf {1}}_{n}^{T} {\varvec{\Omega }} ^{-1}( {\varvec{\kappa }} - {\varvec{X}} {\varvec{\beta }} _x - \sum _{\alpha =1}^L {\varvec{A}} ^{(\alpha )} {\varvec{\gamma }} ^{(\alpha )})}{{\textbf {1}}_{n}^{T} {\varvec{\Omega }} ^{-1}{} {\textbf {1}}_{n}},\frac{1}{{\textbf {1}}_{n}^{T} {\varvec{\Omega }} ^{-1}{} {\textbf {1}}_{n}} \right)\), where \({\varvec{\kappa }} =(\kappa _1,...,\kappa _n)^T\) is an n-dimensional vector of continuous outcomes over all samples.
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\({\varvec{\beta }} _x|- \sim N( {\varvec{\mu }} _{ {\varvec{\beta }} }, {\varvec{\Sigma }} _{ {\varvec{\beta }} })\), where
$$\begin{aligned} {\varvec{\Sigma }} _{ {\varvec{\gamma }} }=( {\varvec{X}} ^T {\varvec{\Omega }} ^{-1} {\varvec{X}} + {\varvec{I}} _{p})^{-1},\,\, {\varvec{\mu }} _{ {\varvec{\gamma }} }= {\varvec{\Sigma }} _{ {\varvec{\gamma }} } {\varvec{X}} ^T {\varvec{\Omega }} ^{-1}( {\varvec{\kappa }} - \mu {\textbf {1}}_{n} - \sum _{l=1}^L {\varvec{A}} ^{(\alpha )} {\varvec{\gamma }} ^{(\alpha )}). \end{aligned}$$ -
\({\varvec{\xi }} _v|-\sim \eta _v N( {\varvec{\mu }} _v, {\varvec{\Sigma }} _v)+(1-\eta _v)\Delta ( {\varvec{0}} )\), where
$$\begin{aligned} {\varvec{\Sigma }} _v=( {\varvec{Z}} _v^T {\varvec{\Omega }} ^{-1} {\varvec{Z}} _v+ {\varvec{K}} ^{-1})^{-1},\,\, {\varvec{\mu }} _v= {\varvec{\Sigma }} _v {\varvec{Z}} _v^T {\varvec{\Omega }} ^{-1}\tilde{ {\varvec{\kappa }} }_v. \end{aligned}$$Here \({\varvec{Z}} _v\) is an \(n\times LH\) matrix whose ith row is given by \((\sum _{k<v}w_{i,(k,v)}^{(1)} {\varvec{\xi }} _k^{(1)T} {\varvec{\Theta }} ^{(1)}+\sum _{v<k}w_{i,(v,k)}^{(1)} {\varvec{\xi }} _k^{(1)T} {\varvec{\Theta }} ^{(1)},..., \sum _{k<v}w_{i,(k,v)}^{(L)} {\varvec{\xi }} _k^{(L)T} {\varvec{\Theta }} ^{(L)}+\sum _{v<k}w_{i,(v,k)}^{(L)} {\varvec{\xi }} _k^{(L)T} {\varvec{\Theta }} ^{(L)})\) and \(\tilde{ {\varvec{\kappa }} }_v\) is an n dimensional vector with its ith entry
$$\tilde{\kappa }_{i} = \kappa _i - \mu - {\varvec{X}} {\varvec{\beta }} _x - \sum _{k<k':k,k'\ne v} \sum _{\alpha =1}^L w_{i,(k,k')}^{(\alpha )} {\varvec{\xi }} _k^{(\alpha )T} {\varvec{\Theta }} ^{(\alpha )} {\varvec{\xi }} _{k'}^{(\alpha )}.$$ -
\(\eta _v|-\sim Ber(\tilde{\eta }_v)\), where
$$\tilde{\eta }_v=\frac{\delta N(\tilde{ {\varvec{\kappa }} }_v| {\varvec{0}} , {\varvec{Z}} _v {\varvec{K}} {\varvec{Z}} _v^T+ {\varvec{\Omega }} )}{\delta N(\tilde{ {\varvec{\kappa }} }_v| {\varvec{0}} , {\varvec{Z}} _v {\varvec{K}} {\varvec{Z}} _v^{T}+ {\varvec{\Omega }} )+(1-\delta )N(\tilde{ {\varvec{\kappa }} }_v| {\varvec{0}} , {\varvec{\Omega }} )}.$$ -
\(\delta |-\sim Beta(a+\sum _{v=1}^V\eta _v,b+V-\sum _{v=1}^V\eta _v)\).
