Bayesian Longitudinal Network Regression With Application to Brain Connectome Genetics

Abstract

The increasing availability of large-scale brain imaging genetics studies enables more comprehensive exploration of the genetic underpinnings of brain functional organizations. However, fundamental analytical challenges arise when considering the complex network topology of brain functional connectivity, influenced by genetic contributions and sample relatedness, particularly in longitudinal studies. In this paper, we propose a novel method named Bayesian Longitudinal Network-Variant Regression (BLNR), which models the association between genetic variants and longitudinal brain functional connectivity. BLNR fills the gap in existing longitudinal genome-wide association studies that primarily focus on univariate or multivariate phenotypes. Our approach jointly models the biological architecture of brain functional connectivity and the associated genetic mixed-effect components within a Bayesian framework. By employing plausible prior settings and posterior inference, BLNR enables the identification of significant genetic signals and their associated brain sub-network components, providing robust inference. We demonstrate the superiority of our model through extensive simulations and apply it to the Adolescent Brain Cognitive Development (ABCD) study. This application highlights BLNR’s ability to estimate the genetic effects on changes in brain network configurations during neurodevelopment, demonstrating its potential to extend to other similar problems involving sample relatedness and network-variate outcomes.

1 ∣. Introduction

With the broad accessibility of neuroimaging and high-throughput genomics data in both healthy and disease studies, the field of brain imaging genomics—linking genetic variations to various brain imaging phenotypes—has experienced rapid growth. This integration is advancing our understanding of the molecular underpinnings of brain function and structure and providing unprecedented opportunities to elucidate the etiology of disorders and the normal trajectories of neurodevelopment and neurodegeneration. Meanwhile, given the large scale and complexity of both imaging traits and genetic variants, it is critical to develop robust analytical frameworks to ensure the validity and interoperability of findings, enabling researchers to uncover meaningful imaging genetic associations that can inform clinical and research applications.

Among the various neuroimaging traits, brain functional connectivity stands out for its ability to characterize brain functional network organizations. By collecting functional magnetic resonance imaging (fMRI) at resting-state or different cognitive tasks, one could aggregate the functional time courses within a node or region of interest (ROI) under a brain atlas, and construct functional connectivity across the nodes with the statistical dependence between each pair of time courses. Given the unique advantage of capturing brain functional underpinnings, functional connectivity has been largely utilized as a neuromarker to develop predictive models that correlate brain characteristics with behavior and disease outcomes with a flurry of successes [1]. Meanwhile, a growing effort has been placed to uncover the genetic bases for functional connectivity endophenotypes under different study populations [2, 3]. One of the fundamental challenges for analyzing brain connectivity is posed by its network topology, making the most dominantly used strategy to extract unique edges from the connectivity network sub-optimal. Particularly, when performing genetic association analyses, a massive univariate analysis that associates each single nucleotide polymorphism (SNP) with each brain connectivity edge (i.e., connection) could dilute genetic effect upon scattered brain connections, leading to less interpretable results. To address this, recent attempts have focused on employing different graphical representations for brain connectivity when linking it with genetic variants [4, 5]. These approaches highlight the necessity of incorporating the biological architecture of brain connectivity into genetic modeling, thereby enhancing the interpretability and relevance of the findings.

Meanwhile, substantial efforts in recent neurocognitive and psychiatric studies have focused on collecting brain imaging scans longitudinally to inform neurodevelopment or neurodegeneration trajectory. These longitudinal scans offer an unprecedented opportunity for quantitative genetic studies to characterize both the static and dynamic genetic influences on brain connectivity phenotypes over time. Traditional approaches for accommodating longitudinal GWAS typically involve linear mixed-effect models with random intercepts or slopes [6]. However, these methods are generally designed for univariate outcomes or vector-variate phenotypes and may not be well-suited for the complex nature of brain functional connectivity data. To the best of our knowledge, only one recent study by Tian et al. (2023) [7] has considered network-variate outcomes with mixed effects in the context of imaging genetics. Their study addresses sample relatedness induced by kinship, which is a different complication than the one discussed in our paper and cannot be directly extended to characterize genetic association across time. Consequently, there remains a critical need for novel analytical frameworks that can handle the complexity of network-variate phenotypes in a longitudinal context.

To bridge the gap, we develop a Bayesian longitudinal network-variant regression framework and implement it to perform genetic association analysis with brain functional connectivity endophenotype. Instead of independently extracting unique functional connections from the connectivity network, we model the unknown modular structure of the connectivity matrix via a stochastic block model (SBM) [8]. The stochastic block assumption upon brain functional network patterns aligns with the converging evidence in neuroscience that brain functional configuration tends to interact through a set of sub-networks or modules [9, 10], and these modules are thought to be fundamental to the brain’s functional architecture, facilitating efficient and specialized information processing. SBM has also been adopted in brain network analyses to parameterize the connectivity matrix in the neurobiologically plausible way [11-13]. In our current study, we design a joint modeling framework that simultaneously uncovers the brain network modular features and their associated mixed effect terms from the genetic variant and other covariates, including time. This approach allows us to identify sub-network modules that are directly influenced by genetic factors, distinguishing them from traditionally constructed brain network modules, such as the canonical functional sub-networks [10]. Building on the modeling framework, we develop a unified Bayesian inference strategy with plausible priors to identify significant genetic-to-brain network associations with valid posterior inference to quantify uncertainty for each effect component. We also apply the proposed method to the latest cohort of the landmark Adolescent Brain Cognitive Development (ABCD) study, investigating the genetic impact on changes in resting-state functional connectivity during the neurodevelopmental stage.

The remainder of the paper is organized as follows. Section 2 describes the joint modeling framework on SBM and mixed-effect modeling for the network-variate phenotype genetic associations. Section 3 introduces our Bayesian paradigm including the prior specifications and posterior computations. In Section 4, we conduct extensive simulations to assess the performance of our proposed method in comparison to existing alternatives along with real data application in Section 5 to investigate the genetic association with repeatedly measured brain functional connectivity. In Section 6, we conclude with discussions.

