Analysis of brain subnetworks within the context of their whole‐brain networks

Abstract

Analyzing the structure and function of the brain from a network perspective has increased considerably over the past two decades, with regional subnetwork analyses becoming prominent in the recent literature. However, despite the fact that the brain, as a complex system of interacting subsystems (i.e., subnetworks), cannot be fully understood by analyzing its constituent parts as independent elements, most studies extract subnetworks from the whole and treat them as independent networks. This approach entails neglecting their interactions with other brain regions and precludes identifying potential compensatory mechanisms outside the analyzed subnetwork. In this study, using simulated and empirical data, we show that the analysis of brain subnetworks within the context of their whole‐brain networks, that is, including their interactions with other brain regions, can yield different outcomes when compared to analyzing them as independent networks. We also provide a multivariate mixed‐effects modeling framework that allows analyzing subnetworks within the context of their whole‐brain networks, and show that it can better disentangle global (whole‐brain) and local (subnetwork) differences when compared to standard t‐test analyses. T‐test analyses may produce misleading results in identifying complex global and local level differences. The provided multivariate model is an extension of a previously developed model for global, system‐level hypotheses about the brain. The modified version detailed here provides the same utilities as the original model—quantifying the relationship between phenotypes and brain connectivity, comparing brain networks among groups, predicting brain connectivity from phenotypes, and simulating brain networks—but for local, subnetwork‐level hypotheses.

1. INTRODUCTION

Network analysis of the brain has catalyzed a fundamental paradigm shift in the way we understand the brain in health and disease. Complex physiological, cognitive, and behavioral responses result from interactions among a vast number of neurons within local circuits of the brain as well as interactions among such local circuits (Sporns, 2010). It is now well‐established that large‐scale brain network processes are essential to high‐level cognitive tasks (Bressler, 1995; Sporns, 2010). Large‐scale brain networks have increasingly been associated with neurological and psychiatric disorders such as Alzheimer's disease (Bahrami & Hossein‐Zadeh, 2015; Stam et al., 2009; Wang et al., 2015; Zhou et al., 2010), schizophrenia (Bassett et al., 2008; Hunt, Kopell, Traub, & Whittington, 2017; Jiang et al., 2018), Parkinson's disease (Gao & Wu, 2016; Tessitore et al., 2017), and depression (Leistedt et al., 2009; Ye et al., 2015).

Recent brain network studies have mainly focused on highly interconnected subnetworks such as the default mode network (DMN) and attention network (ATN), as well as smaller subnetwork components within such subnetworks. However, despite the fact that the brain, as a complex system of interacting subsystems (i.e., subnetworks), cannot be fully understood by analyzing its subsystems as independent elements (Bullmore & Sporns, 2009, Telesford, Simpson, Burdette, Hayasaka, & Laurienti, 2011), most studies extract subnetworks from the whole and analyze them as independent networks. This approach entails neglecting their interactions with other brain regions and precludes identifying potential compensatory mechanisms outside the analyzed subnetwork. Another major issue with subnetwork analysis is the inherent limitations of univariate approaches often employed to investigate the relationships between phenotypic characteristics and brain subnetworks. Even within an independently analyzed subnetwork, univariate methods ignore the contribution of other connections (Cole, Smith, & Beckmann, 2010) and exogenous confounding effects in relating connectivity to a phenotypic variable, and they entail numerous statistical tests (thousands to millions) which can ultimately increase the risk of false positives. Several methods have been introduced for reducing the false positive rates in massive univariate comparisons of brain network connectivity. Meskaldji (Meskaldji et al., 2011) introduced a method that compares subnetworks by distilling them down to a meaningful univariate summary statistic. Zalesky et al., introduced network‐based statistics in (Zalesky, Fornito, & Bullmore, 2010) which provides greater statistical power while guaranteeing control of the family‐wise error rate at the subnetwork level instead of the edge‐wise level. However, this method is limited to identifying differences in networks when these differences form connected component structures.

Multivariate models, on the other hand, allow relating phenotypic variables to all brain connections simultaneously and thus allow capturing the dependence between brain connections while accounting for confounding effects. Additionally, these models dramatically reduce the number of statistical tests conducted on connectivity—or topology—phenotype relationships. However, the development of multivariate modeling frameworks for relating brain networks to phenotypic characteristics and drawing inference from such relationships has lagged behind other methodologies as discussed in (Simpson & Laurienti, 2016).

In this study, using simulated and empirical data from a moderate‐heavy alcohol consumption study, we show that the analysis of brain subnetworks within the context of their whole‐brain networks can yield meaningfully different results when compared to analyzing them as independent networks. We show that the analysis of brain subnetworks within the context of their whole‐brain networks can better disentangle whether an observed population effect is present globally or is restricted to regional subnetworks. Standard t‐test analyses can produce misleading results when attempting to disentangle such differences. They can also lead to multiple testing issues due to the number of statistical tests often necessitated in this context. To address this, we also introduce a promising multivariate framework for analyses of brain subnetworks within the context of their whole‐brain networks. This framework is obtained by extending a two‐part mixed‐effects modeling framework which was originally developed for analyzing whole‐brain networks (Simpson & Laurienti, 2015). While our extension for subnetwork analyses inherently assesses topological differences between groups at the subnetwork and whole‐brain network levels, it can also implicitly identify group level differences in raw connectivity patterns at both levels (even when the edge‐level means and variances are the same) by identifying group differences in the relationships between network measures and the raw connectivity values (as the outcome variable is composed of all edge values). Our framework can also be used to explicitly assess this difference by incorporating a set of indicator variables, representing the edges (or nodes) in the given network, and their interactions with the grouping variable as independent (predictor) variables. This allows directly assessing raw connectivity differences while accounting for the dependence among edges within and outside of the given network.

