Multimodal neuroimaging data integration and pathway analysis

Abstract

With advancements in technology, the collection of multiple types of measurements on a common set of subjects is becoming routine in science. Some notable examples include multimodal neuroimaging studies for the simultaneous investigation of brain structure and function and multi-omics studies for combining genetic and genomic information. Integrative analysis of multimodal data allows scientists to interrogate new mechanistic questions. However, the data collection and generation of integrative hypotheses is outpacing available methodology for joint analysis of multimodal measurements. In this article, we study high-dimensional multimodal data integration in the context of mediation analysis. We aim to understand the roles that different data modalities play as possible mediators in the pathway between an exposure variable and an outcome. We propose a mediation model framework with two data types serving as separate sets of mediators and develop a penalized optimization approach for parameter estimation. We study both the theoretical properties of the estimator through an asymptotic analysis and its finite-sample performance through simulations. We illustrate our method with a multimodal brain pathway analysis having both structural and functional connectivity as mediators in the association between sex and language processing.

1. INTRODUCTION

Modern magnetic resonance imaging (MRI) studies are usually multimodal, in the sense that multiple types of MRI measurements, for example, anatomical MRI, functional MRI (fMRI), and diffusion tensor imaging (DTI), are collected on the same subjects in a single scanner. Integration of these diverse, but scientifically complementary, imaging measurements would strengthen our understanding of brain structure and function, as well as their associations with neurological disorders (see Uludağ and Roebroeck, 2014, for a review).

In this article, we study multimodal neuroimaging data integration in the context of mediation analysis. Mediation analysis seeks to identify and explain the mechanism, or pathway, that underlies an observed relationship between exposure and outcome variables, through the inclusion of an intermediary variable, known as a mediator. Such an analysis often represents a starting point for mechanistic studies. It was originally developed in psychometrics (Baron and Kenny, 1986), and has been extensively studied in statistics (see, eg, Pearl, 2003; Wang et al., 2013; Huang and Pan, 2016, among many others). Recently, mediation analysis has received attention in neuroimaging analysis to understand the roles of brain structure and function as possible mediators between the exposure and some cognitive or behavioral outcome (Caffo et al., 2007; Atlas et al., 2010; Lindquist, 2012; Zhao and Luo, 2016; Chén et al., 2017; Zhao et al., 2020). However, all existing work focused on a single imaging modality as mediators. By contrast, we aim to study the problem of mediation analysis based on multiple high-dimensional imaging modalities.

Our motivation is an investigation of how brain structure and function mediate the relationship between sex and language processing behavior. Numerous studies have observed sex differences in language processing, as well as different structure and function of brain regions related to language processing (Shaywitz et al., 1995; Pinker, 2007). We study a dataset from the Human Connectome Project, where each participant took a picture vocabulary test, a measure of language processing behavior, and both a DTI and a resting-state fMRI imaging scan. DTI is an MRI technique that measures the diffusion of water molecules along white matter fiber bundles, which in turn provides an assessment of brain structural connectivity. fMRI is an MRI technique that measures blood oxygen level as a surrogate of brain neural activity, which in turn assesses brain functional connectivity. Our goal is to develop a new statistical model to integrate brain structural and functional connectivity to identify brain pathways that associate sex with language processing. It should be emphasized that, although motivated by a multimodal neuroimaging application, our method is equally applicable to a variety of multimodal data, such as multi-omics data (Richardson et al., 2016).

The rest of the article is organized as follows. Section 2 presents our multimodal mediation model. Section 3 develops the penalized estimation algorithm, and Section 4 studies the asymptotic properties. Section 5 presents the simulations, and Section 6 revisits our motivating multimodal brain imaging example. Section 7 concludes the paper with a discussion. The supporting information collects the technical proofs and additional results.

2. MODEL

We begin with a description of our multimodal high-dimensional mediation model. Let X denote the exposure variable, and Y the outcome. In our example, X is the participant’s sex, and Y is the picture vocabulary test score. Let $M_1={\left(M_{11},\dots ,M_{1p_1}\right)}^{\top }$ denote the set of potential mediators from the first modality, and $M_2={\left(M_{21},\dots ,M_{2p_2}\right)}^{\top }$ the set of potential mediators from the second modality. In our case, the structural connectivity measures from DTI are M₁, and the functional connectivity measures from fMRI are M₂, with p₁ = 531 and p₂ = 917, respectively. The order of the two sets of mediators is determined by the knowledge that structural connectivity shapes and constrains functional connectivity in adults. More specifically, Hebb’s law portrays that brain regions that communicate frequently are more likely the consequences of direct structural connections (Hebb, 2005). In addition, brain structural connectivity regulates the dynamics of cortical circuits and systems captured by functional connectivity (Sporns, 2007). For n independent and identically distributed observations, we consider the model,

$$\begin{gathered} M_{1j}=X{\beta }_j+{\epsilon }_j,j=1,\dots ,p_1, \\ {M}_{2k}=X{\zeta }_k+\sum _{j=1}^{p_1}{M}_{1j}{\lambda }_{jk}+{\mathit{\vartheta }}_k,k=1,\dots ,p_2, \\ Y=X\delta +\sum _{j=1}^{p_1}{M}_{1j}{\theta }_j+\sum _{k=1}^{p_2}{M}_{2k}{\pi }_k+\xi , \end{gathered}$$ (1)

where $X\in {ℝ}^n$ is the exposure vector stacking all n samples, $Y\in {ℝ}^n$ is the response vector, and $M_{1j}\in {ℝ}^n$, $M_{2k}\in {ℝ}^n$ are two mediator vectors. Moreover, β_j, ζ_k, λ_jk, δ, θ_j, and π_k are scalar coefficients, and ${\epsilon }_j\in {ℝ}^n$, ${\mathit{\vartheta }}_k\in {ℝ}^n$, $\xi \in {ℝ}^n$ are normal random errors with zero means, j = 1, … , p₁, k = 1, … , p₂. For simplicity, we assume the data are centered, and thus drop the intercept terms. We further assume the error term, ε_j, is independent of X, M_2k, ϑ_k is independent of X, M_1j, and ξ is independent of X, M_1j, M_2k. Model (1) is essentially a linear structural equation model (Pearl, 2003); a graphic illustration is given in Figure 1.

FIGURE 1.