2 ∣. Methods

We describe the notations and modeling framework in the context of longitudinal genetic association studies. Assume the study recruits $N$ subjects. For subject $i,i=1,\dots ,N$, let $s_i$ represent a single-nucleotide polymorphism (SNP) dose of interest. For the quantitative phenotypes, brain intrinsic functional activities are repeatedly measured by resting-state fMRI (rs-fMRI) under $T_i$ visits. By registering image scans into the same brain atlas with $V$ nodes or regions of interest (ROI), we aggregate the functional time courses within each node and summarize brain functional connectivity at visit $t$ for subject $i$ by an indirect graph ${𝒢}_{it}=(𝒱,{ℰ}_{it})$. The graph ${𝒢}_{it}$ consists of a vertex set $𝒱=\{1,\dots ,V\}$ and an edge set ${ℰ}_{it}$, with each edge capturing the functional communication between a pair of nodes. We further represent network ${𝒢}_{it}$ by its corresponding weighted adjacency matrix ${\mathbf{A}}_{it}=(a_{it,jl})$, which is symmetric with $a_{it,jl}$ indicating the connected strength between nodes $j$ and $l,0\le j,l\le V$.

Our aim is to identify informative genetic variants for the longitudinal brain connectivity phenotypes. Compared with existing GWAS with longitudinal data that deal with a univariate or multivariate phenotype, our study focuses on a network-variate phenotype generated from functional connectivity. From a neurobiological perspective, converging evidence suggests that brain functional architectures tend to form modular structures across the brain as unique functional processing units. For instance, utilizing rs-fMRI collected from a cohort of healthy adults, canonical brain sub-networks [14] have been constructed with brain regions partitioned into groups or communities; and within each community, functional connections are believed to provide consistent functional involvement. Therefore, instead of marginally modeling individual functional connections which does not accommodate the topological structure, we propose that the functional connectivity phenotype is associated with genetic variants through an unknown set of functional modules.

Specifically, we assume $𝒱$ can be divided into $Q$ unknown latent modules or communities. For each node $v\in 𝒱$, a random vector ${\mathbf{g}}_v={(g_{v1},\dots ,g_{vQ})}^T$ is introduced to capture community allocation with latent indicator $g_{vq}=1$ if node $v$ belongs to module $q$. We further assume ${\mathbf{q}}_v$ follows a Multinomial distribution ${\mathbf{g}}_v\sim \text{Multi}(1,\pi )$ with a vector of allocation probabilities $\pi ={({\pi }_1,\dots ,{\pi }_Q)}^T$, and assemble the community membership as an allocation matrix $\mathit{G}=({\mathbf{g}}_1,\dots ,{\mathbf{g}}_V)\in {\mathbb{R}}^{Q\times V}$. To jointly dissect the module structure of each connectivity network ${\mathbf{A}}_{it}$ and establish the genetic association with the longitudinal modular phenotypic features, we propose the following hierarchical modeling including a weight stochastic block model (SBM) given the latent community structure, and a linear mixed model with latent connectivity weights for $i=1,\dots ,N;t=1,\dots ,T_i;1\le q\le r\le Q$

$$a_{it,jl}\mid g_{jq}=1,g_{lr}=1\sim \mathrm{N}(m_{it,qr},{\sigma }_{qr}^2);$$ (1)

$${\mathbf{m}}_{i,qr}={\mathbf{X}}_i{\beta }_{qr}+{\mathbf{Z}}_i{\mathbf{b}}_{i,qr}+{ϵ}_{i,qr}$$ (2)

In model (1), $m_{it,qr}$ captures the latent subject- and time-specific functional connectivity strength between block $q$ and $r$ with variance parameter ${\sigma }_{qr}^2$. With a symmetric ${\mathbf{A}}_{it}$, we have $m_{it,qr}=m_{it,rq}$ and ${\sigma }_{qr}^2={\sigma }_{rq}^2$, and we also denote ${\mathbf{m}}_{i,qr}={(m_{i1,qr},\dots ,m_{iT,qr})}^T$. We assume a weighted SBM with Normal distributions here instead of a canonical SBM with Bernoulli distributions for a binary network given that in our application, we maintain the continuous scale of functional connections to retain sufficient connectivity information; and those metrics are further normalized through a Fisher’s z-transformation as detailed in Section 5. In model (2), ${\mathbf{X}}_i\in {\mathbb{R}}^{T\times P}$ represents all the fixed effects across visits including the genetic variant; and the set of random effects is denoted by ${\mathbf{Z}}_i\in {\mathbb{R}}^{T\times Q}$. For each connectivity strength between block $q$ and $r$, ${\beta }_{qr}$ and ${\mathbf{b}}_{i,qr}$ are fixed and random effects coefficients, and we also assume ${\mathbf{b}}_{i,qr}\sim \mathrm{N}(0,{\mathbf{D}}_{qr})$ with a general covariance matrix ${\mathbf{D}}_{qr}$ and have a random error vector ${ϵ}_{i,qr}\sim \mathrm{N}(0,{\sigma }_{ϵ,qr}^2\mathbf{I})$. It is worth noting that through the weighted SBM, we achieve a separation of the connectivity topology captured by the community allocation $\mathbf{G}$ with the connectivity strength as measured by $\{{\mathbf{m}}_{i,qr}\}$. Therefore, with topological information fully characterized in Equation (1), those strength parameters can be modeled as individual latent phenotypes in Equation (2). Moreover, the joint modeling of (1) and (2) ensures that the uncovered topological structures are informed by the genetic associations, thereby facilitating the identification of saturated signaling brain modules under the influence of genetic contributions. This approach highlights the advantage of joint modeling of phenotypic network communities and their genetic bases. Compared with directly adopting existing marginal homogeneous functional systems [14] with brain network modules likely to be partially signaling, our approach is expected to enhance the detection power and increase the phenotype-to-genotype effect size.

