11affiliationtext: Department of Biostatistics, Yale School of Public
Health22affiliationtext: Department of Bioengineering, University of
Pennsylvania33affiliationtext: Department of Biostatistics, Epidemiology and
Informatics, University of Pennsylvania

# Establishing group-level brain structural connectivity incorporating
anatomical knowledge under latent space modeling

Selena Wang111The correspondence should be addressed to Selena Wang. Yiting
Wang  Frederick H. Xu  Li Shen  Yize Zhao  for the Alzheimer’s Disease
Neuroimaging Initiative Data used in preparation of this article were obtained
from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database
(adni.loni.ucla.edu). As such, the investigators within the ADNI contributed
to the design and implementation of ADNI and/or provided data, but did not
participate in analysis or writing of this report. A complete listing of ADNI
investigators can be found at: http://adni.loni.usc.edu/wp-
content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf.

###### Abstract

Brain structural connectivity, capturing the white matter fiber tracts among
brain regions inferred by diffusion MRI (dMRI), provides a unique
characterization of brain anatomical organization. One fundamental question to
address with structural connectivity is how to properly summarize and perform
statistical inference for a group-level connectivity architecture, for
instance, under different sex groups, or disease cohorts. Existing analyses
commonly summarize group-level brain connectivity by a simple entry-wise
sample mean or median across individual brain connectivity matrices. However,
such a heuristic approach fully ignores the associations among structural
connections and the topological properties of brain networks. In this project,
we propose a latent space-based generative network model to estimate group-
level brain connectivity. Within our modeling framework, we incorporate the
anatomical information of brain regions as the attributes of nodes to enhance
the plausibility of our estimation and improve biological interpretation. We
name our method the attributes-informed brain connectivity (ABC) model, which
compared with existing group-level connectivity estimations, (1) offers an
interpretable latent space representation of the group-level connectivity, (2)
incorporates the anatomical knowledge of nodes and tests its co-varying
relationship with connectivity and (3) quantifies the uncertainty and
evaluates the likelihood of the estimated group-level effects against chance.
We devise a novel Bayesian MCMC algorithm to estimate the model. We evaluate
the performance of our model through extensive simulations. By applying the
ABC model to study brain structural connectivity stratified by sex among
Alzheimer’s Disease (AD) subjects and healthy controls incorporating the
anatomical attributes (volume, thickness and area) on nodes, our method shows
superior predictive power on out-of-sample structural connectivity and
identifies meaningful sex-specific network neuromarkers for AD.

##  1 Introduction

Brain structural connectivity captures the white matter fiber tracts among
brain regions inferred via diffusion MRI (dMRI) and provides a unique
characterization of brain anatomical network organization. Statistical
analysis at the group and population level of structural connectivity
represents a fundamental component to dissect brain structural alterations and
uncover significant neural substrates for cognitive and disease processes. How
to properly summarize and perform statistical inference for a group-level
connectivity architecture, for instance under different sex groups or disease
cohorts, are important statistical issues that warrant careful considerations.

Most of the existing methods estimating group-level connectivity are in light
of an underlying assumption that all the connections across the brain are
independent from each other. As a result, group-level brain connectivity is
constructed based on the sample mean or median across the group with respect
to each individual edge [1, 54, 52], ignoring the structural dependence and
higher-order topological properties within brain networks [50]. Alternatively,
some methods seek to select the “best” individual network based on certain
selection criteria [41, 32]. However, the processes neglect the heterogeneity
among the individual connectivity, and the resulting group-level brain
connectivity likely includes features not present in the majority of the
subjects. There is still a lack of methods that can properly estimate group-
level connectivity accommodating the topological structure and are capable of
quantification of uncertainty in order to evaluate the likelihood of the
estimates against chance.

In addition, converging evidence indicates that the anatomical attributes
including cortical volume, surface area and cortical thickness of brain
regions are associated with the white matter fiber tracks among them [[,
e.g.,]]feng2021altered,cai2021joint,hodel2020structural,yee2018structural.
However, existing approaches do not consider or offer an intuitive framework
to incorporate anatomical knowledge of the nodes when modeling their
corresponding structural connections. This is particularly crucial when
studying neurodegenerative diseases. For example, [12] found that the surface
area of the brain regions and their connectivity components change in a
coherent way for Alzheimer’s Disease (AD) patients. Uncovering the interplay
between regional and network-level brain structural alternations could provide
an unprecedented opportunity to enhance our understanding on the disease
etiology and neurodegeneration onset. There is an urgent need to integrate
regional and connectivity measures together and investigate their
relationships and how they change under different subpopulations.

With these considerations and challenges, we propose an attributes-informed
brain connectivity (ABC) modeling framework that estimates the group-level
connectivity among the brain regions while incorporating the anatomical
attributes of the brain regions. Different from the existing approaches, we
borrow inspirations from the generative latent space network models that
emphasize modeling the dynamic and complex higher-order topological properties
of the individual networks [30]. Attention has been given to the use of
generative network models in structural and functional analyses [9], and the
group-level brain connectivity has been estimated using exponential random
graph models [36] and the latent distance models [61, 2]. Different from these
existing methods, we propose a joint modeling framework that allows for the
incorporation and investigation of the anatomical attributes of the brain
regions when estimating group-level connectivity, and in the proposed ABC
model, the two modalities regulate and inform each other in the estimation
process. By combining the connectivity and the anatomical attributes into one
analysis, we leverage the strengths of each structural knowledge component to
investigate the connectivity changes. With the added anatomical knowledge, we
expect the proposed method to show superior estimation with better precision
and interpretability.

The proposed ABC model contributes to the current literature in multiple ways.
First, we offer an interpretable latent space representation of the structural
connectivity which accommodates the topological structure. The latent space
shows a clear differentiation of the left and right hemispheres when fitted to
data in line with the intrinsic structural anatomy of the brain. Regions of
the cingulate cortex wrapping around the corpus callosum are found in the
direction separate from the left and right hemispheres reflecting the unique
feature of fiber connectivity around the cingulate cortex. Second, the ABC
model allows us to incorporate the related anatomical knowledge of the brain
regions while estimating group-level brain connectivity. Such a process has
been shown in our numerical studies to improve the estimation of group-level
connectivity, resulting in superior predictive power and interpretability. The
ABC model also allows us to uncover the co-varying relationship between
regional anatomical knowledge and the corresponding structural connectivity.
Furthermore, the ABC model allows us to quantify the uncertainty and evaluate
the likelihood of the estimated group-level effects against chance. Existing
heuristic approaches often apply the unrealistic assumption that the brain
connectivity edges are static and fixed observations. In comparison, the ABC
model, as a generative statistical network model, assumes the connectivity-
induced elements to be realizations of random processes with noise and
proposes a data generation process for the observed ones. This allows us to
assess the uncertainty of the estimated group-level brain connectivity and
investigate the group-level differences against chance without incurring a
large multiple-comparison burden. Finally, we apply the ABC model to study
sex-specific AD neuromarkers based on a joint modeling of structural
connectivity and structural imaging traits from ROIs and identify
interpretable brain regional and network underpinnings related to AD under
different sexes.

The remaining paper is organized as follows. Section 2 describes the data and
the proposed methodology. Section 3 details the optimization routine for
estimating the group-level connectivity. In section 4, we assess whether the
proposed method compares well against the existing methods based on the
predictive power and whether/how the predictive power improves with the added
anatomical knowledge. Section 5 shows the application of our method to
Alzheimer’s Disease Neuroimaging Initiative (ADNI) data. We conclude with a
brief discussion in Section 6.

##  2 Data and Methods

###  2.1 ADNI data preprocessing

Data used in preparation of this article were obtained from the Alzheimer’s
Disease Neuroimaging Initiative (ADNI) database [59, 60]. The ADNI was
launched in 2003 as a public-private partnership with the primary goal of
testing whether serial magnetic resonance imaging, positron emission
tomography, other biological markers and clinical and neuropsychological
assessment can be combined to measure the progression of mild cognitive
impairment and early AD. For up-to-date information, see www.adni-info.org.

Structural connectivity: The ADNI T1-weighted structural MRI (sMRI) and
Diffusion Tensor Imaging (DTI) were obtained from the ADNI Grand Opportunities
and ADNI Phase 2 (ADNI-GO/2) databases. We downloaded the imaging data for 173
subjects along with their demographic information. There are 99 males and 74
females in the data with an average age of 72.9 $\pm$ 7.39 years. No
significant difference in age was detected across different groups (ANOVA: $P$
= 0.805, $F$ = 0.217).

The DTI data were first processed under standard steps including denoising,
motion-correction and distortion-correction using an overcomplete local
principal components analysis [39]. Probabilistic white matter fiber
tractography was performed using a streamline tractography algorithm called
fiber assignment by continuous tracking (FACT)[43]. SMRI scans were registered
to the lower resolution b0 volume of the DTI data using the FLIRT toolbox in
the FMRIB Software Library [31], and the cortical ROIs were defined based on
the Lausanne 2008 parcellation with 68 regions of interest in the native
FreeSurfer space [13]. An average of 3.2 million fibers were extracted between
each ROI pair. The number of the fibers (NOF) connecting each pair of ROIs was
obtained as well as the regions’ surface area (SA). The fiber density-based
structural connectivity was calculated by dividing NOF between two ROIs with
their average surface areas [63, 62]. The structural brain networks were
constructed as the fiber density of tracts connecting pairs of ROIs.

Anatomical attributes: The regional anatomical measurements were acquired from
the University of California San Francisco FreeSurfer 5.1 Cross-Sectional
Study database. The Cross-Sectional study uses ADNIGO/2 T1.5 scans that have
been motion-corrected, B1-corrected and N3-inhomogeneity corrected by
MayoClinic [27]. Each scan was segmented in the native FreeSurfer space
allowing for comparison between subjects at each time point [21]. The cortical
ROI attribute data were obtained from the screening sessions. The cortical
attributes include the cortical volume, surface area, average thickness and
thickness standard deviation. The highly folded nature of the cortex makes
quantifying cortical challenging [22]. FreeSurfer methods address this
difficulty and provide the thickness as an averaged measure across a
reconstructed cortical surface [23], along with its standard deviation both of
which are utilized in our study.