The diagram of the proposed two-modality mediation model: X is the exposure variable, Y is the outcome variable, $M_1={\left(M_{11},\dots ,M_{1p_1}\right)}^{\top }$ collects the mediators from the first modality, and $M_2={\left(M_{21},\dots ,M_{2p_2}\right)}^{\top }$ collects the mediators from the second modality.

The key of Model (1) is that it does not delineate the underlying relationships among the mediators within the same modality. Instead, it focuses on the roles of the mediators between the two sets of modalities on the treatment-outcome pathway. As is shown later, Model (1) does not assume the mediators within the same modality to be conditionally independent, but instead uses correlated errors to account for dependence structures among the mediators. On the other hand, we employ a least-squares type optimization to estimate the mediation effects, and do not estimate the underlying error structures. By contrast, an alternative model to (1) is to fully account for all pathways among the mediators within each modality. More details of this model is given in Section A.1 of the Supporting Information. The main issue with this alternative, however, is that it requires the knowledge of the mechanistic ordering of the mediators within each modality, which is rarely known in practice.

Our model is motivated by Zhao and Luo (2016), but is also distinct in several ways. First, while Zhao and Luo (2016) tackled single-modality pathway analysis, we consider the multimodality scenario, for which there is no existing solution. Given the strong demand for this type of multimodal mediation analysis in brain imaging and many other applications, we offer a timely solution to this important family of scientific problems. Second, even though a straightforward extension conceptually, our multimodal pathway analysis is technically much more involved than the single-modality analysis of Zhao and Luo (2016). In addition, we numerically compare the two methods, and show the superior performance of our method, both through simulations and an application to a multimodal neuroimaging dataset.

Stacking the terms in (1) together, we have $\beta ={\left({\beta }_1,\dots ,{\beta }_{p_1}\right)}_{1\times p_1}$, $\zeta ={\left({\zeta }_1,\dots ,{\zeta }_{p_2}\right)}_{1\times p_2}$, $\Lambda ={\left({\lambda }_{jk}\right)}_{p_1\times p_2}$, $\theta ={\left({\theta }_1,\dots ,{\theta }_{p_1}\right)}_{p_1\times 1}^{\top }$, $\pi ={\left({\pi }_1,\dots ,{\pi }_{p_2}\right)}_{p_2\times 1}^{\top }$, $M_1=\left(M_{11},\dots ,M_{1p_1}\right)\in {ℝ}^{n\times p_1}$, $M_2=\left(M_{21},\dots ,M_{2p_2}\right)\in {ℝ}^{n\times p_2}$, $\epsilon =\left({\epsilon }_1,\dots ,{\epsilon }_{p_1}\right)\in {ℝ}^{n\times p_1}$, and $\mathit{\vartheta }=\left({\mathit{\vartheta }}_1,\dots ,{\mathit{\vartheta }}_{p_2}\right)\in {ℝ}^{n\times p_2}$. We can then rewrite Model (1) in a matrix form,

$$\left(\begin{array}{lll}{M}_1\hfill & M_2\hfill & Y\hfill \end{array}\right)=\left(\begin{array}{lll}X\hfill & M_1\hfill & M_2\hfill \end{array}\right)\left(\begin{array}{lll}\beta \hfill & \zeta \hfill & \mathit{\delta }\hfill \\ 0\hfill & \Lambda \hfill & \theta \hfill \\ 0\hfill & 0\hfill & \pi \hfill \end{array}\right)+\left(\begin{array}{lll}\epsilon \hfill & \mathit{\vartheta }\hfill & \xi \hfill \end{array}\right),$$ (2)

where $\text{vec}(\epsilon )~\mathcal{N}\left(0,{\Sigma }_1\otimes I_n\right)$, $\text{vec}(\mathit{\vartheta })~\mathcal{N}\left(0,{\Sigma }_2\otimes I_n\right)$, and $\xi ~\mathcal{N}\left(0,{\sigma }^2{I}_n\right)$, ${\Sigma }_1\in {ℝ}^{p_1\times p_1}$ and ${\mathbf{\Sigma }}_2\in {ℝ}^{p_2\times p_2}$ are the covariance matrices, I_n is the identity matrix, and ⊗ is the Kronecker product operator. The terms β and ζ summarize the overall treatment effect on the mediators M₁ and M₂, respectively. The (j, k)th element in Λ reveals the overall effect of M_1j on M_2k regardless of the underlying relationships among the mediators in M₂. The error terms in ε and ϑ are dependent, due to the influences among the mediators. Therefore, even though Model (1) does not explicitly model the relationships among the mediators within the same modality, it encapsulates the dependence among the mediators through the correlations between the error terms. Next, we formally define various pathway effects between X and Y under Model (1). Additional assumptions, as discussed in Section A.2 of the Supporting Information, must be made to guarantee a causal interpretation to pathway effects.

Definition 1. Under Model (1), considering two exposure conditions X = x and X = x*, we define the following pathway effects of X on the outcome Y.

1. The indirect pathway effect of X through path X → M_1j → Y is ${\text{IE}}_j^1\left(x,x^{*}\right)={\beta }_j{\theta }_j\left(x-x^{*}\right)$, j = 1, … , p₁. The total indirect pathway effect of X through M₁, but not through M₂, is ${\text{IE}}^1\left(x,x^{*}\right)={\sum }_{j=1}^{p_1}{\beta }_j{\theta }_j\left(x-x^{*}\right)$.

2. The indirect pathway effect of X through path X → M_2k → Y is ${\text{IE}}_k^2\left(x,x^{*}\right)={\zeta }_k{\pi }_k\left(x-x^{*}\right)$, k = 1, … , p₂. The total indirect pathway effect of X through M₂, but not through M₁, is ${\text{IE}}^2\left(x,x^{*}\right)={\sum }_{k=1}^{p_2}{\zeta }_k{\pi }_k\left(x-x^{*}\right)$.

3. The indirect pathway effect of X through path X → M_1j → M_2k → Y is ${\text{IE}}_{jk}^{1,2}\left(x,x^{*}\right)={\beta }_j{\lambda }_{jk}{\pi }_k\left(x-x^{*}\right)$, j = 1, … , p₁, k = 1, … , p₂. The total indirect effect of X through both M₁ and M₂ is ${\text{IE}}^{1,2}={\sum }_{j=1}^{p_1}{\sum }_{k=1}^{p_2}{\beta }_j{\lambda }_{jk}{\pi }_k\left(x-x^{*}\right)$.