In practice, there are various options for the fixed and random effects ${\mathbf{X}}_i$ and ${\mathbf{Z}}_i$. One realization for model (2) we adopt in the data application follows

$$\begin{gathered} m_{it,qr}={\mathbf{x}}_{i,0}^T{\beta }_{0,qr}+{\beta }_{1,qr}{s}_i+{\beta }_{2,qr}{t}_i+{\beta }_{3,qr}{s}_i{t}_i \\ +b_{0i,qr}+b_{1i,qr}{t}_i+{ϵ}_{it,qr} \end{gathered}$$ (3)

where the adjusted covariates ${\mathbf{x}}_{i,0}$ include baseline age, sex and top genetic principle components with nuisance coefficients ${\beta }_{0,qr}$, and coefficients ${\beta }_{1,qr}$, ${\beta }_{2,qr}$ and ${\beta }_{3,qr}$ capture the effects of SNP, time and their interaction. Within model (3), ${\beta }_{1,qr}$ and ${\beta }_{3,qr}$ are of particular interest that characterizes the cross-sectional and longitudinal effect of the genetic variant upon the related connectivity modular phenotype. Along with other parameters, the model estimation and inference will identify significant main and temporal-interaction genetic effects and dissect corresponding phenotypic network components.

3 ∣. Bayesian Framework

3.1 ∣. Prior Specification

Under a Bayesian paradigm, we first describe the prior settings for the joint modeling. Without loss of generality, we focus on the prior specifications for model (3) which can be readily extended to (2). For the main-effect coefficients including the cross-sectional genetic effect, it is anticipated that only a proportion of brain network modular structures generated from whole-brain functional connectivity are impacted by the genotype of interest or changing over time. Thus, we impose sparsity through point mass mixture priors

$$\begin{gathered} {\beta }_{j,qr}\mid {\delta }_{j,qr}\sim (1-{\delta }_{j,qr})I_0+{\delta }_{j,qr}\mathrm{N}(0,{\tau }_j^2); \\ \text{for}j=1,2;1\le q\le r\le Q \end{gathered}$$ (4)

where ${\delta }_{j,qr}\in \{0,1\}$ denotes the selection indicator to include or exclude ${\beta }_{j,qr}$ from the model. When ${\delta }_{j,qr}=1$, we assign a large variance ${\tau }_j^2$ to sample ${\beta }_{j,qr}$ from the non-informative Normal prior. If ${\delta }_{j,qr}=0$, we set ${\beta }_{j,qr}$ to a point mass at zero, denoted by $I_0$, effectively removing this component from the model. Regarding the interaction effect ${\beta }_{3,qr}$, we take into account the heredity constrain that the interaction term is included in the model only if all of its main effects are included and assign

$$\begin{gathered} {\beta }_{3,qr}\mid {\delta }_{1,qr},{\delta }_{2,qr},{\delta }_{3,qr}\sim \left(1-\prod _{j=1}^3{\delta }_{j,qr}\right)I_0+\prod _{j=1}^3{\delta }_{j,qr}\mathrm{N}(0,{\tau }_3^2); \\ \text{for}1\le q\le r\le Q \end{gathered}$$ (5)

Based on Equation (5), the sparsity of ${\beta }_{3,qr}$ is determined by both its own latent selection indicator ${\delta }_{3,qr}$ and main effect ones ${\delta }_{1,qr}$ and ${\delta }_{2,qr}$. When either ${\delta }_{1,qr}$ and ${\delta }_{2,qr}$ is zero indicating at least one of the main effects is not considered in the model, the interaction term is directly excluded from the model. Of note, when heredity is not desired in some applications, one can easily simply prior (5) to ${\beta }_{3,qr}\mid {\delta }_{3,qr}\sim (1-{\delta }_{3,qr})I_0+{\delta }_{3,qr}\mathrm{N}(0,{\tau }_3^2)$ as a special case during implementation.

To perform posterior computation, a common practice to handle point mass mixture models is to represent priors (4) and (5) through equivalent modeling forms as ${\beta }_{j,qr}={\delta }_{j,qr}{\tilde{\beta }}_{j,qr}$ for $j=1,2$ and ${\beta }_{3,qr}={\delta }_{1,qr}{\delta }_{2,qr}{\delta }_{3,qr}{\tilde{\beta }}_{3,qr}$, where ${\tilde{\beta }}_{j,qr}\sim \mathrm{N}(0,{\tau }_j^2),j=1,2,3$ are latent coefficient parameters characterizing the non-zero magnitude. As detailed in Li et al. (2015) [15] such representation induces an independent structure between latent indicators and coefficients, simplifying the posterior computation. It is important to elaborate that for the general model formulation (2), the aforementioned priors can be directly adopted after separating each ${\beta }_{qr}$ into components corresponding to the main and interaction effects, respectively. Similar to Zhao et al. (2023) [16], we can impose sparsity for each interaction term incorporating the selection indicators from its main effects. Without a preference on the prior sparsity, we assign each ${\delta }_{p,qr}$ a Bernoulli prior with 0.5 probability.

For the remaining parameters and hyperparameters, we provide a concise overview of the prior distributions assigned to each, as outlined below:

• For the allocation probabilities parameters $\pi$, we impose a conjugate Dirichlet prior for the weighted SBM allocation probability $\pi$ as $\pi \sim \text{Dir}(p_1,\dots ,p_Q)$.

• For the nuisance parameters ${\beta }_{0,qr}$, we assign ${\beta }_{0,qr}\sim \mathrm{N}(0,{\sigma }_x^2\mathbf{I})$.

• For the variance parameters, we assign each ${\mathbf{D}}_{qr}$ an Inverse Wishart distribution $\text{IW}(\mathbf{I},\nu )$, and each ${\sigma }_{qr}^2$, ${\sigma }_{ϵ,qr}^2$ an Inverse Gamma distribution $\text{IG}(0.01,0.01)$.

• For the hyperparameters ${\tau }_j,j=1,2,3$ and ${\sigma }_x^2$, we assign them a fixed value, choosing a large value (e.g., 10) to minimize their influence on the posterior likelihood.

Finally, we name our model Bayesian Longitudinal Network-variant Regression model (BLNR) as a tool to jointly uncover the modular structure of network phenotype and its associated longitudinal genetic influence.