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\({\varvec{K}} |-\sim IW\left( \nu +\#\{v:\eta _v=1\}, {\varvec{I}} _{LH}+\sum _{\{v:\eta _v =1\}} {\varvec{u}} _v {\varvec{u}} _v^T\right)\), where \({\varvec{u}} _v=( {\varvec{u}} _v^{(1)T},..., {\varvec{u}} _v^{(L)T})^T\).
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$$\theta _{h}^{(\alpha )} \mid - \sim {\left\{ \begin{array}{ll} 1 &{} w.p. \; p_{h,1}^{(\alpha )} \\ 0 &{} w.p. \; p_{h,2}^{(\alpha )} \\ -1 &{} w.p. \; p_{h,3}^{(\alpha )} \end{array}\right. }$$
where \(p_{h,1}^{(\alpha )} = \dfrac{\pi _{h, 1}^{(\alpha )} N( {\varvec{\kappa }} |\mu {\textbf {1}}_{n} + {\varvec{X}} {\varvec{\beta }} _x + {\varvec{A}} ^{(\alpha )} ( {\varvec{\gamma }} ^{(\alpha )})^{(\theta _{h}^{(\alpha )} = 1)}+\sum _{\alpha '\ne \alpha } {\varvec{A}} ^{(\alpha ')} ( {\varvec{\gamma }} ^{(\alpha ')}), {\varvec{\Omega }} )}{S}\), \(p_{h,2}^{(\alpha )} = \dfrac{\pi _{h, 2}^{(\alpha )} N( {\varvec{\kappa }} |\mu {\textbf {1}}_{n} + {\varvec{X}} {\varvec{\beta }} _x + {\varvec{A}} ^{(\alpha )} ( {\varvec{\gamma }} ^{(\alpha )})^{(\theta _{h}^{(\alpha )} = 0)}+\sum _{\alpha '\ne \alpha } {\varvec{A}} ^{(\alpha ')} ( {\varvec{\gamma }} ^{(\alpha ')}), {\varvec{\Omega }} )}{S},\) \(p_{h,3}^{(\alpha )} = \dfrac{\pi _{h, 3}^{(\alpha )} N( {\varvec{\kappa }} |\mu {\textbf {1}}_{n} + {\varvec{X}} {\varvec{\beta }} _x + {\varvec{A}} ^{(\alpha )} ( {\varvec{\gamma }} ^{(\alpha )})^{(\theta _{h}^{(\alpha )} = -1)}+\sum _{\alpha '\ne \alpha } {\varvec{A}} ^{(\alpha ')} ( {\varvec{\gamma }} ^{(\alpha ')}), {\varvec{\Omega }} )}{S}\), where, \(S=\sum _{s\in \{0,1,-1\}}\pi _{h, s}^{(\alpha )} N( {\varvec{\kappa }} |\mu {\textbf {1}}_{n} + {\varvec{X}} {\varvec{\beta }} _x + {\varvec{A}} ^{(\alpha )} ( {\varvec{\gamma }} ^{(\alpha )})^{(\theta _{h}^{(\alpha )} = s)}+\sum _{\alpha '\ne \alpha } {\varvec{A}} ^{(\alpha ')} {\varvec{\gamma }} ^{(\alpha ')}, {\varvec{\Omega }} )\)
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\(\omega _i|- PG(1,\mu + {\varvec{x}} _{i}^{T} {\varvec{\beta }} _x+\sum _{\alpha =1}^L {\varvec{w}} _i^{(\alpha )T} {\varvec{\gamma }} ^{(\alpha )})\), for \(i=1,...,n.\)
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Guha, S., Rodriguez-Acosta, J. & Dinov, I.D. A Bayesian Multiplex Graph Classifier of Functional Brain Connectivity Across Diverse Tasks of Cognitive Control. Neuroinform 22, 457–472 (2024). https://doi.org/10.1007/s12021-024-09670-w
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DOI: https://doi.org/10.1007/s12021-024-09670-w