4. The direct effect of X is DE(x, x*) = δ(x − x*).

By Definition 1, the total effect (TE) of X on Y is decomposed as the sum of the direct effect (DE) and the total indirect effect (IE):

$$\text{TE}\left(x,x^{*}\right)=\text{DE}\left(x,x^{*}\right)+\text{IE}\left(x,x^{*}\right)\equiv \text{DE}\left(x,x^{*}\right)+{\text{IE}}^1\left(x,x^{*}\right)+{\text{IE}}^2\left(x,x^{*}\right)+{\text{IE}}^{1,2}\left(x,x^{*}\right).$$ (3)

3. ESTIMATION

Our goal is to estimate the pathway effects defined in Definition 1 under Model (1). To achieve this, we define the objective function,

$$\mathcal{l}(\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta )=\text{trace}\left\{{\left(M_1-X\mathit{\beta }\right)}^{\top }\left(M_1-X\mathit{\beta }\right)\right\}+\text{trace}\left\{{\left(M_2-X\mathit{\zeta }-M_1\Lambda \right)}^{\top }\left(M_2-X\mathit{\zeta }-M_1\Lambda \right)\right\}+{\left(Y-X\delta -M_1\theta -M_2\pi \right)}^{\top }\left(Y-X\delta -M_1\theta -M_2\pi \right).$$

This objective function in effect sets Σ₁ and Σ₂ to be identity matrices. However, this simplification would not affect the consistency of our least-squares type estimators, as long as all the variables are standardized (to the unit scale, White, 1980).

Next, we introduce a series of penalty functions. One challenge is that the number of the mediators far exceeds the sample size. In our example, p₁ = 531, p₂ = 917, while n = 136. We introduce sparsity in the pathway effects through ${\mathcal{l}}_1$ regularization. Sparsity assumptions are commonly employed in many applications, including neuroscience. This can be justified as human brain anatomy is economically organized (Bullmore and Sporns, 2012), and information processing is often highly clustered and concentrated (Bassett and Bullmore, 2017). We thus consider the following regularized optimization problem

$${\text{minimize}}_{\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta }\mathcal{l}(\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta ),$$

$$\text{ subject to }\sum _{j=1}^{p_1}|{\beta }_j{\theta }_j|\le t_1,\sum _{k=1}^{p_2}|{\zeta }_k{\pi }_k|\le t_2,$$

$$\sum _{j=1}^{p_1}\sum _{k=1}^{p_2}|{\beta }_j{\lambda }_{jk}{\pi }_k|\le t_3,|\delta |\le t_4,$$

where t₁, t₂, t₃, t₄ ≥ 0 are the regularization parameters. The first three ${\mathcal{l}}_1$-penalty functions are not convex. The next lemma presents a convex relaxation. We observe that the mediation effect through both M₁ and M₂ is defined as a three-way product, β_jλ_jkπ_k. Since β_j and π_k are already regularized in the pathway effect through M₁, that is, β_jθ_j, and that through M₂, that is, ζ_kπ_k, respectively, it suffices to regularize λ_jk alone.

Lemma 1.For any ν₁, $v_2\in ℝ\ge 1/2$,

$$\sum _{j=1}^{p_1}\left\{|{\beta }_j{\theta }_j|+v_1\left({\beta }_j^2+{\theta }_j^2\right)\right\},\sum _{k=1}^{p_2}\left\{|{\zeta }_k{\pi }_k|+v_2\left({\zeta }_k^2+{\pi }_k^2\right)\right\},\mathit{\text{ and }}\sum _{j=1}^{p_1}\sum _{k=1}^{p_2}|{\lambda }_{jk}|$$

are convex functions of {β, θ}, {ζ, π}, and Λ, respectively. For any t₁, t₂, $t_3\in ℝ\ge 0$, there exist r₁, r₂, $r_3\in ℝ\ge 0$, such that

$$\{\begin{array}{l}\sum _{j=1}^{p_1}\left\{|{\beta }_j{\theta }_j|+v_1\left({\beta }_j^2+{\theta }_j^2\right)\right\}\le r_1\hfill \\ \sum _{k=1}^{p_2}\left\{|{\zeta }_k{\pi }_k|+v_2\left({\zeta }_k^2+{\pi }_k^2\right)\right\}\le r_2\Rightarrow \hfill \\ \sum _{j=1}^{p_1}\sum _{k=1}^{p_2}|{\lambda }_{jk}|\le r_3\hfill \end{array}\{\begin{array}{l}\sum _{j=1}^{p_1}|{\beta }_j{\theta }_j|\le t_1\hfill \\ \sum _{k=1}^{p_2}|{\zeta }_k{\pi }_k|\le t_2\hfill \\ \sum _{j=1}^{p_1}\sum _{k=1}^{p_2}|{\beta }_j{\lambda }_{jk}{\pi }_k|\le t_3\hfill \end{array}.$$

The proof of Lemma 1 is given in Section C.1 of the Supporting Information. Based on this convex relaxation, we turn to the following optimization problem,

$${\text{minimize}}_{\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta }\left\{\frac{1}{2}\mathcal{l}(\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta )+P_1(\beta ,\theta ,\zeta ,\pi )+P_2(\beta ,\theta ,\zeta ,\pi )+P_3(\Lambda ,\delta )\right\},$$ (4)

where the three penalty functions are of the form

$$P_1(\beta ,\theta ,\zeta ,\pi )={\kappa }_1\left[\sum _{j=1}^{p_1}\left\{|{\beta }_j{\theta }_j|+v_1\left({\beta }_j^2+{\theta }_j^2\right)\right\}\right]+{\kappa }_2\left[\sum _{k=1}^{p_2}\left\{|{\zeta }_k{\pi }_k|+v_2\left({\zeta }_k^2+{\pi }_k^2\right)\right\}\right],$$

$$P_2(\beta ,\theta ,\zeta ,\pi )={\mu }_1\sum _{j=1}^{p_1}\left(|{\beta }_j|+|{\theta }_j|\right)+{\mu }_2\sum _{k=1}^{p_2}\left(|{\zeta }_k|+|{\pi }_k|\right),$$

$$P_3(\Lambda ,\delta )={\kappa }_3\sum _{j=1}^{p_1}\sum _{k=1}^{p_2}|{\lambda }_{jk}|+{\kappa }_4|\delta |,$$

with ν₁, ν₂ ≥ 1∕2, κ₁, κ₂, κ₃, κ₄ ≥ 0, and μ₁, μ₂ ≥ 0 as the tuning parameters. Here ν₁ and ν₂ control the level of convexity relaxation, κ₁, κ₂, κ₃, and κ₄ control the level of penalty on various pathway effects, and μ₁ and μ₂ control the level of sparsity of individual parameters. We note that the tuning parameters κ₁, κ₂, κ₃, μ₁, and μ₂ can also vary with j and k, similarly as in the adaptive Lasso (Zou, 2006). For simplicity, they are kept the same across j = 1, … , p₁ and k = 1, … , p₂. In (4), the penalties P₁ and P₃ work together as a convex relaxation of the ${\mathcal{l}}_1$-regularization on path effects to reduce the estimation bias. The penalty P₂ adds an additional ${\mathcal{l}}_1$-regularization on individual parameters to improve the selection accuracy.