3.2 ∣. Posterior Inference

We develop a posterior inference algorithm based on Markov chain Monte Carlo (MCMC) for the proposed BLNR model. Given the observed data $𝒪=({\mathbf{A}}_{it},t_i,s_i,{\mathbf{x}}_{i,0},t=1,\dots ,T_i;i=1,\dots ,n)$, the unknown parameters $\Theta =({\{{\mathbf{g}}_v\}}_{v=1}^V,\pi ,{\mathbf{b}}_{i,qr},m_{it,qr},{\{{\tilde{\beta }}_{j,qr},{\delta }_{j,qr}\}}_{j=1}^3,{\beta }_{0,qr},{\sigma }_{qr}^2,{\sigma }_{ϵ,qr}^2,{\mathbf{D}}_{qr};t=1,\dots ,T;1\le q\le r\le Q;i=1,\dots ,n)$ follow the joint conditional posterior distribution:

$$\begin{gathered} f\{\Theta \mid 𝒪\}\propto \prod _{i=1}\prod _t\prod _{1\le j\ne l\le Q}f(a_{it,jl}\mid m_{it,qr},{\sigma }_{qr}^2,{\{{\mathbf{g}}_v\}}_{v=1}^V) \\ \times \prod _{i=1}\prod _t\prod _{1\le q\le r\le Q}f(m_{it,qr}\mid s_i,t_i,{\mathbf{x}}_{i,0},{\mathbf{b}}_{i,qr},{\{{\delta }_{j,qr},{\tilde{\beta }}_{j,qr}\}}_{j=1}^3, \\ \times {\beta }_{0,qr},{\sigma }_{qr}^2,{\sigma }_{ϵ,qr}^2) \\ \times \prod _{i=1}\prod _{1\le q\le r\le Q}f({\mathbf{b}}_{i,qr}\mid {\mathbf{D}}_{qr})f({\mathbf{D}}_{qr})f({\beta }_{0,qr})f({\sigma }_{qr}^2)f({\sigma }_{ϵ,qr}^2) \\ \times f({\{{\tilde{\beta }}_{j,qr}\}}_{j=1}^3\mid {\{{\delta }_{j,qr}\}}_{j=1}^3)f({\{{\delta }_{j,qr}\}}_{j=1}^3)f({\{{\mathbf{g}}_v\}}_{v=1}^V\mid \pi )f(\pi ) \end{gathered}$$

The posterior computation can be achieved through Gibbs samplers based on the full conditional distribution of each parameter. A brief layout of each update step is shown below with the detailed algorithm provided in the Supporting Information:

• For the community allocation vector ${\{{\mathbf{g}}_v\}}_{v=1}^V\mid {\mathbf{A}}_{it},\pi ,m_{it,qr},{\sigma }_{qr}^2$, update it from its posterior multinomial distribution.

• For the vector of allocation probabilities $\pi \mid {\{{\mathbf{g}}_v\}}_{v=1}^V$, update it from the posterior Dirchlet distribution.

• For the subject/measurement-level connection strength $m_{it,qr}\mid {\mathbf{A}}_{it},{\{{\mathbf{g}}_v\}}_{v=1}^V,{\sigma }_{qr}^2,{\beta }_{0,qr},{\{{\tilde{\beta }}_{j,qr},{\delta }_{j,qr}\}}_{j=1}^3,s_i,t_i,{\mathbf{x}}_{i,0},{\mathbf{b}}_{i,qr},{\sigma }_{ϵ,qr}^2$, update it from the posterior normal distribution.

• For the selection indicator ${\delta }_{j,qr}\mid s_i,t_i,{\mathbf{x}}_{i,0},m_{it,qr},{\sigma }_{ϵ,qr}^2,{\mathbf{b}}_{i,qr},{\{{\delta }_{j^{\prime },qr}\}}_{j^{\prime }\ne j},{\{{\tilde{\beta }}_{j,qr}\}}_{j=1}^3,{\beta }_{0,qr}$, which relate to genetic exposure and time coefficient, update each element from the corresponding posterior Bernoulli distributions.

• Given the current value of ${\{{\delta }_{j,qr}\}}_{j=1}^3$, update the latent coefficient parameters characterizing the non-zero magnitude ${\{{\tilde{\beta }}_{j,qr}\}}_{j=1}^3$ from the posterior normal distribution.

• For the covariates coefficients ${\beta }_{0,qr}\mid s_i,t_i,{\mathbf{x}}_{i,0},m_{it,qr},{\sigma }_{ϵ,qr}^2,{\mathbf{b}}_{i,qr},{\{{\tilde{\beta }}_{j,qr},{\delta }_{j,qr}\}}_{j=1}^3$, update it from its posterior Multivariate Normal distribution.

• For the random effects ${\mathbf{b}}_{i,qr}\mid s_i,t_i,{\mathbf{x}}_{i,0},m_{it,qr},{\mathbf{D}}_{qr},{\{{\tilde{\beta }}_{j,qr},{\delta }_{j,qr}\}}_{j=1}^3,{\sigma }_{ϵ,qr}^2$, update it from its posterior Multivariate Normal distribution.

• For covariance matrix of random effects, update ${\mathbf{D}}_{qr}$ from its corresponding posterior Inverse Wishart distribution.

• Update each ${\sigma }_{qr}^2$, ${\sigma }_{ϵ,qr}^2$ from their corresponding posterior IG distribution.

Under random initialization, the above procedure will be repeated iteratively with the convergence of MCMC assessed by trace plots and GR method [17] on the posterior samples. To determine the number of blocks $Q$, we utilize a grid search and select the optimal value of $Q$ using the Bayesian Information Criterion (BIC). Our numerical experiments demonstrate that this approach effectively identifies the true or nearly true block number.