###  2.2 The ABC model

In this study, we have a set of brain structural connectivity matrices
$\\{\boldsymbol{X}_{1},\boldsymbol{X}_{2},...,\boldsymbol{X}_{N}\\}$, each
with $V\times V$ dimensions, where $V$ is the number of nodes or brain
regions, and N is the sample size. For subject $i$, each element $x_{u,v,i}$
in the connectivity matrix measures the strength of the anatomical connection
between brain regions $u$ and $v$, $u<v,u,v=1,2,..,V$; and we have
$x_{u,v,i}=x_{v,u,i}$ following the symmetry of connectivity matrix and
$x_{u,u,i}=0$. Simultaneously, we also measure a series of anatomical
information e.g. volume, area and thickness upon each of the nodes for each
subject and summarize them by a set of anatomical attribute matrices
$\\{\boldsymbol{Y}_{1},\boldsymbol{Y}_{2},...,\boldsymbol{Y}_{N}\\}$, each
with $V\times P$ dimensions, where $P$ is the number of anatomical metrics, or
attributes. We assume the structural connections on the edges interact with
the anatomical structure of the nodes. Given these two data components, the
likelihood of the ABC model can be written as:

$\displaystyle
p(\boldsymbol{X}_{1},\boldsymbol{X}_{2},...,\boldsymbol{X}_{N},\boldsymbol{Y}_{1},\boldsymbol{Y}_{2},...,\boldsymbol{Y}_{N}|\boldsymbol{Z},a_{1},a_{2},...,a_{N},b_{1},b_{2},...,b_{N},\sigma^{2},\tau^{2},\boldsymbol{\Theta},\boldsymbol{\Sigma})$
$\displaystyle=\prod_{i=1}^{N}p(\boldsymbol{X}_{i},\boldsymbol{Y}_{i}|\boldsymbol{Z},a_{i},b_{i},\sigma^{2},\tau^{2},\boldsymbol{\Theta},\boldsymbol{\Sigma}),$
(2.1)

where we assume that the individual brain connectivity and attribute matrices
are generated from a set of shared latent variables $\boldsymbol{Z}$ and
$\boldsymbol{\Theta}$. To characterize the latent topological structure as
well as link, we introduce latent space modeling for the connectivity
component, the attribute component and the joint component linking the
previous two components.

For subject $i$, the connectivity between nodes $u$ and $v$ is modeled by

$\displaystyle
x_{u,v,i}=\boldsymbol{w}_{i}^{T}\boldsymbol{\beta}+a_{i}+\boldsymbol{z}_{u}^{T}\boldsymbol{z}_{v}+e_{u,v,i},\qquad
e_{u,v,i}\overset{iid}{\sim}N(0,\sigma^{2}),$ (2.2)

where $\boldsymbol{z}_{u}$ is the $K$-dimensional,
$\boldsymbol{z}_{u}\in\mathbb{R}^{K}$, $K\ll V$ vector containing the latent
variable values for node $u$; $a_{i}$ is the fixed intercept for individual
$i$ modeling the overall connection, and for identification purposes, the sum
of the intercepts across subjects is constrained to be zero; $e_{u,v,i}$ is
the error term associated with the connectivity edge between nodes $u$ and $v$
for individual $i$; and $\sigma^{2}$ is the error variance. We also adjust for
$Q$ covariates denoted by $\boldsymbol{w}_{i}$ with the effects of the
covariates on the connectivity characterized by $\boldsymbol{\beta}$. In real
practice, we standardize each connection value across subjects making the
node-level additive effects obsolete. In matrix notations, we use
$\boldsymbol{Z}$ to denote the $V\times K$ matrix of the latent variable
values and $\boldsymbol{E}$ to denote the $V\times V$ matrix of errors. The
approximation of the posterior distributions for the unknown quantities is
facilitated by setting a
$\text{MVN}(\boldsymbol{\mu}_{\beta},\boldsymbol{\Sigma}_{\beta}),\boldsymbol{\mu}_{\beta}=(0,0,...,0,0)^{T},\boldsymbol{\Sigma}_{\beta}=\boldsymbol{I}_{Q}$
prior distribution for $\boldsymbol{\beta}$, noninformative priors including a
$\text{gamma}(1/2,1/2)$ distribution for $\sigma_{e}^{-2}$ and a
$\text{N}(0,1)$ distribution for $a_{i}$. We leave the description on the
prior distributions for the latent network variables afterwards.

For subject $i$, the attribute value of node $u$ on attribute $p$ is modeled
by

$\displaystyle
y_{u,p,i}=\boldsymbol{h}_{i}^{T}\boldsymbol{\gamma}+b_{i}+\theta_{u,p}+\epsilon_{u,p,i},\qquad\epsilon_{u,p,i}\overset{iid}{\sim}N(0,\tau^{2}),$
(2.3)

where $b_{i}$ is the fixed intercept for individual $i$ accounting the overall
difference across subjects, and for identification purposes, its sum across
subjects is zero. The latent variable $\theta_{u,p}$ represents the average
attribute value for node $u$ and attribute $p$ across subjects. In the vector
form, $\boldsymbol{\theta}_{u}$ is a $P$-dimensional vector that follows a
Multivariate Normal distribution defined in the joint component. The error
term is denoted by $\epsilon_{u,p,i}$; and $\tau^{2}$ is the error variance.
We adjust for $Q^{\prime}$ individual-specific covariates,
$\boldsymbol{h}_{i}$, which can be the same as or different from
$\boldsymbol{w}_{i}$ in model (2.2). The effects of the covariates on the
attributes of the brain regions are characterized by $\boldsymbol{\gamma}$.
Many elements of the attribute model mirror those of the connectivity model
except for the latent variables. Despite that the vector products of the
latent variables are often used to model network connectivity data in order to
capture the topological properties of the network [57], they are not
applicable for the attributes data. The attributes data usually do not have
the same dependence structures as the networks, and when there are only a few
attributes, latent dimensions are not warranted [58], as is the case for our
study. Approximation of the posterior distribution for the item parameters is
facilitated by setting a
$\text{MVN}(\boldsymbol{\mu}_{\gamma},\boldsymbol{\Sigma}_{\gamma}),\boldsymbol{\mu}_{\gamma}=(0,0,...,0,0)^{T},\boldsymbol{\Sigma}_{\gamma}=\boldsymbol{I}_{Q^{\prime}}$
prior distribution for $\boldsymbol{\gamma}$. Similarly, $\text{N}(0,1)$ and
$\text{gamma }(1/2,1/2)$ prior distributions are assigned for $b_{i}$ and
$\tau^{-2}_{i}$, respectively. The prior distributions for the latent
variables of the attributes are described in the following joint modeling.

Figure 1: ABC pipeline.

To further characterize the association between structural connections and
regional anatomical information, we integrate the latent components from the
two parts through a joint model written as:

$\displaystyle(\boldsymbol{z}_{u},\boldsymbol{\theta}_{u})^{T}\overset{iid}{\sim}\text{MVN}\left(\begin{pmatrix}\boldsymbol{0}_{K}\\\
\boldsymbol{0}_{N}\end{pmatrix},\boldsymbol{\Sigma}\right),\qquad\boldsymbol{\Sigma}=\begin{pmatrix}\boldsymbol{\Lambda}_{z}&\boldsymbol{\Lambda}_{zy}^{T}\\\
\boldsymbol{\Lambda}_{zy}&\boldsymbol{\Lambda}_{y}\end{pmatrix},$ (2.4)

where $\boldsymbol{\Lambda}_{z}$ is the $K\times K$ covariance matrix for the
connectivity, and $\boldsymbol{\Lambda}_{y}$ is the $P\times P$ covariance
matrix for the attributes. The off-diagonal $P\times K$ matrix
$\boldsymbol{\Lambda}_{zy}$ captures the association between the connectivity
and the node attributes, where each of their entries describes the
relationship between a latent connectivity variable and a latent attribute
variable. When there are non-zero elements in the $\boldsymbol{\Lambda}_{zy}$
matrix, the connectivity and node attribute estimations regulate and inform
each other, providing a more informative and plausible result. We assign a
$\text{Wishart}(\boldsymbol{I}_{K+P},K+P+2)$ prior distribution for
$\boldsymbol{\Sigma}$, facilitating nonzero posterior correlation between
brain structural connectivity and node anatomical structures.

To summarize, we demonstrate the ABC framework in Figure 1. The ABC model
takes both brain connectivity and regional attributes as the inputs. Within
the model, through shared group-level latent variables, we induce shared brain
connectivity components across subjects. As illustrated by the existing
precedents in network literature [25, 58], such a approach using shared group-
level latent variables is a succinct way to synthesize information across
subjects and to improve the estimation and prediction of the model. To link
brain connectivity and regional attributes, we propose a shared distribution
of the latent variables and allow their relationship to be represented in an
intuitive way through covariance parameters. The benefit of this approach is
that the relationship is more realistically represented as co-varying, where
both the regional attributes and the brain connectivity influence each other
instead of only allowing one to influence the other. Current literature
supports this representation with studies showing that structural connectivity
explains regional attributes [64], and vice versa [12, 28, 64]. Using the ABC
model, we produce three types of outputs: (1) the attributes-informed group-
level connectivity and its latent space representation, (2) the connectivity
difference between subject groups and (3) the dependence between the brain
connectivity and the attributes. These outputs will be illustrated with the
application.

##  3 Bayesian Inference

In this section, we propose a Bayesian posterior inference algorithm to
estimate the proposed joint model using Markov chain Monte Carlo (MCMC)
methods. With sufficient iterations, we obtain stable Markov chains to
approximate various quantities of the targeted posterior distributions. With
random initial values on the unknown parameters, we conduct posterior
computation by iterating the following steps. We also briefly summarize the
procedure in Algorithm 1.

  * •

For connectivity parameters,

    * –

simulate $\boldsymbol{\beta},\boldsymbol{a}$ from their full conditional
distributions.