The objective function in (4) consists of a differentiable loss function, $\mathcal{l}/2$, and an indifferentiable regularization function, (P₁ + P₂ + P₃). We next develop an alternating direction method of multipliers (ADMM; Boyd et al., 2011) to solve (4). The ADMM form of the optimization problem (4) is

$${\text{minimize}}_{\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta ,\tilde{\mathit{\beta }},\tilde{\mathit{\theta }},\tilde{\mathit{\zeta }},\tilde{\mathit{\pi }}}\frac{1}{2}\mathcal{l}(\beta ,\theta ,\zeta ,\pi ,\Lambda ,\delta )+P_1(\tilde{\mathit{\beta }},\tilde{\mathit{\theta }},\tilde{\mathit{\zeta }},\tilde{\mathit{\pi }})+P_2(\tilde{\mathit{\beta }},\tilde{\mathit{\theta }},\tilde{\mathit{\zeta }},\tilde{\mathit{\pi }})+P_3(\Lambda ,\delta ),$$

$$\text{subject to }\beta =\tilde{\mathit{\beta }},\theta =\tilde{\mathit{\theta }},\zeta =\tilde{\mathit{\zeta }},\pi =\tilde{\mathit{\pi }},$$

where $\tilde{\mathit{\beta }}\in {ℝ}^{1\times p_1}$, $\tilde{\mathit{\theta }}\in {ℝ}^{p_1}$, $\tilde{\mathit{\zeta }}\in {ℝ}^{1\times p_2}$, and $\tilde{\mathit{\pi }}\in {ℝ}^{p_2}$ are the newly introduced parameters. This way, the objective function is decomposed into a convex loss function plus

Algorithm 1.

The optimization algorithm for (5)

Input: (X, M1, M2, Y).

1: initialization: $\left\{{\beta }^{(0)},{\theta }^{(0)},{\zeta }^{(0)},{\pi }^{(0)},{\Lambda }^{(0)},{\delta }^{(0)},{\tilde{\mathit{\beta }}}^{(0)},{\tilde{\mathit{\theta }}}^{(0)},{\tilde{\mathit{\zeta }}}^{(0)},{\tilde{\mathit{\pi }}}^{(0)},{\tau }_1^{(0)},{\tau }_2^{(0)},{\tau }_3^{(0)},{\tau }_4^{(0)}\right\}.$

2: repeat for iteration s = 0,1,2, …

3:  $\text{update }{\beta }^{(s+1)}={\left(X^{\top }X+\rho \right)}^{-1}\left\{X^{\top }{M}_1-{\tau }_1^{(s)\top }+\rho {\tilde{\mathit{\beta }}}^{(s)}\right\}.$

4:  $\text{update }{\theta }^{(s+1)}={\left(M_1^{\top }{M}_1+\rho I\right)}^{-1}\left[M_1^{\top }\left\{Y-X{\delta }^{(s)}-M_2{\mathit{\pi }}^{(s)}\right\}-{\tau }_2^{(s)}+\rho {\tilde{\mathit{\theta }}}^{(s)}\right].$

5:  $\text{update }{\zeta }^{(s+1)}={\left(X^{\top }X+\rho \right)}^{-1}\left[X^{\top }\left\{M_2-M_1{\Lambda }^{(s)}\right\}-{\tau }_3^{(s)\top }+\rho {\tilde{\mathit{\zeta }}}^{(s)}\right].$

6:  $\text{update }{\pi }^{(s+1)}={\left(M_2^{\top }{M}_2+\rho I\right)}^{-1}\left[M_2^{\top }\left\{Y-X{\delta }^{(s)}-M_1{\theta }^{(s+1)}\right\}-{\mathit{\tau }}_4^{(s)}+\rho {\tilde{\mathit{\pi }}}^{(s)}\right].$

7:  $\text{update }{\delta }^{(s+1)}={\left(X^{\top }X\right)}^{-1}\text{Soft}\left[X^{\top }\left\{Y-M_1{\theta }^{(s+1)}-M_2{\pi }^{(s+1)}\right\},{\kappa }_4\right].$

8:  for k = 1 to p2 do update ${\Lambda }_k^{(s+1)}$ by solving (6). end for

9:  for j = 1 to p1 do update $\left\{{\tilde{\beta }}_j^{(s+1)},{\tilde{\theta }}_j^{(s+1)}\right\}$ by solving (7). end for

10:  for k = 1 to p2 do update $\left\{{\tilde{\zeta }}_k^{(s+1)},{\tilde{\pi }}_k^{(s+1)}\right\}$ by solving (8). end for

11:  $\text{update }{\tau }_r^{(s+1)}={\tau }_r^{(s)}+\rho h_r\left\{{{\rm Y}}^{(s+1)},{\widetilde{\mathbf{{\rm Y}}}}^{(s+1)}\right\},r=1,\dots ,4.$

12: until the objective function converges.