4 ∣. Simulation Studies

We evaluate the finite sample performance of the proposed BLNR method and compare it with existing alternatives by simulations. We consider two different sample sizes with N = 500 and 1 000. For each subject, brain fMRI images are measured repeatedly across three visits. To cover the sizes of the commonly used brain atlases, we set the number of nodes V = {100, 200}, and upon each set of nodes, brain functional connectivity is generated for each subject. For the genetic variant, we sample its value from {0, 1, 2} to represent homozygous recessive, homozygous dominant, and heterozygous genotypes. We further assume the genetically related functional modular structures contain $Q=5$ blocks with brain nodes randomly assigned to these blocks under an equal probability. The number of blocks chosen here follows the results of our data application in the following section, which is based on BIC and is under a similar scale as the number of canonical brain sub-networks [14]. For the signal sparsity level associated with each of the main effects, we consider both 50% and 90% zeros within $\{{\delta }_{j,qr}=0,1\le q\le r\le Q\}$ with $j=1,2,3$. For the non-zero main effect coefficients, we directly set them to 0.15 and generate the random effects ${\mathbf{b}}_{i,qr}\sim \text{MVN}\left(0,\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0.3\hfill \\ \hfill 0.3\hfill & \hfill 1\hfill \end{array}\right]\right)$ and set the random error variance ${\sigma }_{ϵ,qr}^2=1$. To generate the repeatedly measured connectivity matrices, following model (1), we consider both a low-noise scenario with ${\sigma }_{qr}^2=3$ and a high-noise one with ${\sigma }_{qr}^2=10$. In total, there are 16 simulated settings that vary by sample size, atlas size, sparsity level, and noise scenario, and we generate 100 Monte Carlo datasets for each setting.

To implement the proposed BLNR, following the hyper-prior settings detailed in Section 3.1, we set $\mu =3$ in the IW prior for ${\mathbf{D}}_{qr}$ and large variances ${\sigma }_x^2={\tau }_j=10,j=1,2,3$ with non-informative prior support. The block number $Q$ is searched from {3, 4, 5, 6} based on BIC with $Q$ set to 5 in the majority of the simulated datasets. Under random initials, MCMC is performed for 5,000 iterations, where the first 2,000 iterations are burn-in, with the convergence confirmed by trace plots and the GR method. For the competing approaches, given that there are no existing GWAS methods to accommodate a repeatedly measured network-variant phenotype, we directly extract the unique edges from the network by the upper diagonal elements in each connectivity matrix. Individual longitudinal connection is then treated as a phenotype and modeled using existing methods, including a simple linear regression model (LM), a linear mixed-effect model (LMM) implemented by R package lme4, and a linear mixed-effects kinship model (lmekin) implemented by R package coxme. The detailed formulas for each of the models can be found in the Supporting Information.

To evaluate the accuracy of both model estimation and network phenotypic feature selection, we consider performance metrics including the root mean predicted square error (RMSE) of ${\mathbf{A}}_{it}$ across different visits, and sensitivity (Sen), specificity (Spe) for identifying signal elements captured by the non-zero elements in ${\{{\beta }_{j,qr}\}}_{j=1}^3$ as well as the area under the curve (AUC) summarized from the marginal posterior likelihood of ${\{{\delta }_{j,qr}\}}_{j=1}^3$. For the competing methods, RMSE of ${\mathbf{A}}_{it}$ is summarized directly from regression analysis; signal elements are identified by p-values (< 0.05) which is used for the sensitivity and specificity calculation; and AUC is determined by the absolute value of the magnitude of the coefficient values. The computational cost to complete the posterior inference for BLNR is around 5 h for V=200 and N=500 settings under Yale High-Performance Computing (one CPU core, 8GB RAM). All the simulation results are summarized in Table 1.

TABLE 1 ∣.

Simulation results under different settings evaluated by root mean predicted square error (RMSE) of ${\mathbf{A}}_{it}$ across visits (RMSE), sensitivity (Sen), specificity (Spe) and area under the curve (AUC) for signal detection.