    * –

simulate $\sigma^{2}$ given $\boldsymbol{\beta},\boldsymbol{a}$, $\tau^{2}$,
$\boldsymbol{\gamma},\boldsymbol{b}$, $\boldsymbol{Z}$, $\boldsymbol{\Theta}$,
$\boldsymbol{\Sigma}$, $\boldsymbol{X},\boldsymbol{Y}$.

  * •

For attribute parameters,

    * –

simulate $\boldsymbol{\gamma},\boldsymbol{b}$ from their full conditional
distributions.

    * –

simulate $\tau^{2}$ given $\boldsymbol{\beta},\boldsymbol{a}$, $\sigma^{2}$,
$\boldsymbol{\gamma},\boldsymbol{b}$, $\boldsymbol{Z}$, $\boldsymbol{\Theta}$,
$\boldsymbol{\Sigma}$, $\boldsymbol{X},\boldsymbol{Y}$.

  * •

For the joint latent variables and the covariance,

    * –

simulate $\\{\boldsymbol{Z}\text{ and }\boldsymbol{\Theta}\\}$ from their full
conditional distributions.

    * –

simulate $\boldsymbol{\Sigma}$ from its full conditional distribution.

###  3.1 Full Conditionals of Connectivity Parameters

To derive the full conditional distributions of $\boldsymbol{\beta}_{Q\times
1}$ and $\boldsymbol{a}_{N\times 1}$, we look at the connectivity component of
the joint model without the vector product. For each connectivity matrix for
each individual $\boldsymbol{X}_{i}$, we subtract
$\boldsymbol{Z}^{T}\boldsymbol{Z}$ and obtain
$\boldsymbol{R}_{i}=\boldsymbol{X}_{i}-\boldsymbol{Z}^{T}\boldsymbol{Z}$. We
extract the unique off-diagonal elements of the resulting symmetric matrix and
form a $V(V-1)/2\times 1$ vector. We stack $N$ such vectors and form a
$NV(V-1)/2\times 1$ vector, $\boldsymbol{r}$, $\boldsymbol{r}_{NV(V-1)/2\times
1}=\boldsymbol{G}\boldsymbol{a}+\boldsymbol{W}\boldsymbol{\beta}+\boldsymbol{e}$,
where $\boldsymbol{G}$ is the $NV(V-1)/2\times N$ matrix,
$\boldsymbol{G}=(\boldsymbol{I}_{N}\otimes\boldsymbol{1}_{V(V-1)/2}^{T})^{T}$,
with $\boldsymbol{I}_{N}$ being the $N\times N$ identity matrix and
$\boldsymbol{1}_{V(V-1)/2}$ being the $V(V-1)/2\times 1$ vector of $1$s, and
$\boldsymbol{W}$ is the $NV(V-1)/2\times Q$ appropriate design matrix. We can
further transform $\boldsymbol{r}$ such that the transformed error term is a
standard normal distribution using
$\boldsymbol{\tilde{r}}=c\boldsymbol{r},c=\sigma^{-1}$. Therefore, the model
can be written as
$\boldsymbol{\tilde{r}}=c\boldsymbol{G}\boldsymbol{a}+c\boldsymbol{W}\boldsymbol{\beta}+\boldsymbol{\tilde{e}}$,
where elements in $\boldsymbol{\tilde{e}}$ follow a standard normal
distribution.

The first step of the iteration is implemented by first simulating
$\boldsymbol{\beta}$ conditional on
$\\{\boldsymbol{\tilde{r}},\boldsymbol{W}\\}$ and then simulating
$\boldsymbol{a}$ conditional on
$\\{\boldsymbol{\tilde{r}},\boldsymbol{W},\boldsymbol{\beta}\\}$. The latter
distribution can be derived by letting
$\boldsymbol{m}=\boldsymbol{\tilde{r}}-c\boldsymbol{W}\boldsymbol{\beta}$, and
thus it can be shown that

$\displaystyle p(\boldsymbol{a}|\boldsymbol{m})$ $\displaystyle\propto
p(\boldsymbol{m}|\boldsymbol{a})p(\boldsymbol{a})$
$\displaystyle\propto\exp\left(-\frac{1}{2}\left(\boldsymbol{m}-c\boldsymbol{G}\boldsymbol{a}\right)^{T}\left(\boldsymbol{m}-c\boldsymbol{G}\boldsymbol{a}\right)\right)\exp\left(-\frac{1}{2}\boldsymbol{a}^{T}\Sigma_{0}^{-1}\boldsymbol{a}\right)$
$\displaystyle\propto\exp\left(-\frac{1}{2}\boldsymbol{a}^{T}\left(c^{2}\boldsymbol{G}^{T}\boldsymbol{G}+\Sigma^{-1}_{0}\right)\boldsymbol{a}+\boldsymbol{a}^{T}\left(c\boldsymbol{G}^{T}\boldsymbol{m}\right)\right).$
(3.1)

This is the kernel of a multivariate normal distribution with variance
$\text{Var}[\boldsymbol{a}|\boldsymbol{m}]=(c^{2}\boldsymbol{G}^{T}\boldsymbol{G}+\Sigma^{-1}_{0})^{-1}$
and mean
$\mathrm{E}[\boldsymbol{a}|\boldsymbol{m}]=c(c^{2}\boldsymbol{G}^{T}\boldsymbol{G}+\Sigma^{-1}_{0})^{-1}\boldsymbol{G}^{T}\boldsymbol{m}$.
The distribution of $\boldsymbol{\beta}$ conditional on
$\\{\boldsymbol{\tilde{r}},\boldsymbol{W}\\}$ is the product of
$p(\boldsymbol{\beta})$ and
$p(\boldsymbol{\tilde{r}}|\boldsymbol{\beta},\boldsymbol{W})$, and it can be
shown that

$\displaystyle p(\boldsymbol{\tilde{r}}|\boldsymbol{\beta},\boldsymbol{W})$
$\displaystyle\propto\int
p(\boldsymbol{\tilde{r}}|\boldsymbol{\beta},\boldsymbol{W},\boldsymbol{a})p(\boldsymbol{a})\;d\boldsymbol{a}$
$\displaystyle\propto\int\exp\left(-\frac{1}{2}\left(\boldsymbol{m}-c\boldsymbol{G}\boldsymbol{a}\right)^{T}\left(\boldsymbol{m}-c\boldsymbol{G}\boldsymbol{a}\right)\right)\exp\left(-\frac{1}{2}\boldsymbol{a}^{T}\Sigma_{0}^{-1}\boldsymbol{a}\right)\;d\boldsymbol{a}$
$\displaystyle\propto\exp\left(-\frac{1}{2}\mathrm{E}[\boldsymbol{a}|\boldsymbol{m}]^{T}\text{Var}[\boldsymbol{a}|\boldsymbol{m}]^{-1}\mathrm{E}[\boldsymbol{a}|\boldsymbol{m}]\right)$
$\displaystyle\propto\exp\left(-\frac{1}{2}\boldsymbol{\beta}^{T}\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{W}\right)\boldsymbol{\beta}+\boldsymbol{\beta}^{T}\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{r}\right)\right),$
(3.2)

where
$\boldsymbol{S}=\boldsymbol{G}(c^{2}\boldsymbol{G}^{T}\boldsymbol{G}+\Sigma^{-1}_{0})^{-1}\boldsymbol{G}^{T}$.
Given that the prior distribution of $\boldsymbol{\beta}$ is proportional to
$\exp\left(-\frac{1}{2}\boldsymbol{\beta}^{T}\boldsymbol{S}_{0}\boldsymbol{\beta}+\boldsymbol{\beta}^{T}\boldsymbol{S}_{0}\boldsymbol{\beta}_{0}\right).$
The distribution of $\boldsymbol{\beta}$ conditional on
$\\{\boldsymbol{\tilde{r}},\boldsymbol{W}\\}$ thus follows a multivariate
normal distribution with variance
$\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{W}+\boldsymbol{S}_{0}\right)^{-1}$
and mean
$\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{W}+\boldsymbol{S}_{0}\right)^{-1}\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{r}+\boldsymbol{S}_{0}\boldsymbol{\beta}_{0}\right)$.
The posterior for $\sigma^{-2}$ is
$\mbox{Gamma}(\frac{N*V*(V-1)/2+1}{2},1/2*(1+\sum_{i=1}^{N}\sum_{u=1}^{V}\sum_{u<v}^{V}e_{u,v,i}^{2}))$.

###  3.2 Full Conditionals of Attribute Parameters

The full conditional distributions of the attribute parameters can be
similarly obtained as those of the connectivity parameters. In particular, we
stack $N$ vectorized attribute matrices and form a $NVP\times 1$ vector,
$\boldsymbol{r}^{\prime}$, $\boldsymbol{r}^{\prime}_{NVP\times
1}=\boldsymbol{G^{\prime}}\boldsymbol{b}+\boldsymbol{H}\boldsymbol{\gamma}+\boldsymbol{\epsilon}$,
where $\boldsymbol{G^{\prime}}$ is the $NVP\times N$ matrix,
$\boldsymbol{G}=(\boldsymbol{I}_{N}\otimes\boldsymbol{1}_{VP}^{T})^{T}$, with
$\boldsymbol{I}_{N}$ being the $N\times N$ identity matrix and
$\boldsymbol{1}_{VP}$ being the $VP\times 1$ vector of $1$s, and
$\boldsymbol{H}$ is the $NVP\times Q^{\prime}$ appropriate design matrix. We
further transform $\boldsymbol{r^{\prime}}$ such that the transformed error
term is a standard normal distribution using
$\boldsymbol{\tilde{r^{\prime}}}=d\boldsymbol{r^{\prime}},d=\tau^{-1}$.
Therefore, the model can be written as
$\boldsymbol{\tilde{r^{\prime}}}=d\boldsymbol{G^{\prime}}\boldsymbol{b}+d\boldsymbol{H}\boldsymbol{\gamma}+\boldsymbol{\tilde{\epsilon}}$,
where elements in $\boldsymbol{\tilde{\epsilon}}$ follow a standard normal
distribution.