Output: $(\widehat{\mathit{\beta }},\widehat{\mathit{\theta }},\widehat{\mathit{\xi }},\widehat{\mathit{\pi }},\widehat{\Lambda },\widehat{\delta }).$

a convex, but nondifferentiable regularization function, which we handle separately. Let ϒ = (β, θ, ζ, π) and $\widetilde{\mathbf{{\rm Y}}}=(\tilde{\mathit{\beta }},\tilde{\mathit{\theta }},\tilde{\mathit{\zeta }},\tilde{\mathit{\pi }})$; the augmented Lagrangian function to enforce the constraints is

$$\frac{1}{2}\mathcal{l}({\rm Y},\Lambda ,\delta )+P_1(\widetilde{\mathbf{{\rm Y}}})+P_2(\widetilde{\mathbf{{\rm Y}}})+P_3(\Lambda ,\delta )+\sum _{r=1}^4\left(\langle h_r({\rm Y},\widetilde{\mathbf{{\rm Y}}}),{\mathit{\tau }}_r\rangle +\frac{\rho }{2}\Vert h_r{({\rm Y},\widetilde{{\rm Y}})\Vert }_2^2\right),$$ (5)

where $h_1(\mathbf{{\rm Y}},\widetilde{\mathbf{{\rm Y}}})=\beta -\tilde{\beta }$, $h_2(\mathbf{{\rm Y}},\widetilde{\mathbf{{\rm Y}}})=\theta -\tilde{\mathit{\theta }}$, $h_3(\mathbf{{\rm Y}},\widetilde{\mathbf{{\rm Y}}})=\zeta -\tilde{\mathit{\zeta }}$, $h_4(\mathbf{{\rm Y}},\widetilde{\mathbf{{\rm Y}}})=\pi -\tilde{\mathit{\pi }}$, τ₁, ${\tau }_2\in {ℝ}^{p_1}$, τ₃, ${\tau }_4\in {ℝ}^{p_2}$, and ρ > 0 is the augmented Lagrangian parameter. We propose to update the parameters in (5) iteratively, and summarize our estimation procedure in Algorithm 1.

A few remarks are in order. The explicit forms of Steps 3 to 6 of Algorithm 1 are derived in Section B of the Supporting Information. In Step 7, Soft(a, b) = sgn(a) max{|a| − b, 0} is the soft-thresholding function. Step 8 is to update Λ, one column at a time. Its kth column, k = 1, … , p₂, can be obtained by

$${\text{minimize}}_{{\Lambda }_k\in {ℝ}^{p_1}}\frac{1}{2}{\Vert M_{2k}-X{\zeta }_k^{(s+1)}-M_1{\Lambda }_k\Vert }_2^2+{\kappa }_3{\Vert {\Lambda }_k\Vert }_1.$$ (6)

This is a standard Lasso problem with $\left\{M_{2k}-X{\zeta }_k^{(s+1)}\right\}$ as the response and M₁ as the predictor. Step 9 is to update $(\tilde{\mathit{\beta }},\tilde{\mathit{\theta }})$, also one column pair at a time, and the jth column pair $\left({\tilde{\beta }}_j,{\tilde{\theta }}_j\right)$, j = 1, … , p₁, can be obtained by

$${\text{minimize}}_{\left({\tilde{\beta }}_j,{\tilde{\theta }}_j\right)}v\left\{{\tilde{\beta }}_j,{\tilde{\theta }}_j;{\kappa }_1,{\mu }_1,2{\kappa }_1{v}_1+\rho ,2{\kappa }_1{v}_1+\rho ,{\tau }_{1j}^{(s)}+\rho {\beta }_j^{(s+1)},{\tau }_{2j}^{(s)}+\rho {\theta }_j^{(s+1)}\right\}.$$ (7)

Similarly, Step 10 is to update $\left(\tilde{\mathit{\zeta }},{\tilde{\mathit{\pi }}}_k\right)$, one column pair at a time, and the kth column pair $\left({\tilde{\mathit{\zeta }}}_k,{\tilde{\mathit{\pi }}}_k\right)$, k = 1, … , p₂, can be obtained by

$${\text{minimize}}_{\left({\tilde{\zeta }}_k,{\tilde{\pi }}_k\right)}v\left\{{\tilde{\zeta }}_k,{\tilde{\pi }}_k;{\kappa }_2,{\mu }_2,2{\kappa }_2{v}_2+\rho ,2{\kappa }_2{v}_2+\rho ,{\tau }_{3k}^{(s)}+\rho {\zeta }_k^{(s+1)},{\tau }_{4k}^{(s)}+\rho {\pi }_k^{(s+1)}\right\}.$$ (8)

In both (7) and (8), the function (a₁, a₂) is of the form

$$v\left(a_1,a_2;b_1,b_2,b_3,b_4,b_5,b_6\right)=b_1|a_1{a}_2|+b_2|a_1|+b_2|a_2|+\frac{1}{2}{b}_3{a}_1^2+\frac{1}{2}{b}_4{a}_2^2-b_5{a}_1-b_6{a}_2.$$

Its optimization has a closed-form solution (see Zhao and Luo, 2016, Lemma 3.2).

Our method involves a number of tuning parameters. For ν₁ and ν₂, our simulations have found that the final estimators are not overly sensitive to their values (see Section D.2 of the Supporting Information). The same phenomenon was observed in Zhao and Luo (2016). For ρ, the augmented Lagrangian parameter, it can be fixed at a constant level (Boyd et al., 2011). We thus fix ν₁ = ν₂ = 2 and ρ = 1. For (κ₁, κ₂, κ₃, κ₄, μ₁, μ₂), for simplicity, we set ${\kappa }_1={\kappa }_2={\kappa }_3={\kappa }_4=\tilde{\kappa }$, and ${\mu }_1={\mu }_2=\tilde{\mu }$. A grid search is then run to minimize a modified Bayesian information criterion (BIC) to select $\tilde{\kappa }$ and $\tilde{\mu }$,

$$\text{BIC}=-2\text{ log }L(\widehat{\mathit{\beta }},\widehat{\mathit{\theta }},\widehat{\mathit{\zeta }},\widehat{\mathit{\pi }},\widehat{\mathbf{\Lambda }},\widehat{\delta })+\text{log}(n)\left(|{\widehat{\mathcal{A}}}_1|+|{\widehat{\mathcal{A}}}_2|+|{\widehat{\mathcal{A}}}_3|\right),$$ (9)

where the estimators are obtained under a given set of tuning parameters, ${\widehat{\mathcal{A}}}_1=\left\{j:{\widehat{\beta }}_j{\widehat{\theta }}_j\ne 0\right\}$, ${\widehat{\mathcal{A}}}_2=\left\{k:{\widehat{\zeta }}_k{\widehat{\pi }}_k\ne 0\right\}$, ${\widehat{\mathcal{A}}}_3=\left\{(j,k):{\widehat{\beta }}_j{\widehat{\lambda }}_{jk}{\widehat{\pi }}_k\ne 0\right\}$, and $|{\widehat{\mathcal{A}}}_j|$ is the cardinality of set ${\widehat{\mathcal{A}}}_j$.