	Sparsity	Noise	Model	N = 500				N = 1000
				RMSE	Sen	Spe	AUC	RMSE	Sen	Spe	AUC
V = 100	50%	low	BLNR	3.11 (0.00)	0.76 (0.13)	0.83 (0.14)	0.79 (0.06)	3.11 (0.01)	0.94 (0.03)	1.00 (0.00)	0.98 (0.02)
			LM	16.83 (0.13)	0.44 (0.13)	0.77 (0.12)	0.51 (0.01)	16.84 (0.04)	0.44 (0.09)	0.86 (0.10)	0.52 (0.01)
			LMM	7.50 (0.01)	0.45 (0.13)	0.72 (0.12)	0.53 (0.01)	7.57 (0.01)	0.45 (0.10)	0.80 (0.09)	0.55 (0.02)
			lmekin	8.73 (0.02)	0.43 (0.09)	0.76 (0.10)	0.52 (0.01)	8.74 (0.02)	0.43 (0.11)	0.84 (0.08)	0.53 (0.01)
	50%	high	BLNR	10.04 (0.03)	0.83 (0.08)	0.81 (0.10)	0.80 (0.05)	10.04 (0.02)	0.93 (0.03)	1.00 (0.00)	0.97 (0.02)
			LM	107.55 (0.08)	0.42 (0.08)	0.60 (0.08)	0.50 (0.01)	107.73 (0.07)	0.41 (0.06)	0.64 (0.06)	0.51 (0.02)
			LMM	91.64 (0.10)	0.42 (0.09)	0.60 (0.09)	0.50 (0.01)	92.83 (0.11)	0.41 (0.10)	0.64 (0.10)	0.60 (0.02)
			lmekin	96.99 (0.09)	0.41 (0.11)	0.62 (0.11)	0.50 (0.00)	97.02 (0.07)	0.42 (0.11)	0.63 (0.11)	0.51 (0.00)
	90%	low	BLNR	3.11 (0.01)	0.97 (0.02)	0.64 (0.04)	0.83 (0.02)	3.11 (0.01)	1.00 (0.01)	1.00 (0.00)	0.99 (0.01)
			LM	16.79 (0.08)	0.86 (0.11)	0.31 (0.10)	0.51 (0.03)	16.84 (0.10)	0.88 (0.09)	0.38 (0.08)	0.52 (0.02)
			LMM	7.50 (0.20)	0.87 (0.12)	0.26 (0.10)	0.53 (0.03)	7.57 (0.02)	0.88 (0.08)	0.30 (0.09)	0.55 (0.03)
			lmekin	8.74 (0.02)	0.87 (0.06)	0.24 (0.14)	0.52 (0.02)	8.74 (0.02)	0.87 (0.06)	0.37 (0.18)	0.54 (0.03)
	90%	high	BLNR	10.04 (0.03)	0.97 (0.02)	0.56 (0.04)	0.80 (0.04)	10.03 (0.03)	0.99 (0.01)	1.00 (0.00)	0.98 (0.01)
			LM	107.57 (0.11)	0.86 (0.06)	0.15 (0.15)	0.50 (0.02)	107.72 (0.12)	0.86 (0.05)	0.17 (0.17)	0.51 (0.03)
			LMM	91.63 (0.09)	0.86 (0.08)	0.15 (0.08)	0.50 (0.04)	92.82 (0.10)	0.86 (0.09)	0.17 (0.07)	0.51 (0.02)
			lmekin	96.98 (0.09)	0.87 (0.06)	0.14 (0.07)	0.50 (0.00)	97.02 (0.06)	0.87 (0.05)	0.16 (0.06)	0.51 (0.01)
V = 200	50%	low	BLNR	3.11 (0.01)	0.75 (0.11)	0.84 (0.11)	0.80 (0.04)	3.11 (0.01)	0.93 (0.03)	1.00 (0.00)	0.97 (0.02)
			LM	16.79 (0.11)	0.41 (0.10)	0.78 (0.12)	0.51 (0.03)	16.84 (0.09)	0.44 (0.12)	0.85 (0.08)	0.52 (0.02)
			LMM	7.50 (0.01)	0.42 (0.11)	0.73 (0.12)	0.53 (0.02)	7.57 (0.10)	0.46 (0.10)	0.80 (0.12)	0.55 (0.02)
			lmekin	8.94 (0.02)	0.41 (0.11)	0.75 (0.11)	0.52 (0.01)	8.74 (0.02)	0.45 (0.10)	0.83 (0.02)	0.56 (0.01)
	50%	high	BLNR	10.03 (0.02)	0.73 (0.08)	0.82 (0.09)	0.77 (0.02)	10.03 (0.02)	0.94 (0.02)	1.00 (0.00)	0.98 (0.02)
			LM	107.56 (0.11)	0.39 (0.08)	0.63 (0.07)	0.50 (0.02)	107.71 (0.12)	0.41 (0.10)	0.64 (0.09)	0.51 (0.01)
			LMM	91.61 (0.09)	0.39 (0.07)	0.63 (0.08)	0.50 (0.02)	92.83 (0.09)	0.41 (0.08)	0.64 (0.08)	0.51 (0.02)
			lmekin	96.87 (0.07)	0.38 (0.09)	0.64 (0.09)	0.50 (0.00)	97.20 (0.06)	0.41 (0.11)	0.63 (0.11)	0.51 (0.00)
	90%	low	BLNR	3.11 (0.00)	0.97 (0.02)	0.62 (0.03)	0.82 (0.02)	3.10 (0.01)	1.00 (0.02)	1.00 (0.00)	0.99 (0.01)
			LM	16.85 (0.11)	0.86 (0.11)	0.32 (0.10)	0.52 (0.03)	16.85 (0.10)	0.86 (0.08)	0.44 (0.11)	0.53 (0.03)
			LMM	7.49 (0.02)	0.86 (0.07)	0.26 (0.07)	0.53 (0.02)	7.57 (0.03)	0.87 (0.07)	0.35 (0.08)	0.56 (0.02)
			lmekin	8.73 (0.02)	0.87 (0.05)	0.25 (0.14)	0.52 (0.02)	8.78 (0.02)	0.87 (0.05)	0.42 (0.16)	0.54 (0.02)
	90%	high	BLNR	10.03 (0.07)	0.97 (0.02)	0.56 (0.08)	0.80 (0.02)	10.03 (0.07)	0.99 (0.02)	1.00 (0.00)	0.98 (0.02)
			LM	107.57 (0.12)	0.87 (0.10)	0.14 (0.13)	0.50 (0.02)	107.69 (0.10)	0.86 (0.09)	0.17 (0.13)	0.51 (0.01)
			LMM	91.62 (0.09)	0.87 (0.09)	0.14 (0.04)	0.50 (0.02)	92.83 (0.08)	0.86 (0.09)	0.17 (0.07)	0.51 (0.01)
			lmekin	96.89 (0.07)	0.88 (0.06)	0.14 (0.07)	0.50 (0.00)	97.21 (0.08)	0.88 (0.06)	0.16 (0.06)	0.51 (0.00)

Note: The Monte Carlo standard deviation is included in the parentheses.

Based on the results in Table 1, we conclude that under all simulated longitudinal brain connectivity genetics settings, our proposed BLNR model consistently outperforms competing methods in uncovering genetic effects and identifying associated phenotypic network configurations. Specifically, BLNR demonstrates significantly lower RMSE, indicating high estimation accuracy, along with higher sensitivity, specificity, and AUC, which reflect its excellent imaging genetics signal detection performance. This superior performance is anticipated as our method integrates the topological information of the functional connectivity phenotype and explicitly accounts for noise components within network features with little association with the genetic variant. Among the competing methods, LM performs the worst due to its neglect of correlation among repeated measurements. LMM and lmekin display similar performance, serving as the current standard options for longitudinal univariate outcome analyses and longitudinal GWAS. Across different settings, we observe a consistent improvement in feature selection accuracy for all methods as sample size increases and noise level decreases. RMSE, a measure of parameter estimation performance, is particularly sensitive to noise levels, showing a dramatic deterioration for competing methods under higher noise levels. However, our method, despite some increase in RMSE, maintains reasonable results, demonstrating robustness to noise. Lastly, the signal sparsity level has a minor impact on overall model performance. Although specificity for all methods decreases with the inclusion of more noise features, the overall selection accuracy, as captured by AUC, remains stable across different sparsity levels.

To further evaluate the feasibility of our model in a higher-dimensional setting, we conducted additional simulations with 400 nodes under N = 500 and N = 1 000, while keeping all other data generation settings unchanged. As shown in Table S1 in the Supporting Information, the BLNR framework continues to demonstrate satisfactory performance even with this increased feature dimension. It consistently outperforms competing methods, achieving lower RMSE and higher sensitivity, specificity, and AUC. Additionally, to assess the reliability and robustness of BIC in selecting the optimal number of blocks $(Q)$, we performed sensitivity analyses using an expanded grid search under simulated settings with 200 nodes and 500 individuals. We also reanalyzed the data using DIC as the selection criterion. As detailed in the Supporting Information, the results under BIC and DIC are highly consistent, with BIC performing slightly better, successfully identifying the true $Q$ in 75%–80% of cases. These findings confirm the robustness of using BIC to select the optimal tuning parameter in our analyses.