Let
$\boldsymbol{m^{\prime}}=\boldsymbol{\tilde{r^{\prime}}}-d\boldsymbol{H}\boldsymbol{\gamma}$.
Following similar derivation steps as before, we can show that the
distribution of $\boldsymbol{b}$ conditional on
$\\{\boldsymbol{\tilde{r^{\prime}}},\boldsymbol{H},\boldsymbol{\gamma}\\}$
follows a multivariate normal distribution with variance
$\text{Var}[\boldsymbol{b}|\boldsymbol{m^{\prime}}]=(d^{2}\boldsymbol{G^{\prime}}^{T}\boldsymbol{G^{\prime}}+\Sigma^{-1}_{0})^{-1}$
and mean
$\mathrm{E}[\boldsymbol{b}|\boldsymbol{m^{\prime}}]=d(d^{2}\boldsymbol{G^{\prime}}^{T}\boldsymbol{G^{\prime}}+\Sigma^{-1}_{0})^{-1}\boldsymbol{G^{\prime}}^{T}\boldsymbol{m^{\prime}}$.
It can be further concluded that the distribution of $\boldsymbol{\gamma}$
conditional on $\\{\boldsymbol{\tilde{r^{\prime}}},\boldsymbol{H}\\}$ also
follows a multivariate normal distribution with variance
$\left(d^{2}\boldsymbol{H}^{T}\boldsymbol{S^{\prime}}\boldsymbol{H}+\boldsymbol{S^{\prime}}_{0}\right)^{-1}$
and mean
$\left(d^{2}\boldsymbol{H}^{T}\boldsymbol{S^{\prime}}\boldsymbol{H}+\boldsymbol{S^{\prime}}_{0}\right)^{-1}\left(d^{2}\boldsymbol{H}^{T}\boldsymbol{S^{\prime}}\boldsymbol{r^{\prime}}+\boldsymbol{S^{\prime}}_{0}\boldsymbol{\gamma}_{0}\right)$,
where
$\boldsymbol{S}^{\prime}=\boldsymbol{G}^{\prime}(d^{2}\boldsymbol{G^{\prime}}^{T}\boldsymbol{G^{\prime}}+\Sigma^{-1}_{0})^{-1}\boldsymbol{G^{\prime}}^{T}$.
The posterior for $\tau^{-2}$ is
$\mbox{Gamma}\left(\frac{N*V*P+1}{2},1/2*(1+\sum_{i=1}^{N}\sum_{u=1}^{V}\sum_{p=1}^{P}\epsilon_{u,p,i}^{2})\right)$.

###  3.3 Full Conditionals of the Joint Latent Variables and the Covariance

The brain region attributes are related to the brain connectivity through the
dependence between $\boldsymbol{\Theta}$ and $\boldsymbol{Z}$. We are
interested in the joint full conditional distribution of $\boldsymbol{\Theta}$
and $\boldsymbol{Z}$. Let us take a look at the probability model for the
$u$th rows of $\boldsymbol{Z}$ and $\boldsymbol{\Theta}$—$\boldsymbol{z}_{u}$
and $\boldsymbol{\theta}_{u}$. We first reparameterize the covariance matrix
$\boldsymbol{\Sigma}$ as the block matrx:
$\boldsymbol{\Sigma}=\begin{pmatrix}\Lambda_{z}&\Lambda_{zy}^{T}\\\
\Lambda_{zy}&\Lambda_{y}\end{pmatrix},$ and the inverse of this block matrix
is

$\displaystyle\boldsymbol{\Sigma}^{-1}=\begin{pmatrix}\Lambda_{z}&\Lambda_{zy}^{T}\\\
\Lambda_{zy}&\Lambda_{y}\end{pmatrix}^{-1}=\begin{pmatrix}(\Lambda_{z}-\Lambda_{zy}^{T}\Lambda_{y}^{-1}\Lambda_{zy})^{-1}&-(\Lambda_{z}-\Lambda_{zy}^{T}\Lambda_{y}^{-1}\Lambda_{zy})^{-1}\Lambda_{zy}^{T}\Lambda_{y}^{-1}\\\
-(\Lambda_{y}-\Lambda_{zy}\Lambda_{z}^{-1}\Lambda_{zy}^{T})^{-1}\Lambda_{zy}\Lambda_{z}^{-1}&(\Lambda_{y}-\Lambda_{zy}\Lambda_{z}^{-1}\Lambda_{zy}^{T})^{-1}\end{pmatrix}.$
(3.3)

For the ease of notations, we further define the inverse of the covariance
matrix as another block matrix:
$\boldsymbol{\Sigma}^{-1}=\begin{pmatrix}Q_{z}&Q_{yz}\\\
Q_{zy}&Q_{y}\end{pmatrix}$ with each component as a function of $\Lambda$s.

For individual $i$, the connectivity component of the joint model with only
the vector products and the error term is:

$\displaystyle\boldsymbol{F}_{i}=\boldsymbol{X}_{i}-a_{i}-\boldsymbol{w}_{i}\boldsymbol{\beta}=\boldsymbol{Z}\boldsymbol{Z}^{T}+\boldsymbol{E}_{i}$
(3.4)

We can transform $\boldsymbol{F}_{i}$ in such a way that the transformed error
term is a standard normal distribution using
$\boldsymbol{\tilde{F}}_{i}=c\boldsymbol{F}_{i}$, where $c=\sigma_{e}^{-1}$.
Therefore, the model for $\boldsymbol{\tilde{F}}_{i}$ can be written as
$\boldsymbol{\tilde{F}}_{i}=c\boldsymbol{Z}\boldsymbol{Z}^{T}+\boldsymbol{\tilde{E}}_{i},$
where $\tilde{e}_{u,v,i}$ follows a standard normal distribution. Consider the
$u$th row of the matrix $\boldsymbol{\tilde{F}}_{i}$:

$\displaystyle\tilde{f}_{u,1,i}$
$\displaystyle=c\boldsymbol{z}_{u}^{T}\boldsymbol{z}_{1}+\tilde{e}_{p,1,i},$
$\displaystyle\tilde{f}_{u,2,i}$
$\displaystyle=c\boldsymbol{z}_{u}^{T}\boldsymbol{z}_{2}+\tilde{e}_{p,2,i},$
$\displaystyle...$ $\displaystyle\tilde{f}_{u,V,i}$
$\displaystyle=c\boldsymbol{z}_{u}^{T}\boldsymbol{z}_{V}+\tilde{e}_{p,V,i},$
(3.5)

where $\boldsymbol{z}_{u}$ is the $K\times 1$ vector of coordinates on the
latent network dimensions for node $u$. In this way, the connectivity
component is written in the more familiar simple regression form.

Similarly, we can rewrite the attribute component into another regression
form. First, we consider the matrix form of the attribute component without
the individual-specific intercepts and the covariate effects:

$\boldsymbol{T}_{i}=\boldsymbol{Y}_{i}-b_{i}-\boldsymbol{h}_{i}^{T}\boldsymbol{\gamma}=\boldsymbol{\Theta}+\boldsymbol{\Psi}_{i},$
(3.6)

where, as mentioned before, $\boldsymbol{\Theta}$ is the $V\times P$ matrix of
latent variables; and $\boldsymbol{\Psi}_{i}$ is the $V\times P$ matrix of
random error. Consider the $u$th row of $\boldsymbol{T}_{i}$:

$\displaystyle\boldsymbol{t}_{u,i}$
$\displaystyle=\boldsymbol{\theta}_{u}+\boldsymbol{\epsilon}_{u,i},$ (3.7)

where $\boldsymbol{\theta}_{u}$ is the $P\times 1$ vector of latent variable
values for node $v$, and $\boldsymbol{t}_{u,i}$ is the $P\times 1$ vector of
attribute values for node $u$ and individual $i$.

The joint full conditional distribution of $\boldsymbol{z}_{u}$ and
$\boldsymbol{\theta}_{u}$ is:

$\displaystyle p\left(\begin{pmatrix}\boldsymbol{z}_{u}\\\
\boldsymbol{\theta}_{u}\end{pmatrix}|\boldsymbol{t}_{u,i},\boldsymbol{\tilde{f}}_{u,i},\boldsymbol{\Sigma},\tau^{2}\right)\propto$
$\displaystyle\prod_{i=1}^{N}p(\boldsymbol{t}_{u,i}|\boldsymbol{\theta}_{u},\tau^{2})\prod_{i=1}^{N}p(\boldsymbol{\tilde{f}}_{u,i}|\boldsymbol{z}_{u})p\left(\begin{pmatrix}\boldsymbol{z}_{u}\\\
\boldsymbol{\theta}_{u}\end{pmatrix}|\boldsymbol{\Sigma}\right)$
$\displaystyle\propto$
$\displaystyle\exp\left(-\frac{1}{2}\tau^{-2}\sum_{i=1}^{N}(\boldsymbol{t}_{u,i}-\boldsymbol{\theta}_{u})^{T}(\boldsymbol{t}_{u,i}-\boldsymbol{\theta}_{u})\right)\exp\left(-\frac{1}{2}\sum_{i=1}^{N}\sum_{v=1,v\neq
u}^{V}(\tilde{f}_{u,v,i}-c\boldsymbol{z}_{u}^{T}\boldsymbol{z}_{v})^{2}\right)$
$\displaystyle\exp\left(-\frac{1}{2}\begin{pmatrix}\boldsymbol{z}_{u}\\\
\boldsymbol{\theta}_{u}\end{pmatrix}^{T}\boldsymbol{\Sigma}^{-1}\begin{pmatrix}\boldsymbol{z}_{u}\\\
\boldsymbol{\theta}_{u}\end{pmatrix}\right).$