4. THEORY

We next study the asymptotic properties of our proposed estimator. To simplify the presentation and focus on the mediation pathways, assume that the direct effect, δ, is known, and let V = Y − Xδ. Let Θ* = (β*, θ*, ζ*, π*, Λ*) denote the true parameters, and $\widehat{\mathbf{\Theta }}=(\widehat{\mathit{\beta }},\widehat{\mathit{\theta }},\widehat{\zeta },\widehat{\mathit{\pi }},\widehat{\mathbf{\Lambda }})$ the global minimizer of the optimization in (4). Let ${\mathit{\varsigma }}^{*}={\Lambda }^{*}{\pi }^{*}\in {ℝ}^{p_1}$. Further, let ${\mathcal{S}}_1=\left\{j:{\beta }_j^{*}\ne 0\right\}$, ${\mathcal{S}}_2=\left\{j:{\theta }_j^{*}\ne 0\right\}$, ${\mathcal{S}}_3=\left\{k:{\zeta }_k^{*}\ne 0\right\}$, ${\mathcal{S}}_4=\left\{k:{\pi }_k^{*}\ne 0\right\}$, and ${\mathcal{S}}_5=\left\{j:{\varsigma }_j^{*}\ne 0\right\}$ denote the support of β*, θ*, ζ*, π*, and ς*, respectively, and $s_l=\left|{\mathcal{S}}_l\right|$ the cardinality of set ${\mathcal{S}}_l$, l = 1, … , 5. Consider a set of regularity conditions.

• (C1)The distribution of X has a finite variance, and |X_i| ≤ c₀ almost surely, i = 1, … , n.
• (C2)The penalty functions, evaluated at the true parameters, are bounded. That is, for P₁, ${\sum }_{j=1}^{p_1}\left\{|{\beta }_j^{*}{\theta }_j^{*}|+v_1\left({\beta }_j^{*2}+{\theta }_j^{*2}\right)\right\}\le c_{11}$, ${\sum }_{k=1}^{p_2}\left\{|{\zeta }_k^{*}{\pi }_k^{*}|+v_2\left({\zeta }_k^{*2}+{\pi }_k^{*2}\right)\right\}\le c_{12}$; for P₂, ${\sum }_{j=1}^{p_1}\left(|{\beta }_j^{*}|+|{\theta }_j^{*}|\right)\le c_{21}$, ${\sum }_{k=1}^{p_2}\left(|{\zeta }_k^{*}|+|{\pi }_k^{*}|\right)\le c_{22}$; and for P₃, ${\sum }_{j=1}^{p_1}{\sum }_{k=1}^{p_2}|{\lambda }_{jk}^{*}|\le c_3$.
• (C3)All the entries of the error variance terms are bounded by c₄.

Condition (C1) is a standard regularity condition on the design matrix in high-dimensional regression settings; when X is binary or categorical, (C1) is satisfied. (C2) regulates the sparsity levels of β*, θ*, ζ*, and π*, and (C3) is the finite variance condition on the model errors, both of which are again common in the literature.

We evaluate the accuracy of our estimator by the mean squared prediction error

$$\text{MSPE}=\frac{1}{n}\sum _{i=1}^n{\left({\widehat{V}}_i-V_i^{*}\right)}^2,$$

where V = Y − Xδ, and ${\widehat{V}}_i$ and $V_i^{*}$ are the corresponding values under the estimated parameter $\widehat{\Theta }$ and the true parameter Θ*, respectively, for subject i, i = 1, … , n. The predicted pathway effects under our multimodal mediation model are defined as follows.

Definition 2. For a treatment condition X = x, define

1. the predicted outcome through M₁, but not through M₂, is ${\widehat{V}}_1=x\widehat{\mathit{\beta }}\widehat{\mathit{\theta }}=x\left({\sum }_{j=1}^{p_1}{\widehat{\beta }}_j{\widehat{\theta }}_j\right)$;

2. the predicted outcome through M₂, but not through M₁, is ${\widehat{V}}_2=x\widehat{\mathit{\zeta }}\widehat{\mathit{\pi }}=x\left({\sum }_{k=1}^{p_2}{\widehat{\zeta }}_k{\widehat{\pi }}_k\right)$.

3. the prediction through both M₁ and M₂ is ${\widehat{V}}_3=x\widehat{\mathit{\beta }}\widehat{\mathbf{\Lambda }}\widehat{\mathit{\pi }}=x\left({\sum }_{j=1}^{p_1}{\sum }_{k=1}^{p_2}{\widehat{\beta }}_j{\widehat{\lambda }}_{jk}{\widehat{\pi }}_k\right)$;

4. the total prediction of $\widehat{V}={\widehat{V}}_1+{\widehat{V}}_2+{\widehat{V}}_3$.

Theorem 1. Suppose regularity conditions (C1)-(C3) hold. Assume $\mathbb{E}{X}^2=c_5>0$. Then,

$$\mathbb{E}(\text{MSPE})\le 2c_0{c}_4\left\{c_{11}{c}_{21}\sqrt{\frac{2\text{ log}\left(2p_1\right)}{n}}+c_{12}{c}_{22}\sqrt{\frac{2\text{ log}\left(2p_2\right)}{n}}+c_3\sqrt{c_{11}{c}_{12}/v_1{v}_2}\left(1+c_3{c}_{22}\right)\sqrt{\frac{2\text{ log}\left(2p_1\right)}{n}}\right\},$$

$${\Vert \widehat{\mathit{\beta }}\widehat{\mathit{\theta }}-{\beta }^{*}{\mathit{\theta }}^{*}\Vert }_2^2\le \frac{1}{c_5}\left\{8c_{11}^2{c}_0^2\sqrt{\frac{2\text{ log}(2)}{n}}+2c_0{c}_4{c}_{11}{c}_{21}\sqrt{\frac{2\text{ log}\left(2p_1\right)}{n}}\right\},$$

$${\Vert \widehat{\mathit{\zeta }}\widehat{\mathit{\pi }}-{\zeta }^{*}{\pi }^{*}\Vert }_2^2\le \frac{1}{c_5}\left\{8c_{12}^2{c}_0^2\sqrt{\frac{2\text{ log}(2)}{n}}+2c_0{c}_4{c}_{12}{c}_{22}\sqrt{\frac{2\text{ log}\left(2p_2\right)}{n}}\right\},$$

$${\Vert \widehat{\mathit{\beta }}\widehat{\mathbf{\Lambda }}\widehat{\mathit{\pi }}-{\beta }^{*}{\Lambda }^{*}{\pi }^{*}\Vert }_2^2\le \frac{1}{c_5}\left\{8c_3^2\left(c_{11}{c}_{12}/v_1{v}_2\right)c_0^2\sqrt{\frac{2\text{ log}(2)}{n}}+2c_0{c}_4{c}_3\sqrt{c_{11}{c}_{12}/v_1{v}_2}\left(1+c_3{c}_{22}\right)\sqrt{\frac{2\text{ log}\left(2p_1\right)}{n}}\right\}.$$