5 ∣. Real Data Application

We finally apply the proposed BLNR to the motivated imaging genetics dataset collected from the landmark Adolescent Brain Cognitive Development (ABCD) study. The ABCD study launched in 2015 is the largest prospective study on brain development and child health in the United States (https://abcdstudy.org/). This ongoing study collects a variety of psychosocial, neurocognitive, genetics, and neuroimaging data from nearly 12,000 preadolescents across the United States with a goal to investigate how brain structural and functional changes impact cognitive behaviors from childhood till early adulthood [18]. In this analysis, we work with the recent 4.0 release of the ABCD data that includes longitudinal fMRI data conducted at baseline and the first follow-up two years afterward. For the resting-state fMRI data, participants were scanned across different sites using harmonized imaging protocols to remove the site effects. Each child completed 4–5 five-minute resting-state functional imaging sessions to ensure at least eight minutes of relatively low-motion data. After imaging acquisition, standard preprocessing procedures—including gradient-nonlinearity distortions, inhomogeneity correction for structural data, gradient-nonlinearity distortion correction, motion correction, and field map correction—were carried out by the ABCD Data Analysis and Informatics Core. Details of the imaging acquisition protocol and processing pipeline are detailed in Casey et al. [18] and Hagler et al. [19]. The subsequent processing is conducted using BioImage Suite [20], adhering to the standard procedures outlined in previous studies [21-23]. Initially, all fMRI images are realigned to correct for motion and registered to MNI space. They are then anatomically parcellated using a 268-node whole-brain atlas, which includes the cortex, subcortex, and cerebellum [24]. Covariates of no interest including linear, quadratic, and cubic drifts, 24 motion parameters [25], mean cerebrospinal fluid signal, mean white matter signal and overall global signal are regressed from the data. Finally, the data are temporally smoothed with a Gaussian filter with $\sigma =1.95$. To construct 268 × 268 functional connectivity for each participant at each visit, Pearson’s correlation between the regional time course of each pair of nodes is computed and scaled to be Normally distributed by a Fisher’s Z transformation. In our analysis, the eligible subjects are those who have collected qualified scans at both visits. Finally, 1881 subjects were included in our analysis.

For the genotyping data, the saliva and blood samples are collected as part of the biospecimen collection initiative for genotyping [26]. After downloading the raw genetics data, we perform standard data quality by excluding individuals with more than 10% missing genotyping and SNPs that have more than 10% missing genotyping rate, failed the Hardy-Weinberg test at the threshold of 1 × 10⁻⁵, have minor allele frequency (MAF) ≥ 0.05, or are in approximate linkage equilibrium with each other with a pairwise threshold of 0.2. This results in a total of 153,461 SNPs. To mitigate the computational cost, we perform a pre-screening GWAS step by taking the average value of functional connections for each subject at each visit as the phenotype, and conduct an association analysis for each SNP after adjusting for age, gender, and the top 10 components from genetic principal components. Eventually, there are 11 SNPs associated with the phenotype significantly with a p-value less than 1 × 10⁻²³. Those SNPs are included in our formal analyses given their potential overall correspondence with functional connectivity at either visit.

The goal is to identify the association between genetic markers and temporally changed brain functional connectivity. Following the implementation details as the simulation studies, we apply the proposed BLNR to our ABCD longitudinal imaging genetic dataset. Eventually, we identify significant genetic associations linked to SNP-related main and interaction terms from 9 SNPs. After performing a functional annotation, these SNPs belong to the following unique gene variants–FMNL2, LINC02098, MSR1, MTUS2, and PLB1, and are located at seven distinct chromosomes. Among them, FMNL2 as a member of the formin family, has been implicated in the regulation of cytoskeletal dynamics particularly in the developing brain, which is crucial for synaptic plasticity and neuronal connectivity [27, 28]; MSR1 has been associated with neuroinflammatory responses, which are known to affect neuronal connectivity and brain function [29]; and MTUS2 has a role to orchestrate cellular dynamics and consistently been detected in neurodegenerative studies [30].

For the constructed brain network modular structures corresponding to each of the association analyses, the brain nodes are allocated into five sub-networks based on the optimal BIC. Notably, for each SNP, the constructed modular allocations within brain functional nodes are different yet largely overlapped, reflecting the dissected brain sub-networks with strong genetic impact. After integrating the genetics main effect and temporal interaction term, in Figure 1, we display the sub-networks that are strongly associated with the genotype under each visit. As we can see, some of the genetic variants including rs59675461 and rs72701041 display a stationary impact on brain functional networks, while the rest genetic risk biomarkers exhibit varying brain network associations at different visits. To further investigate the network anatomy of the genetically associated brain sub-networks, we compare our constructed sub-networks with the canonical neural networks and the macroscale brain regions. This comparison includes the number of nodes within each sub-network, the most overlapping canonical networks and macroscale regions, and the number of shared nodes as shown in Figure 2. From the result, we observe that our constructed sub-networks are relatively consistent across SNPs with most overlapped canonical networks including the default mode and the somato-motor and the most overlapped macroscale regions including cerebellum, motor strip, occipital, prefrontal, and temporal regions.

FIGURE 1 ∣.

Significant sub-networks identified by genetic effects across two visits (left four panel for visit 1 and right four panel for visit 2).

FIGURE 2 ∣.

Circular plot of a number of brain nodes within sub-networks with the most overlapped macroscale regions and canonical networks.