Using the previously defined block matrix, we can write
$p\left(\begin{pmatrix}\boldsymbol{z}_{u}\\\
\boldsymbol{\theta}_{u}\end{pmatrix}|\boldsymbol{\Sigma}\right)$ as
$\exp(-\frac{1}{2}(\boldsymbol{z}_{u}^{T}Q_{z}\boldsymbol{z}_{u}+\boldsymbol{z}_{u}^{T}Q_{yz}\boldsymbol{\theta}_{u}+\boldsymbol{\theta}_{u}^{T}Q_{zy}\boldsymbol{z}_{u}+\boldsymbol{\theta}_{u}^{T}Q_{y}\boldsymbol{\theta}_{u}))$.
Extracting relevant terms from $p\left(\begin{pmatrix}\boldsymbol{z}_{u}\\\
\boldsymbol{\theta}_{u}\end{pmatrix}|\boldsymbol{t}_{u,i},\boldsymbol{\tilde{f}}_{u,i},\boldsymbol{\Sigma},\tau^{2}\right)$,
we can see that the full conditional distribution of $\boldsymbol{z}_{u}$ is

$\displaystyle
p\left(\boldsymbol{z}_{u}|\boldsymbol{\tilde{f}}_{u,i},\boldsymbol{\Sigma},\boldsymbol{\theta}_{u}\right)$
$\displaystyle\propto$
$\displaystyle\exp\left(-\frac{1}{2}\boldsymbol{z}_{u}^{T}(\sum_{v=1,v\neq
u}^{V}Nc^{2}\boldsymbol{z}_{v}\boldsymbol{z}_{v}^{T}+Q_{z})\boldsymbol{z}_{u}+\boldsymbol{z}_{u}^{T}(\sum_{i=1}^{N}\sum_{v=1,v\neq
u}^{V}c\tilde{f}_{u,v,i}\boldsymbol{z}_{v}-\frac{1}{2}Q_{yz}\boldsymbol{\theta}_{u}-\frac{1}{2}Q_{zy}^{T}\boldsymbol{\theta}_{u})\right).$
(3.8)

This is a multivariate normal distribution, with variance $(\sum_{v=1,v\neq
u}^{V}Nc^{2}\boldsymbol{z}_{v}\boldsymbol{z}_{v}^{T}+Q_{z})^{-1}$ and mean
$(\sum_{v=1,v\neq
u}^{V}Nc^{2}\boldsymbol{z}_{v}\boldsymbol{z}_{v}^{T}+Q_{z})^{-1}(\sum_{i=1}^{N}\sum_{v=1,v\neq
u}^{V}c\tilde{f}_{u,v,i}\boldsymbol{z}_{v}-\frac{1}{2}Q_{yz}\boldsymbol{\theta}_{u}-\frac{1}{2}Q_{zy}^{T}\boldsymbol{\theta}_{u})$.
Similarly, the full conditional of $\boldsymbol{\theta}_{u}$ is

$\displaystyle
p\left(\boldsymbol{\theta}_{u}|\boldsymbol{t}_{u,i},\boldsymbol{\Sigma},\boldsymbol{z}_{u},\tau^{2}\right)$
$\displaystyle\propto$
$\displaystyle\exp\left(-\frac{1}{2}\boldsymbol{\theta}_{u}^{T}(N\tau^{-2}\boldsymbol{I}+Q_{y})\boldsymbol{\theta}_{u}+\boldsymbol{\theta}_{u}^{T}(\sum_{i=1}^{N}\tau^{-2}\boldsymbol{t}_{u,i}-\frac{1}{2}Q_{yz}^{T}\boldsymbol{z}_{u}-\frac{1}{2}Q_{zy}\boldsymbol{z}_{u})\right).$
(3.9)

This is a multivariate normal distribution, with variance
$(N\tau^{-2}\boldsymbol{I}+Q_{y})^{-1}$ and mean
$(N\tau^{-2}\boldsymbol{I}+Q_{y})^{-1}(\sum_{i=1}^{N}\tau^{-2}\boldsymbol{t}_{u,i}-\frac{1}{2}Q_{yz}^{T}\boldsymbol{z}_{u}-\frac{1}{2}Q_{zy}\boldsymbol{z}_{u})$.

Recall that prior for $\boldsymbol{\Sigma}$ is
$\boldsymbol{\Sigma}^{-1}\sim\text{Wishart}(\boldsymbol{I}_{K+P},K+P+2)$. Let
$\boldsymbol{F^{\prime}}$ be a $V\times(K+P)$ matrix with $u$th row as
$(\boldsymbol{z}_{u}^{T},\boldsymbol{\theta}_{u}^{T})$. The full conditional
for $\boldsymbol{\Sigma}$ follows a inverse-Wishart
$(\boldsymbol{I}_{K+P}+\boldsymbol{F^{\prime}}^{T}\boldsymbol{F^{\prime}},V+K+P+2)$.

1: Initialize $\sigma^{2}$, $\boldsymbol{\beta},\boldsymbol{a}$, $\tau^{2}$,
$\boldsymbol{\gamma},\boldsymbol{b}$, $\boldsymbol{Z}$, $\boldsymbol{\Theta}$
and $\boldsymbol{\Sigma}$;

2: while $t<N_{\text{iter}}$ do

3: Sample $\boldsymbol{a}$ from a multivariate normal distribution with
variance $(c^{2}\boldsymbol{G}^{T}\boldsymbol{G}+\Sigma^{-1}_{0})^{-1}$ and
mean
$c(c^{2}\boldsymbol{G}^{T}\boldsymbol{G}+\Sigma^{-1}_{0})^{-1}\boldsymbol{G}^{T}\boldsymbol{m}$;

4: Sample $\boldsymbol{\beta}$ from a multivariate normal distribution with
variance
$\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{W}+\boldsymbol{S}_{0}\right)^{-1}$
and

mean
$\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{W}+\boldsymbol{S}_{0}\right)^{-1}\left(c^{2}\boldsymbol{W}^{T}\boldsymbol{S}\boldsymbol{r}+\boldsymbol{S}_{0}\boldsymbol{\beta}_{0}\right)$;

5: Sample $\sigma^{-2}$ from a gamma distribution
$\left(\frac{N*V*(V-1)/2+1}{2},1/2*(1+\sum_{i=1}^{N}\sum_{u=1}^{V}\sum_{u<v}^{V}e_{u,v,i}^{2})\right)$;

6: Sample $\boldsymbol{b}$ from a multivariate normal distribution with
variance
$(d^{2}\boldsymbol{G^{\prime}}^{T}\boldsymbol{G^{\prime}}+\Sigma^{-1}_{0})^{-1}$
and mean
$d(d^{2}\boldsymbol{G^{\prime}}^{T}\boldsymbol{G^{\prime}}+\Sigma^{-1}_{0})^{-1}\boldsymbol{G^{\prime}}^{T}\boldsymbol{m^{\prime}}$;

7: Sample $\boldsymbol{\gamma}$ from a multivariate normal distribution with
variance
$\left(d^{2}\boldsymbol{H}^{T}\boldsymbol{S^{\prime}}\boldsymbol{H}+\boldsymbol{S^{\prime}}_{0}\right)^{-1}$
and

mean
$\left(d^{2}\boldsymbol{H}^{T}\boldsymbol{S^{\prime}}\boldsymbol{H}+\boldsymbol{S^{\prime}}_{0}\right)^{-1}\left(d^{2}\boldsymbol{H}^{T}\boldsymbol{S^{\prime}}\boldsymbol{r^{\prime}}+\boldsymbol{S^{\prime}}_{0}\boldsymbol{\gamma}_{0}\right)$;

8: Sample $\tau^{-2}$ from a gamma distribution
$\left(\frac{N*V*P+1}{2},1/2*(1+\sum_{i=1}^{N}\sum_{u=1}^{V}\sum_{p=1}^{P}\epsilon_{u,p,i}^{2})\right)$;

9: Sample $\boldsymbol{\theta}_{u}$ from a multivariate normal distribution,
with variance $(\sum_{v=1,v\neq
u}^{V}Nc^{2}\boldsymbol{z}_{v}\boldsymbol{z}_{v}^{T}+Q_{z})^{-1}$

and mean $(\sum_{v=1,v\neq
u}^{V}Nc^{2}\boldsymbol{z}_{v}\boldsymbol{z}_{v}^{T}+Q_{z})^{-1}(\sum_{i=1}^{N}\sum_{v=1,v\neq
u}^{V}c\tilde{f}_{u,v,i}\boldsymbol{z}_{v}-\frac{1}{2}Q_{yz}\boldsymbol{\theta}_{u}-\frac{1}{2}Q_{zy}^{T}\boldsymbol{\theta}_{u})$;

10: Sample $\boldsymbol{z}_{u}$ from a multivariate normal distribution with
variance $(N\tau^{-2}\boldsymbol{I}+Q_{y})^{-1}$ and mean
$(N\tau^{-2}\boldsymbol{I}+Q_{y})^{-1}(\sum_{i=1}^{N}\tau^{-2}\boldsymbol{t}_{u,i}-\frac{1}{2}Q_{yz}^{T}\boldsymbol{z}_{u}-\frac{1}{2}Q_{zy}\boldsymbol{z}_{u})$;

11: Sample $\boldsymbol{\Sigma}$ from a inverse-Wishart
$(\boldsymbol{I}_{K+P}+\boldsymbol{F^{\prime}}^{T}\boldsymbol{F^{\prime}},V+K+P+2)$;

12: $t\leftarrow$ $t+1$;

13: end while

14: return posterior samples of $\sigma^{2}$,
$\boldsymbol{\beta},\boldsymbol{a}$, $\tau^{2}$,
$\boldsymbol{\gamma},\boldsymbol{b}$, $\boldsymbol{Z}$, $\boldsymbol{\Theta}$
and $\boldsymbol{\Sigma}$ as approximated target distributions;

Algorithm 1 ABC Estimation procedure

Following equation 2.2, it can be proved that $\boldsymbol{Z}$ is identified
up to rotations while $\boldsymbol{Z}^{T}\boldsymbol{Z}$ is directly
identified following the condition that $\boldsymbol{Z}$ is centered. To
resolve the rotation indeterminacy issue, we perform a procrustean
transformation of the estimated $\boldsymbol{Z}$ at each iteration to its
initial position, $\boldsymbol{Z}_{0}$. For each $\boldsymbol{Z}$ sampled from
the posterior, we find the rotated $\boldsymbol{Z}^{*}$ of $\boldsymbol{Z}$
such that $\boldsymbol{Z}^{*}$ has the smallest sum of squared deviations from
the target orientation $\boldsymbol{Z}_{0}$. We freely estimate the covariance
matrix $\Sigma$.