The proof is given in Section C.2 of the Supporting Information. Theorem 1 shows that the mean squared prediction error converges. In addition, all three types of total pathway effect estimators are consistent in ${\mathcal{l}}_2$-norm. Particularly, the convergence rate of the total indirect effect IE¹ and IE² are $\sqrt{\text{log}\left(p_1\right)/n}$ and $\sqrt{\text{log}\left(p_2\right)/n}$, respectively, which are consistent with the rate under only one set of mediators as studied in Zhao and Luo (2016). When considering the indirect effect through both M₁ and M₂, ς = Λπ summarizes the post-M₁ pathway effect, and the problem degenerates to the case with a single set of p₁ mediators. As such, the convergence rate is $\sqrt{\text{log}\left(p_1\right)/n}$, and depends on the number of nonzero elements in ς.

5. SIMULATIONS

We next investigate the finite-sample performance. We first study the method under different mediator dimensions and sample sizes. We then compare with the method of Zhao and Luo (2016) under different mediation scenarios. We also study the performance under different sparsity levels, and report those results in Section D.3 of the Supporting Information.

First, the data are simulated following Model (1), where the detailed data generation scheme is reported in Section D.1 of the Supporting Information. Consider two mediator dimension sizes p₁ = 20, p₂ = 30, with the sparsity level set at 0.9, and p₁ = p₂ = 100, with the sparsity level at 0.95. Also consider two sample sizes, n = 50 and n = 500. We employ a similar tuning strategy as in Zou and Hastie (2005) to tune $\tilde{\kappa }$ and $\tilde{\mu }/\tilde{\kappa }$, and select the tuning parameters that minimize the modified BIC criterion in (9). Table 1 reports the average results based on 200 data replications. The evaluation criteria include the estimation bias and mean squared error of the estimated total indirect effect, plus the sensitivity and specificity of the identified pathways with nonzero effects. From this table, we see that the proposed method yields a competitive performance in the finite-sample setting, and the performance improves as the sample size increases, complying with our theory.

TABLE 1.

Performance under varying mediator dimensions and sample sizes Note. Reported are the estimation bias and mean squared error (MSE) of the total indirect effect, and the sensitivity and specificity, as well as the standard error (SE) of the measures, of the pathway selection

Mediator dimension	Sample size	Bias	MSE	Sensitivity (SE)	Specificity (SE)
p1 = 20, p2 = 30	n = 50	0.361	28.06	0.912 (0.112)	0.935 (0.022)
	n = 500	0.077	3.108	1.000 (0.000)	0.989 (0.003)
p1 = 100, p2 = 100	n = 50	0.210	10.50	0.668 (0.153)	0.998 (0.001)
	n = 500	0.027	1.223	1.000 (0.000)	0.999 (0.000)

Next, we compare our multimodality pathway method with the single-modality method of Zhao and Luo (2016). Different pathway scenarios are considered in Figure 2, while fixing the mediator dimension at p₁ = p₂ = 20 and the sample size at n = 50. More data generation details are given in Section D.4. For scenario (a), only M₁ has a direct path to Y, and for (b), only M₂ has a direct path to Y. For (c), both M₁ and M₂ have paths directly to Y, while we further consider two scenarios, (c-1) with the effect size of the first modality dominating the second, and (c-2) vice versa. For the single-modality method, we simply concatenate the two modalities of mediators as $M=\left(M_1,M_2\right)\in {ℝ}^{n\times \left(p_1+p_2\right)}$. Table 2 reports the average results based on 200 data replications. From this table, it is seen that, for scenarios (a) and (b) where only one modality of mediators has a direct path to the response, the multimodality approach works almost as well as, and sometimes even slightly better, than the single-modality approach. For scenario (b), the multimodality approach can distinguish both types of path effect, that is, X → M₁ → M₂ → Y and X → M₂ → Y, whereas the single-modality approach cannot. Generally, if these two types of path effect are of the same magnitude, but opposite signs, then the single-modality approach cannot identify this path as the total path effect is zero. However, the multimodality approach is still able to identify both paths. We consider such a case in the simulation. The single-modality approach identifies a zero-effect pathway, while the multimodality approach identifies two nonzero effect pathways. For scenario (c) where both modalities of mediators have paths directly to the response, our multimodality approach clearly outperforms the single-modality approach in both estimation and selection accuracy, especially when the effect of the first modality dominates the second. Overall, we find the multimodality approach achieves a superior numerical performance, and can identify and distinguish pathways that could have been missed by the single-modality approach.

FIGURE 2.

Comparison of single- and multimodality mediation methods under different scenarios: A, Only M₁ has a direct path to Y; B, Only M₂ has a direct path to Y; C, Both M₁ and M₂ have direct paths to Y.

TABLE 2.

Performance comparison between the single and multimodality methods Note. Reported are the estimation bias and mean squared error (MSE) of the total indirect effect, and the sensitivity and specificity, as well as the standard error (SE) of the measures, of the pathway selection

Scenario	Method	Bias	MSE	Sensitivity (SE)	Specificity (SE)
(a)	Multimodality	0.293	3.538	0.997 (0.035)	0.909 (0.030)
	Single-modality	0.613	2.208	0.995 (0.050)	0.898 (0.044)
(b)	Multimodality	0.069	23.95	0.966 (0.073)	0.935 (0.017)
	Single-modality	0.306	22.91	0.998 (0.023)	0.929 (0.043)
(c-1)	Multimodality	0.420	3.552	0.814 (0.151)	0.952 (0.011)
	Single-modality	2.524	7.874	0.341 (0.302)	0.923 (0.045)
(c-2)	Multimodality	−0.015	14.55	0.814 (0.110)	0.958 (0.012)
	Single-modality	1.219	13.23	0.638 (0.121)	0.982 (0.020)