Although several other methods have been introduced that aim to analyze brain networks within a multivariate framework, including multivariate distance matrix regression (MDMR) (Shehzad et al., 2014), exponential random graphs (ERGs) (Simpson et al., 2011; Simpson, Moussa, & Laurienti, 2012), and permutation network frameworks (PNFs) (Simpson, Lyday, Hayasaka, Marsh, & Laurienti, 2013), they suffer from important limitations. For example, MDMR does not account for the potential correlation (dependence) between connections across brain regions and cannot be used for prediction purposes and simulating brain networks. PNFs can only be used for group‐based inferential purposes, and not for quantifying the relationship between networks and continuous covariates or for prediction purposes. Finally, ERGs have limited utility for multiple subject comparisons. The two‐part mixed‐effects model addresses these limitations by providing a framework for multiple analyses, including identifying the association of phenotypic characteristics with whole‐brain connectivity patterns while controlling for endogenous (e.g., spatial distance between brain regions) and exogenous (e.g., gender) confounding effects, simulating brain networks, and predicting an outcome from brain networks, and vice versa. This framework has been used in (Bahrami et al., 2017) to study the impacts of pesticide and nicotine on brain networks. Also, a user‐friendly toolbox has recently been released for the application of the original framework (Bahrami, Laurienti, & Simpson, 2019), as well as an extension that enables studying system‐level brain properties across multiple tasks (Simpson et al., 2019).

To show how analyzing brain subnetworks within the context of their whole‐brain networks is different from analyzing them as independent networks, we modeled simulated connectivity matrices (representing brain networks) with known “population” differences at global and local (i.e., subnetwork) levels through two different approaches. Simulated data sets were first modeled while including subnetworks within the context of their whole‐brain networks (i.e., modeling with regional subnetworks), and then by treating subnetworks as independent networks (i.e., modeling without regional subnetworks). We also used the standard t‐test analysis commonly used in network studies (Ginestet, Fournel, & Simmons, 2014) to find the simulated differences to further demonstrate the utility of the introduced method in identifying complex network differences where standard t‐tests might fail.

Finally, we used the new method for assessing the impacts of a moderate‐heavy alcohol consumption lifestyle on the default mode network (DMN) in older adults. Our analyses yielded different outcomes when the DMN was analyzed within the context of the whole brain network compared to analyzing it as an independent network, which again demonstrates the importance of analyzing subnetworks within the context of their global networks, and also the promise of the proposed extensions in this work.

2. MATERIALS AND METHODS

The original two‐part mixed‐effects modeling framework is described briefly in Appendix A and the methodology developed for subnetwork analysis is described in Appendix B. To make the original framework useable for analyses of brain subnetworks, we devised a procedure to include subnetworks as additional fixed‐effects covariates within the original models. This provided a new modeling framework for both global (i.e., system‐level or whole‐brain) and local (i.e., regional or subnetwork) analyses of brain networks. Therefore, the new multivariate models in this study not only allow investigating each brain subnetwork as a subsystem of the whole that interacts with other subsystems (i.e., subnetworks), and thus allow including the contribution of other brain regions and potential compensatory connections outside the subnetwork of interest, but also allow controlling for confounding effects (e.g., spatial distance between brain regions and gender). Additionally, this extension can be used for simulating whole‐brain networks and subnetworks as well as predicting an outcome from brain connectivity and topology. We strongly recommend reading these two sections. Simulated data sets are described in section 2.1, and finally the imaging data from the moderate‐heavy alcohol consumption study is described in section 2.2.

In this study we used a very limited number of subnetworks (mostly a single subnetwork) in both the simulation and empirical data due to the following reasons:(a) using simple models with a limited number of subnetworks was critical for understanding the main utility of the fairly complex methodology developed for analyzing subnetworks within their whole‐brain networks and interpreting the results; (b) these simple models were sufficient to demonstrate the difference in results that occur when analyzing brain subnetworks within the context of their whole‐brain networks compared with analyzing them as independent networks; and (c) simulating networks with more complex population differences in multiple subnetworks (and smaller networks within such subnetworks) was not feasible. However, it is important to note that that modeling a larger number of subnetworks will create more parameters and increase the computational cost. Thus, this model could be most useful for modeling a limited number of large, well‐known subnetworks. Most studies of functional brain networks over the past two decades have studied a very limited number of well‐known subnetworks such as DMN and attention networks. This framework can be easily used for modeling this limited number (~5) of subnetworks. Nevertheless, as each subnetwork is mainly modeled via a binary variable and its interaction with a covariate of interest, and the contrast statements are estimated by using the already estimated residuals, the computational cost does not increase rapidly enough to make it nonapplicable for even larger numbers of subnetworks (or subnetwork components). Additionally, we will add GUIs to an already developed user‐friendly toolbox (Simpson et al., 2019) to make the application of this framework more accessible to those interested in using it. The appropriate data reduction methods already implemented in this toolbox, and its interface with strong statistical programming software packages such as SAS and R will make modeling even larger numbers of subnetworks feasible. We will also provide further documentation with explicit examples after implementing it in the WFU_MMNET toolbox to aid in interpretation.