##  4 Simulation

We conduct simulation studies to evaluate whether the proposed ABC model
compared with the existing alternatives will facilitate a better prediction of
brain connectivity from new subjects, and to assess how incorporating node-
level attributes improves the performance. We compare the ABC model with two
existing approaches, namely the commonly used averaging method (Average),
where the group-level connectivity is estimated as the entry-wise sample mean
of individual connectivity matrices, and the multiplex stochastic block model
(MSBM, [6]), a generative network model for estimating the group-level
connectivity with discrete latent variables. To assess the improvement on
estimation by incorporating the regional anatomical attributes, we compare the
ABC model against the same model without incorporating node-wise attributes,
denoted as the brain connectivity (BC) model.

The data are generated as follows. For simplicity and consistency, the number
of attributes and the number of latent variables are both assigned as two in
all generated data. We first generate the connectivity latent variables as
well as the latent variables of the attributes from the multivariate normal
distribution with the mean zero and the pre-specified correlation matrix (the
variances are assumed to be 1). Following existing literature on the joint
models with multiple sources of information [[,
e.g.,]]wang2019joint,gollini2016joint, we believe that the benefit of adding
attributes of the brain regions depends on the relationship between
connections and attributes. Thus, we specify the correlation matrix between
connectivity and attributes based on three levels of dependence: (a) they are
independent, (b) they have moderate correlations of 0.5 and (c) they have
strong correlations of 0.9. To create symmetric positive definite matrices, we
assign correlation values to the corresponding dimensions between the
connectivity and the attributes of the brain regions, e.g., for the strong
dependence condition, we assign the 0.9 correlation between the first
dimensions of connectivity and attributes and between the second dimensions of
the connectivity and attributes. With these three levels of dependence, we
randomly generate latent variable values for both the connectivity and the
attributes.

We consider two sample sizes, $N=50$ and $N=100$ and two conditions for the
number of nodes $V=20$ and $V=70$, and we specify three levels of the signal-
to-noise (S/N) ratio, $0.05$, $0.02$ and $0.01$ controlled by modifying the
error variance while keeping the variance of the latent variables constant. To
compare ABC with BC, we add three additional levels of the signal-to-noise
ratio, 0.1, 1 and 2, and we also add a smaller sample size $N=20$. These
additional conditions allow us to assess the predictive power of ABC with a
fuller range of parameters. We randomly sample the errors for the connectivity
from the normal distribution with the mean $0$ and the variance defined by the
desired S/N ratio. The errors for the attributes are sampled from the normal
distribution with the mean $0$ and variance 0.5. The individual-specific
intercepts are set as $1$ or $-1$ with a sum of zero across subjects. In
total, we will consider 24 different scenarios combining from different
connectivity and attribute correlations, sample sizes, node numbers and S/Ns.
Under each scenario, we simulate the data $100$ times.

To evaluate prediction, each simulated dataset is randomly divided by an
equally sized training set and test set. We implement the ABC model on the
training set based on the developed MCMC algorithm with the convergence of the
posterior computation confirmed by trace plots of the key parameters. As
comparisons, we also apply the competing methods Average, MSBM and BC on the
same training set. Specifically, for the Average method, we average over the
individual connectivity in the training set and use the averaged connectivity
as the estimated group connectivity; for MSBM, we fit the model using the sbm
package [6] in R; and for BC, we implement it following the same process as
ABC model except that the attribute component is excluded. We estimate the
Mean Squared Error (MSE) of the predicted connectivity on the test set by
calculating the average squared difference between the upper diagonal elements
of the predicted values and corresponding observed values. We also calculate
the correlation between the upper diagonal elements of the predicted values
and corresponding observed values. To estimate the predicted values, we input
data with test subjects as NA into ABC model. The predicted values are the sum
of the group-level connectivity (shared by everyone in the group) and
individual intercepts.

Figure 2:  The difference in the correlational predictive power between ABC
and BC when (a) $V=20$ and (b) $V=70$. Positive values indicate that ABC
outperforms BC with higher correlations between the predicted and the observed
brain connectivity. Negative values indicate that BC outperforms ABC with
higher correlations between the predicted and the observed brain connectivity.

Table 1 shows the MSEs and the correlations of the proposed ABC method against
those of the Average method and the MSBM. The standard deviations of the
squared differences between the predicted and the observed connectivity across
replications are reported along with the standard deviations of the
correlations. As an illustrate, we report the results from the independence
and strong dependence conditions here, and the results for the moderate
dependence condition are shown in Figure 2.

Table 1: Prediction of the group-level connectivity

|  |  |  | MSE | Correlation
---|---|---|---|---|---
$N$ | $V$ | Dependence | $S/N$ | ABC | Average | MSBM | ABC | Average
|  | 0.05 | 0.71 (0.31) | 0.82 (0.09) | 1.90 (0.86) | 0.90 (0.06) | 0.82 (0.07)
|  | IND | 0.02 | 0.87 (0.34) | 2.02 (0.21) | 1.90 (0.91) | 0.81 (0.11) | 0.67 (0.10)
|  |  | 0.01 | 1.12 (0.39) | 4.02 (0.40) | 1.93 (0.90) | 0.69 (0.16) | 0.54 (0.10)
| 20 |  | 0.05 | 0.68 (0.28) | 0.82 (0.09) | 1.94 (0.80) | 0.91 (0.06) | 0.82 (0.06)
|  | DEP | 0.02 | 0.80 (0.31) | 2.01 (0.20) | 2.00 (0.96) | 0.86 (0.08) | 0.68 (0.09)
|  |  | 0.01 | 0.94 (0.35) | 4.01 (0.40) | 2.01 (0.96) | 0.80 (0.10) | 0.56 (0.10)
50 |  |  | 0.05 | 0.57 (0.13) | 0.82 (0.04) | 1.44 (0.42) | 0.97 (0.01) | 0.84 (0.03)
|  | IND | 0.02 | 0.62 (0.13) | 2.02 (0.07) | 1.80 (0.36) | 0.95 (0.02) | 0.70 (0.04)
|  |  | 0.01 | 0.71 (0.14) | 4.02 (0.13) | 2.03 (0.50) | 0.92 (0.02) | 0.57 (0.05)
| 70 |  | 0.05 | 0.56 (0.14) | 0.82 (0.04) | 1.36 (0.29) | 0.97 (0.02) | 0.84 (0.03)
|  | DEP | 0.02 | 0.59 (0.14) | 2.02 (0.07) | 1.73 (0.35) | 0.96 (0.02) | 0.69 (0.04)
|  |  | 0.01 | 0.66 (0.15) | 4.02 (0.13) | 1.99 (0.48) | 0.93 (0.02) | 0.57 (0.04)
|  |  | 0.05 | 0.63 (0.29) | 0.40 (0.04) | 1.88 (0.85) | 0.93 (0.04) | 0.90 (0.05)
|  | IND | 0.02 | 0.72 (0.29) | 0.99 (0.09) | 1.90 (0.90) | 0.89 (0.07) | 0.79 (0.08)
|  |  | 0.01 | 0.86 (0.30) | 1.98 (0.19) | 1.91 (0.90) | 0.82 (0.11) | 0.68 (0.10)
| 20 |  | 0.05 | 0.63 (0.27) | 0.40 (0.04) | 1.92 (0.79) | 0.94 (0.04) | 0.90 (0.04)
|  | DEP | 0.02 | 0.70 (0.28) | 0.99 (0.09) | 1.98 (0.97) | 0.91 (0.07) | 0.80 (0.07)
|  |  | 0.01 | 0.79 (0.30) | 1.98 (0.19) | 1.99 (0.97) | 0.86 (0.10) | 0.69 (0.08)
100 |  |  | 0.05 | 0.55 (0.13) | 0.41 (0.02) | 1.36 (0.30) | 0.98 (0.01) | 0.91 (0.02)
|  | IND | 0.02 | 0.57 (0.13) | 1.00 (0.03) | 1.76 (0.35) | 0.97 (0.02) | 0.81 (0.03)
|  |  | 0.01 | 0.63 (0.14) | 2.00 (0.06) | 2.03 (0.50) | 0.95 (0.02) | 0.70 (0.04)
| 70 |  | 0.05 | 0.54 (0.13) | 0.41 (0.02) | 1.36 (0.35) | 0.98 (0.02) | 0.91 (0.02)
|  | DEP | 0.02 | 0.55 (0.14) | 1.00 (0.03) | 1.75 (0.34) | 0.97 (0.01) | 0.80 (0.03)
|  |  | 0.01 | 0.60 (0.14) | 2.00 (0.06) | 1.98 (0.48) | 0.96 (0.02) | 0.70 (0.04)

Based on Table 1, we can see that the group-level connectivity is more
precisely and accurately recovered using ABC than the Average method and MSBM.
The improvement becomes more evident when the estimation of the group-level
connectivity becomes more difficult. As the signal-to-noise ratio decreases,
the MSE of the Average method substantially increases, which can be attributed
to the fact that the Average method does not and cannot separate signals from
noise. As the noise becomes stronger, the Average method performs worse. This
is true for both independent and dependent conditions. The correlations
between the predicted and the observed connectivity show a similar trend.
Between ABC and MSBM, ABC consistently outperforms MSBM in predicting the
group-level connectivity. There are two potential reasons for this. First, the
stochastic block model is inferior for capturing the topological properties of
brain structural connectivity as generated by our settings. The block model is
often used to capture potential clusters in the network, effective for
networks with high levels of stochastic equivalence and less effective for
networks with high levels of homophily [29]. Second, the ABC model outperforms
because it incorporates the additional attributes information. Between the
independence and strong dependence conditions, the predictive power of ABC
improves while the predictive powers of MSBM stay the same. This provides
evidence for the benefit of incorporating the attributes information.