6. A MULTIMODAL NEUROIMAGING STUDY

We revisit the motivation example in Section 1. A publicly available dataset from the recent S1200 release of the Human Connectome Project is analyzed, with n = 136 participants. The outcome of interest is a language behavior measure, which is a composite score of a picture vocabulary test evaluating vocabulary comprehension (ranging between 98.23 and 145.27 in the data). In the test, an audio recording of a word and four photographic images were presented to the participants on a screen, who responded by choosing the image that most closely matches the meaning of the word. More details are presented in Section E.1 of the Supporting Information. The two sets of mediators are DTI and resting-state fMRI measures. The DTI images is preprocessed following the pipeline of Zhang et al. (2019). From each DTI scan, a symmetric structural connectivity matrix is constructed, with nodes corresponding to the brain regions-of-interest based on the Desikan Atlas (Desikan et al., 2006), and the edges recording the number of white fiber pathways, which measures the structural connectivity between pairs of brain regions. The regions with zero connectivity in over 25% subjects are removed and the upper triangular connectivity matrix is vectorized to obtain a p₁ = 531-dimensional vector of DTI measures. The fMRI images are preprocessed following the pipeline of Glasser et al. (2013). From each fMRI scan, a symmetric functional connectivity matrix is constructed, with nodes corresponding to the brain regions based on the Harvard-Oxford Atlas of FSL (Smith et al., 2004), and the edges recording the z-transformed Pearson correlation. The focus is on the brain regions corresponding to those of DTI, vectorizing the upper triangular connectivity matrix, and obtaining a p₂ = 917-dimensional vector of fMRI measures. Moreover, following Rosenbaum (2002), the mediator measures from both DTI and fMRI, as well as the picture vocabulary outcome, are all age adjusted.

A significant sex total effect (TE 3.831, standard error 1.567, P-value 0.016) is observed under a linear model. Consider the proposed method to this data. This total effect can be decomposed following (3). Specifically, the penalized estimate of direct effect is zero, suggesting that the sex difference is fully explained by the variations in brain connectivity. The estimated total indirect effects due to the structural connectivity alone, the functional connectivity alone, and both connectivities are 3.322, 0.297, and −0.109, respectively. Figure 3 presents the identified brain pathways through both the structural and functional connectivities by the penalized estimation. The 95% confidence interval of the path effects is reported using a residual bootstrap method in Section E.3 of the Supporting Information. The single-modality method is also applied, with the results presented in Section E.2 of the Supporting Information. The multimodality method finds about twice as many paths, meanwhile identifies nearly all the paths found by the single-modality method.

FIGURE 3.

The estimated pathways with picture vocabulary test performance as the outcome (Y) when comparing male (X = 1) versus female (X = 0). The nodes in purple are diffusion-based brain structural connectivity, and nodes in orange are brain functional connectivity measures. The edges in red indicate positive effects, and the ones in blue indicate negative effects. This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

Among the pathways found by the multimodality method, one group involves the structural connectivity between the left postcentral gyrus (postCentral_L) and left superior parietal lobule (SupP_L), then through the functional connectivity between numerous brain regions. Figure 4 top path shows a subset of the pathways through the structural connectivity and functional connectivity between the left superior parietal lobule and left cuneus (Cuneus_L), and between the left precuneus (Precuneus_L) and left lingual gyrus (Lingual_L). These pathways are related to working memory. In particular, the postcentral gyrus and cuneus have been identified in visual processing. The cuneus is a mid-level visual processing area and has been found to be modulated by working memory (Salmon et al., 1996). The precuneus, as part of the default mode network, is involved in working memory, especially for tasks related to verbal processing (Wallentin et al., 2006). The lingual gyrus, located in the occipital lobe, plays an important role in visual processing. The left lingual gyrus is found activated during memorization (Kozlovskiy et al., 2014), and is involved in tasks related to naming and word recognition (Mechelli et al., 2000).

FIGURE 4.

The identified brain pathways related to working memory and language. The working memory pathway includes DTI postcentral gyrus (Left) – superior parietal lobule (Leff) → fMRI superior parietal lobule (Left) – cuneus (Left) and fMRI precuneus (Left) – lingual gyrus (Left). The language pathway includes DTI inferior temporal gyrus (Left) – precentral gyrus (Left) → fMRI middle temporal gyrus (Left) – postcentral gyrus (Left). This figure appears in color in the electronic version of this article, and any mention of color refers to that version.

The other pathway involves the structural connectivity between the left inferior temporal gyrus (IT_L) and left precentral gyrus (preCentral_L), then through the functional connectivity between the left middle temporal gyrus (MT_L) and left postcentral gyrus. The bottom path of Figure 4 shows this pathway, which is suggestive of a language mechanism. The inferior temporal gyrus, as part of the inferior longitudinal and inferior occipitofrontal fasciculi, is crucial for semantic processing and for word naming (Race et al., 2013). The middle temporal gyrus is typically viewed as part the language network (Ficek et al., 2018).

7. DISCUSSION

In this article, we propose a method of multimodal mediation analysis. It requires the ordering of the multiple modalities on the mediation pathways, but it does not require the ordering of the potential individual mediators within each modality. We define three types of indirect pathway effects, and develop a regularized estimation algorithm. Both the asymptotic and empirical behaviors of the method are studied.

The proposed method neatly fits within the context of multimodal brain imaging analysis for combining diffusion weighted MRI with fMRI data. Abstracting the setting, the framework interrogates the idea that, across a sample of subjects, an exposure or treatment impacts neural wiring, and the consequent changes in structural wiring impacts brain functional activity, which in turn impacts behavior. As our model is agnostic to domain, one might also consider applying the same approach where an exposure is postulated to impact epigenetic measurements, then subsequently genomic measurements, such as RNA expression, and subsequently changes in behavior or clinical outcome.

In our data analysis, by integrating structural and functional imaging, we have postulated mechanistic pathways, including ones related to working memory and language, that mediate sex related differences in language behavior. However, care must be taken, as any mechanistic interpretation would be highly dependent on a variety of modeling assumptions, such as the path analysis ordering, the linearity, and the confounders.

Our work points to a number of potential extensions. The model vectorizes the connectivity measures and does not exploit the symmetric and positive definite matrix structure. Recent developments in covariance regression (Sun and Li, 2017; Zhao et al., 2019) are potentially useful. Our current work focuses on penalized point estimation, whereas both statistical inference and postselection inference are crucial problems that require further investigation. Moreover, we have not yet considered the cases where the exposure and/or the outcome are themselves are high dimensional. We plan to pursue these lines of work as future research.