We finally investigate the associated brain subnetwork components for each of the identified genetic associations at the brain node-level. Visualization of genetically associated brain nodes and connections are displayed in Figure 3, where the left three panels correspond to the results based on visit 1, and the right three panels display the results corresponding to visit 2. The nodes are highlighted if they are in the blocks with corresponding absolute values of the effect size greater than 0.01. To ensure clear visualization, edges with corresponding absolute values of average brain functional connections greater than 0.07 are highlighted. We observed temporal consistency in the distribution of significant brain nodes across two visits for some of the SNPs including rs59675461, rs57196961, and rs72701041 while the variation in connectivity strength underscores the differential impact of genetic variants on brain networks. Additionally, the estimated genetic effect on functional connectivity for each biomarker across brain nodes at each visit is shown in the pairs of heatmaps (left panel for visit 1 and right panel for visit 2) in Figure S1 in the Supporting Information, demonstrating its temporal dynamics. The color gradient reflects the strength and direction of the genetic effect, with red indicating positive effects, blue indicating negative effects, and white representing near-zero effects. The general pattern for the identified phenotypic sub-networks tends to be consistent over time with a small number of sub-network elements diminishing to or aggregating from null while the effect strength varies between different SNPs. Lastly, Figure S2 in the Supporting Information shows the location and distribution of significant changes in functional connectivity between brain regions color-coded by anatomical location across two visits (left panel for visit 1 and right panel for visit 2) with the connection colors indicating positive (red) or negative (cyan) genetic effects. We can see that the genetic association pattern of brain functional connectivity exhibits longitudinal changes with complex connections across different lobes. Specifically, there is a change in sign for the estimated effects for some of the genetic variants including rs7582333, rs5967461, rs57196961, and rs72701041 between the two visits. The genetic effects on connectivity are concentrated in specific brain lobes including the prefrontal, temporal, and limbic regions across multiple SNPs which suggest these areas may be more susceptible to longitudinal genetic influences and may be key regions for understanding changes in genetic contributions to brain function and behavior development. Our finding aligns closely with previous literature, which has demonstrated that genetic variants are associated with longitudinal transitions in brain structure and functions across the lifespan [8].

FIGURE 3 ∣.

Brain functional connectivity associated with significant brain nodes (left three panels associated with visit 1 and right three panels associated with visit 2). Nodes are selected based on genetic effects and edges highlighted with average FC greater than 0.07 (edge size corresponds to connectivity strength).

6 ∣. Discussion

In this paper, we present an innovative Bayesian network-response mixed-effect model to estimate genetic association with longitudinally measured brain functional connectivity. Our model develops an integrative approach to jointly dissect the phenotypic network modular structures while uncovering the associated longitudinal genetic effects. To accommodate the biological architecture within functional connectivity, we incorporate a stochastic block model with unknown community allocations alongside a mixed-effect model to characterize the genetic contribution and its interaction with time to the longitudinal brain network developmental pattern. By implementing an MCMC algorithm, the model can identify significant genetic associations that could vary temporally with phenotypic brain network configurations. Extensive simulations demonstrate the superiority of the proposed method compared with existing approaches. By applying our approach to the latest ABCD cohort with repeated brain fMRI scans, we obtain interesting results on the genetic underpinnings of brain functional connectivity over time during this brain developmental stage.

The modular structures detected by our model in the real data analysis represent functional subdivisions of the brain that appear to be coherently influenced by genetic variation during neurodevelopment, reinforcing the potential link between genetic factors and neuronal connectivity. While our analysis establishes strong associations between specific SNPs and brain network patterns, it is important to note that the identified SNPs cannot yet be considered causal without further rigorous validation. However, they serve as strong candidates for future experimental or observational studies to confirm causality. In addition, our current data application demonstrates the effectiveness of our approach using two visits of brain imaging measurements from the ABCD study. With the recent release of ABCD 5.1 (May 2024), which includes imaging measurements at a third visit, we anticipate initiating the preprocessing of this new imaging data to construct functional connectivity. This addition will allow us to refine our analyses and validate our findings by incorporating data from the third visit. Including more visits will enable us to better characterize the dynamics of brain functional networks and capture critical neurodevelopmental stages.

Beyond studying genetic associations of brain functional connectivity, our proposed method offers a general regression framework for mixed-effect models with network-variate outcomes. This approach is readily applicable to other epidemiological and social science studies where data collection processes increasingly involve repeatedly measured graphical or network-related outcomes. Additionally, our method can be extended to consider potential non-linear relationships between covariates and the outcome variable through techniques such as kernel estimation or neural networks.

In our current modeling framework, similar to the traditional GWAS, we associate each genetic variant individually with the network phenotype. A potential future extension is to jointly consider high-dimensional SNPs as covariates using a mixed-effects polygenic regression while also accommodating higher-order terms, including inter-SNP and SNP-time interactions. Although a polygenic model could effectively incorporate the joint predictive effects from genome-wide SNPs, its computational demands grow exponentially with the number of SNPs, making whole-genome analysis impractical, especially given our complex network-variate phenotype. Furthermore, jointly modeling multiple SNPs introduces interpretability challenges, as disentangling individual SNP effects becomes increasingly difficult in the presence of related samples and network-level outcomes, potentially limiting the clarity and utility of the results. Similarly, increasing the data dimension with the number of nodes larger than 1 000 or more, while theoretically feasible, presents practical challenges due to potential poor mixing and prohibitive computation in uncovering the stochastic block allocations. In these cases, MCMC methods may not be the most efficient for posterior inference. Instead, optimization-based algorithms such as expectation-maximization or variational inference might be more suitable. Additionally, label switching has always been an issue for MCMC-based Bayesian clustering algorithms. In our model, following the strategy adopted by most previous studies [31], we put ordering restrictions on the SBM mean parameters $\{{\mathbf{m}}_{i,qr}\}$ to avoid any invariance during each MCMC iteration. Alternatively, one could also restrict inference for posterior distributions to ensure identifiability [32]. Finally, this study employs the BIC criterion to select the optimal number of blocks, which achieves satisfactory performance in numerical studies. As an alternative, more dynamical approaches could be explored to determine the number of blocks during learning [33], which may induce a more adaptive block selection in more complex analytical scenarios.

Supplementary Material

Additional supporting information can be found online in the Supporting Information section.

Data Availability Statement

Data used in the preparation of this article were obtained from the Adolescent Brain Cognitive Development (ABCD) Study (https://abcdstudy.org), held in the NIMH Data Archive (NDA).