In Figure 2, we present the difference in the predictive power between ABC and
BC when (a) $V=20$ and (b) $V=70$. Positive values indicate that ABC
outperforms BC with higher correlations between the predicted and the observed
brain connectivity. The comparison between ABC and BC illustrates the
improvement on connectivity prediction by incorporating node-wise attributes.
Figure 2 shows that adding the attributes information is beneficial for
estimating the group-level connectivity when the attributes are related to the
connectivity with both moderate (red) and strong (green) dependence. The
stronger the dependence, the more beneficial it is to incorporate attributes.
When they are independent (blue), there is no observable improvement. When
there is dependence, the level of improvement depends on how difficult it is
to estimate the group-level connectivity or how much information there is to
estimate the group-level connectivity. There is more benefit for incorporating
the attributes when the signal-to-noise ratio is small and when the sample
size is small. The results for when $V=20$ show a similar trend as the results
for $V=70$.

##  5 Application

We apply the proposed ABC model to the ADNI data described in section 2.1. The
purpose of the analyses is three-fold. First, we estimate the group-level
(including AD female, AD male, healthy female, and healthy male) structural
connectivity while incorporating the brain node attributes information on the
cortical volume, surface areas, average cortical thickness and standard
deviation of the cortical thickness. Each of the four groups has 30, 49, 45,
and 50 subjects, respectively. Second, we compare the estimated connectivity
across different groups and identify their differences. Lastly, we
characterize and perform inference on the correlation between structural
connections and regional anatomical information under different subgroups to
understand their interplay.

###  5.1 Estimation and Model Fit

We perform posterior inference based on the MCMC algorithm described in
section 3. We generate $20,000$ iterations with a burn-in period of 500 and
thinning every 10th sample. Each model takes about 1 hour to complete with
193.67 MB memory and one computing node with three cores on the Yale HPC
cluster. The trace plots show no obvious signs of non-convergence; see
examples in the supplementary material.

Table 2: The correlation between the predicted and the observed connectivity Dimension ($K$) | AD Female | AD Male | Healthy Female | Healthy Male
---|---|---|---|---
$2$ | 0.396 | 0.370 | 0.368 | 0.424
$3$ | 0.448 | 0.466 | 0.459 | 0.453
$4$ | 0.515 | 0.515 | 0.523 | 0.529
$5$ | 0.547 | 0.526 | 0.556 | 0.542
$6$ | 0.414 | 0.306 | 0.513 | 0.344
$7$ | 0.363 | 0.352 | 0.239 | 0.200
$8$ | 0.323 | 0.437 | 0.191 | 0.228
Test | $0.508$ | $0.561$ | $0.540$ | $0.570$

The samples in each group are randomly divided into 3 sets: training ($90\%$),
validation ($5\%$) and test set ($5\%$). To select the number of connectivity
dimensions, each ABC model is trained on the training set with
$K=2,3,4,5,6,7,8$. To compare the fit of the model with varying numbers of
dimensions, we assess the recovery of the observed connectivity based on the
validation set. The number of dimensions with the best recovery of the
observed connectivity in the validation set is selected. Lastly, after the
dimensions are selected, the model fit index is calculated based on the test
set. In Table 2, we present the predictive power, estimated as the correlation
between the predicted and the observed connectivity, of the ABC model with
varying dimensions based on the validation set. The results show that the
predictive power increases as the number of dimensions increases from $2$ to
$5$, and then decreases as $K$ continues to increase. The highest predictive
power is observed with $K=5$ suggesting that there are likely 5 connectivity
dimensions. This result applies to all four subject groups. To assess the fit
of the model to data, we calculate the predictive power of the ABC model with
$K=5$ based on the test data. As shown in Table 2, these results show
satisfactory fit of the ABC model to data.

###  5.2 Attributes-informed Group-level Connectivity

Figure 3: Five estimated latent connectivity dimensions from (a) the lateral
view of the right hemisphere, (b) the superior view of the brain and (c) the
lateral view of the left hemisphere. Brain regions are colored based on the
values of the latent dimensions with red indicating positive values and green
indicating negative values.  Figure 4: The latent space representations of the
structural connectivity between (a) dimensions 1 and 2, (b) dimensions 3 and 5
and (c) dimensions 1 and 4. Each dot of the latent space represents a brain
region. The color differentiates the left hemisphere (black) from the right
hemisphere (red).

Figure 3 shows five estimated latent dimensions of the structural connectivity
from (a) the lateral view of the right hemisphere, (b) the superior view of
the brain, and (c) the lateral view of the left hemisphere. These latent
dimensions are estimated based on the healthy female subjects, similar to
those based on the healthy male subjects (see the supplementary materials for
their similarities). The brain regions are colored based on the values of the
latent dimensions with red indicating positive values and green indicating
negative values. Each row of Figure 3 shows the estimated values of the
corresponding latent dimension. For example, Figure 3a,1 shows the estimated
values of the first latent dimension from the lateral view of the right
hemisphere. Figure 4 shows the latent space of the structural connectivity
between (a) dimensions 1 and 2, (b) dimensions 3 and 5 and (c) dimensions 1
and 4. See the latent spaces between each pair of dimensions in the
supplementary materials. Each dot of the latent space represents a brain
region. The color differentiates the left hemisphere (black) from the right
hemisphere (red). A vector drawn from the origin $(0,0)$ to the dot is the
vector presentation of the brain region. We use capital letters to refer to
directions of the latent space represented by the region vectors. Following
the model equation, the structural connectivity between two brain regions is
represented by the vector product of the two regions’ vectors. Strong
connectivity is represented by the large vector product, and vise versa. Thus,
both the length of the vector and the angle between two vectors signal
connectivity strength with a small angle indicating a strong connection. For a
given brain region, the same vector length is used in the presentation of its
connection with all other regions. Therefore, the vector length indicates an
overall connectivity strength for the region.

In Figure 4, a grouping of the brain regions can be observed when these
regions share a similar direction, distinct from directions of the other
regions. For example, between dimensions 1 and 2 in Figure 4a, three groups of
brain regions are found in the directions of A,B and C. In direction A, we
find most of the regions in the left hemisphere; and in direction B, we find
most of the regions in the right hemisphere. This differentiation of the left
and right hemispheres can also be seen in rows 1 and 2 in Figure 3. In these
two rows, edges in the same hemisphere are estimated with latent variable
values of the same sign, shown with the same color; and edges in different
hemispheres are estimated with latent variable values of opposite signs, shown
with opposite colors. Therefore, the first two latent dimensions differentiate
the left from the right hemisphere. This differentiation suggests that regions
in the same hemisphere are more connected with higher fiber density than
regions from different hemispheres. This could be attributed to the
association fibers interconnecting areas of the cerebral cortex in the same
hemisphere. These association fibers may be short, connecting adjacent
regions, or long, connecting more distant areas of the cortex. In direction C,
we mostly find regions of the cingulate cortex. As we know, the
interconnections between hemispheres are based on the commissural fibers.
Their bands link the hemispheres and include the corpus callosum, which is
wrapped by the cingulate gyrus like a “belt”. Our results reflect the distinct
nature of the fiber connectivity around regions of the cingulate. This
distinction is captured by the latent space that would have been lost
otherwise.

In Figure 4b, between dimensions 3 and 5 of the latent space, four groupings
of regions are found in the directions of D,E,F and G. In directions D and G,
regions in the posterior parietal cortex, the temporal and occipital lobes are
found from the right (D) and left (G) hemispheres. In directions E and F,
regions in the frontal cortex and the primary somatosensory cortex are found
from the right (F) and left (E) hemispheres. The differentiation of the
posterior parietal cortex, the temporal and occipital lobes (directions D and
G) from the frontal cortex and the primary somatosensory cortex (directions F
and E) can also be observed in rows 3 and 5 of Figure 3. We anticipate that
directions D and G reflect the long association fibers including the inferior
longitudinal fasciculus connecting the temporal and the occipital lobes and
the superior longitudinal fasciculus connecting the parietal, occipital and
temporal lobes with the frontal lobe [19]. Meanwhile, directions E and F
reflect the connectivity of the frontal cortex and the primary somatosensory
cortex. The short association fibers interconnect the prefrontal cortex, the
premotor area and the motor cortex with the primary somatosensory cortex [3].

In Figure 4c, dimension 4 of the latent space differentiates the prefrontal
cortex regions including lateral orbitofrontal, medial orbitofrontal and
frontal pole from the parietal lobe including postcentral, superior parietal
and supramarginal. This differentiation of the prefrontal cortex from the
parietal lobe can also be seen in row 4 of Figure 3. In directions of H and I,
the posterior cingulate regions are found with the parietal lobe regions. The
posterior cingulate cortex connects regions of parietal lobe that receive
inputs from the dorsal visual stream and somatosensory areas [49, 56], and
posterior cingulate cortex is concerned with spatial representation, spatial
processing, orientation and certain types of memory [7, 4].

###  5.3 Group-level Connectivity Differences

Figure 5: Estimated attributes-informed brain connectivity for (a) healthy
male subjects, (b) male subjects with AD, and (c) female subjects with AD.
Figure 6:  Gender-based difference in AD phenotype for the structural
connectivity between regions of (a) pericalcarine and precuneus (b) lingual
and cuneus, (c) lingual and pericalcarine, and (d) pericalcarine and cuneus.

Figure 5 shows the estimated attributes-informed group-level structural
connectivity for (a) healthy male subjects, (b) male subjects with AD and (c)
female subjects with AD. The estimated connectivity for healthy female
subjects is similar to that of the healthy male subjects and can be found in
the supplementary material. The estimated connectivity values are scaled to a
range between 0 and 1 for comparison between different subject groups.
Compared with the healthy male subjects, AD male subjects show increased
connectivity for certain brain regions in the occipital lobe, highlighted in
Figure 5b. Meanwhile, a different set of regions is found with increased
connectivity for AD female subjects in certain regions of the parietal lobe,
highlighted in Figure 5c. This result suggests that there is gender-based
difference in the neuromarkers of AD.

To assess whether these gender differences associated with AD are significant,
we estimate the uncertainty of the estimated edge values based on the
posterior sampling. We selectively investigate the edges that show large
connectivity differences between different subject groups based on Figure 5.
Each edge is investigated based on the posterior distributions, and those that
show significant gender differences are shown in Figures 7 and 6. In Figures 7
and 6, the first column shows the locations of the brain regions involved in
the edges from the lateral view of the left hemisphere. The second column
shows the posterior distributions of the associated edges for AD female, AD
male, healthy female and healthy male subjects.

Among all edges identified with significant gender differences, the associated
brain regions are found in the left hemisphere. This could be attributed to
the asymmetric thinning of the cerebral cortex between the left and right
hemispheres exacerbated by AD [48]. [48] found that the two hemispheres
deteriorate at different rates with the left hemisphere shrinking faster in
patients with AD. It is thus not surprising that large structural connectivity
differences are observed between subject groups in the left hemisphere.

Figure 6 shows gender-based difference in the neuromarkers of AD for the
structural connectivity between (a) pericalcarine and precuneus (b) lingual
and cuneus, (c) lingual and pericalcarine and (d) pericalcarine and cuneus.
Among these four edges, AD male subjects have the highest fiber density
connectivity, followed by healthy male subjects. Females tend to have lower
fiber density connectivity than males, and there does not seem to be a
difference between healthy females and AD females. It is interesting to note
that the regions associated with these edges are uniformly located in the
medial aspect of the occipital lobe and the precueus of the parietal lobe,
which includes the primary visual cortex. AD patients often report symptoms of
visual deficits, which are linked to damages to the occipital lobe [8, 16,
44]. The abnormality in the precuneus and the occipital lobe for the male AD
patients could also be linked to psychosis [47, 14, 53, 65], which is more
often observed with male patients [5, 15].

Figure 7: Gender-based difference in AD phenotype for the structural
connectivity between regions of (a) superior parietal and supramarginal, (b)
inferior parietal and supramarginal, and (c) inferior parietal and superior
parietal.

Figure 7 shows gender-based differences in the structural connectivity between
regions of (a) superior parietal and supramarginal, (b) inferior parietal and
supramarginal, and (c) inferior parietal and superior parietal. Among these
three edges, healthy female subjects have the highest fiber density
connectivity, followed by AD female, healthy male and AD male subjects.
Between males and females, females have higher fiber density connectivity in
these regions; and both genders suffer a loss of connectivity with AD. It is
interesting to note that the regions associated with these edges are found in
the parietal lobe, more specifically, the posterior parietal lobe, a region
implicated in spatial attention and eye movements [26].

The parietal lobe is known to have structural differences between males and
females [[,
e.g.,]]salinas2012sex,koscik2009sex,frederikse1999sex,studholme2020motion.
Females have a larger ratio of parietal lobe cortex to white matter than males
in children between 7 and 17 years of age [51]. This gender-based structural
difference in the parietal lobe can also be traced back to the early
developmental stages, with females having greater relative proportion of the
brain surface area occupied by the parietal lobe than males in both fetuses
and neonates [55, 37]. The parietal lobe is involved in the visuospatial
processing, which is known to suffer damage with AD. Compared with the healthy
elderly, AD patients tend to experience visuospatial deficits [17, 46, 11, 10,
18], faring worse on visuospatial tasks such as the tests of mental rotation
[34, 40, 38]. It is thus not surprising that the parietal lobe is found to
suffer extensive damage in AD patients, in line with existing research on AD
such as [38].

The proposed ABC model provides consistent and interpretable evidence for the
gender-based difference in the neuromarkers of AD such as the role of the
parietal lobe in AD. By contrast, existing research generally show a lack of
success in finding consistent gender-based visuospatial difference for AD
patients [35]. Some show no significant gender differences in visuospatial
deficits for patients with mild AD [45], and others show male superiority on
some visuospatial tasks, but not others [42]. Thus, it can be seen that the
proposed ABC model is a comprehensive tool for identifying group-level
differences based on gender and disease types.

###  5.4 Dependence between Connectivity and Attributes

Lastly, we discuss the dependence between the regional anatomical attributes
and structural connections among them reported by the ABC model. Through ABC,
we estimate the covariances (and correlations) between each of the attributes
and the latent connectivity dimensions for each group of subjects. Among the
four anatomical attributes, the cortical volume was consistently found to be
significantly related to the structural connectivity among the brain regions.
The $95\%$ credible intervals for the correlation in absolute values between
the cortical volume and the fourth dimension of the brain connectivity are
$[0.056,0.484],[0.063,0.484],[0.096,0.509],$ and $[0.036,0.452]$ for AD
female, AD male, healthy female and healthy male, respectively. The result
shows that the cortical volume is significantly related to the white matter
fiber density with a relatively stronger correlation for healthy females. The
corresponding credible intervals of the other three attributes contain zero,
and further research is needed to conclude their relationship with structural
connectivity.

##  6 Discussion

The ABC model outlined in this article constitutes a principal strategy along
with a valid inference process for estimating group-level brain connectivity
while incorporating node attributes. Characterizing brain connectivity
specific to a subpopulation in an analytically and biologically plausible way
plays a crucial role to understand neurological architectures under different
phenotypic conditions. Filling a literature gap, we have proposed a generative
latent space network model as a comprehensive and extendable strategy for
estimating attributes-informed brain connectivity. Based on the proposed
modeling framework, we dissect interpretable latent space structures of group-
level structural connectivity, informed by regional anatomical knowledge.
Through a valid inference process, we quantify the uncertainty of each unknown
connectivity and regional component and evaluate the likelihood of group-level
difference against chance. Through extensive simulations and applications on
AD, we have shown the superior performance of our model in understanding and
comparing group-level brain structural networks and their interplay with
anatomical structures, as well as predicting connectivity traits for new
subjects.

Predicting structural connectivity is an important task with many studies
having access to MRI for anatomical information but without or with missing
DTI data. In such cases, being able to predict missing structural connectivity
based on available anatomical information allows us to fully utilize
accessible information to deal with a missing data issue and to improve
prediction accuracy. The ABC model outperforms existing approaches in
predictive tasks by incorporating anatomical knowledge as demonstrated by the
simulation studies. In particular, the advantages of the ABC model become more
evident as the estimation task becomes more difficult with decreasing signal
to noise ratio and smaller sample size. The Average method is not able to
differentiate signal from noise, and its predictive power deteriorates when
more noise is found in the data. Meanwhile, the ABC method shows minimal loss
in predictive power with increasing noise outperforming both the Average
method and the MSBM. In practice, rarely do we observe negligible noise in the
brain imaging data, thus making the use of the Average method problematic.

We have demonstrated the practical benefits of the ABC model using the ADNI
data. The estimated latent structural connectivity space offers a reduced-
dimension and interpretable representation of the fiber density structural
connectivity. For future direction, the ABC model can also be applied to the
Adolescent Brain Cognitive Development (ABCD) study to investigate the
biological basis of cognitive development through adolescence into young
adulthood. The model allows us to estimate group-level connectivity to
identify brain region changes following adolescents’ cognitive developmental
stages. The ABC model allows us to quantify the uncertainty of the potential
group-level differences and test the possible dependence between cognition and
anatomy of the brain.

Although our focus is on jointly modeling structural connectivity and
anatomical attributes, converging literature suggests that anatomical
attributes may also impact functional connectivity [61]. To investigate the
relationship between anatomical attributes and functional connectivity, we can
apply the ABC model to estimate the group-level functional brain connectivity
while incorporating the anatomical attributes of the corresponding brain
regions. We can identify and compare the estimated brain connectivity between
different genders or disease conditions and test the impact of the anatomical
attributes on functional connectivity. In addition, we also intend to consider
regional functional attributes including measurements from PET imaging (e.g.
AV45, FDG-PET) when modeling group-level functional connectivity. This will
allow us to characterize how regional metrics from PET interact with
functional connections measured by functional MRI, which has received growing
attention and become an active research direction for AD study, specially with
more multi-modal PET imaging data becoming available.

Future research will also be considered to extend our analytical framework to
accommodate alternative data conditions. For example, the ABC model can be
extended to model dynamic functional connectivity in order to query how
connectivity metric changes along the time course. The temporal dependence
between the functional brain imaging data can be accounted for to facilitate
the latent connectivity space to be time-dependent, and then the dynamics of
connectivity across time windows can be explored. The ABC model can also be
extended to incorporate behavior measurements by allowing multiple latent
dimensions in the attributes model. In general mental health and behavior
studies, behavior outcomes are typically collected from multiple spectra, and
we would expect different data spectra to be highly correlated while offering
distinct information. To integrate different attribute spectra, multi-
dimensional latent space modeling needs to be developed to characterize their
shared latent components. Finally, the ABC model is currently limited to
continuous connection metrics, and it can be easily extended to binary metrics
with Bernoulli distributions or count data with Poisson distributions.

## Acknowledgments

This study was supported in part by NIH grants RF1 AG068191, P30 AG066508, UL1
TR0001863, T32 AG076411, U01 AG066833 and U01 AG068057; and NSF grant IIS
1837964. Data collection and sharing for this project was funded by the
Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of
Health Grant U01 AG024904) and DOD ADNI (Department of Defense award number
W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the
National Institute of Biomedical Imaging and Bioengineering, and through
generous contributions from the following: AbbVie, Alzheimer’s Association;
Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.;
Biogen; Bristol-Myers Squibb Company; CereSpir, Inc.; Cogstate; Eisai Inc.;
Elan Pharmaceuticals, Inc.; Eli Lilly and Company; EuroImmun; F. Hoffmann-La
Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE
Healthcare; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research &
Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development
LLC.; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.;
NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals
Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical
Company; and Transition Therapeutics. The Canadian Institutes of Health
Research is providing funds to support ADNI clinical sites in Canada. Private
sector contributions are facilitated by the Foundation for the National
Institutes of Health (www.fnih.org). The grantee organization is the Northern
California Institute for Research and Education, and the study is coordinated
by the Alzheimer’s Therapeutic Research Institute at the University of
Southern California. ADNI data are disseminated by the Laboratory for Neuro
Imaging at the University of Southern California.

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