2.1. Simulated connectivity matrices

We simulated connectivity matrices that were sufficient to illustrate (a) the importance of analyzing a subnetwork within the context of the entire network, and (b) the utility of employing our mixed modeling approach instead of the standard t‐test approach for subnetwork analyses. We simulated 40 connectivity matrices with 90 regions for the first group, and applied population differences on this simulated data set to obtain data sets for the second group. The 90 regions were derived from the AAL atlas (Tzourio‐Mazoyer et al., 2002) so that the simulated data could be represented in brain space. We only used simulations of the brain connection strength (Equation A5) as the results can be generalized to simulations of the brain connection probability (Equation A4). Connectivity values were simulated using a normal distribution. As FZT of connectivity values are used in Equation A5, simulated connectivity values were considered to be FZT of connectivity values. We included the spatial distance constraint in our simulations to account for the fact that connectivity drops with distance, and added a single random value to each connectivity matrix to generate the random effect—that is, to create different connectivity averages across subjects. These single random values also came from a normal distribution to meet the normality assumption of the random effects distributions in mixed models. More specifically, let C_{ijk, gr1} denote the simulated (FZT‐) connection strength between node j and node k for subject i in the first group, and sdist_jk denote the spatial distance between node j and node k in the AAL space; thus, we will have:

$${\mathit{C}}_{\mathit{ijk},\mathit{gr}1}=\left(w_{\mathit{ijk}}-{\alpha }_{\mathit{ijk}}{\mathit{\text{sdist}}}_{\mathit{jk}}\right)+b_i,w_{\mathit{ijk}}\sim N\left(\mu =0.02,\sigma =0.4\right),b_i\sim N\left(\mu =0,\sigma =0.08\right)$$ (1)

where i = 1, 2, …, 40, j = 1, 2, …, 90, k = 1, 2, …, 90, C_{ijk, gr1} = C_{ikj, gr1}, and C_{ijk, gr1} = 0 for j = k.w_ijk is a random value that comes from a normal distribution, α_ijk is a random coefficient for the jkth spatial distance that comes from a uniform distribution in (0.006, 0.008), and b_i is the single random value coming from a normal distribution added to all elements of the simulated connectivity values of subject i to create the random effect. This simulation provided 40 matrices, which were used as the simulated data set for the first group. Connectivity matrices for the second group were obtained by applying population differences to C _ijk,gr1 (details are presented in the following sections). The averages and standard deviations of the normal distributions and intervals for the uniform distribution were selected from results of modeling a real resting‐state fMRI data (Simpson et al., 2019; Simpson & Laurienti, 2015) to have more realistic simulations of brain connection strength. We recognize that these relatively simple simulations do not account for many properties of the brain networks or other sources of correlation. However, as pointed out earlier, simulating such data sets were not necessary for our comparative purposes as for each simulation the same data sets were used in models with regional subnetworks and models without them, as well as in the t‐test analyses. Simulated data sets only had to meet the basic requirements of being sufficient to illustrate (a) the importance of analyzing a subnetwork within the context of the entire network, and (b) the utility of employing our mixed modeling approach instead of the standard t‐test approach for subnetwork analyses. It is important to note that one of the important applications of the original (and extended) mixed models is simulating brain networks. However, we are not focused on the simulation utility of the mixed models here.

We made known differences in simulated connectivity matrices of the first group to create data sets with different local and global connection strength and different network feature values (global efficiency) for the second group, as well as data sets with exogenous confounding effects (gender). Details are presented in the corresponding sections. In analyses of the connectivity differences we used simple models that only included the covariates with the known differences—that is, for simulations of the (local and global) connection strength differences and simulations of the confounding effects of gender on connection strength, global efficiency was not included.

For the t‐test analysis, connection strength values were averaged within individuals, yielding two vectors of averaged values (vector size: 40 × 1), and subsequently submitted to a two‐tailed t‐test analysis—that is, the t‐test analysis was used for a simple comparison of average values of connection strength between the two groups. We used MATLAB R2016b (The MathWorks Inc., Natick, MA) for simulating the connectivity data sets and SAS v.9.4 for fitting the mixed models. Simulated data sets and implemented SAS scripts are provided as supplementary files accompanying this article. Simulated data sets and codes are provided on the mendeley data repository (https://data.mendeley.com/datasets/wpxk9s6wbf/1).

2.2. Imaging data and network generation for moderate‐heavy alcohol consumption lifestyle study

To further demonstrate the utility of the developed models, we examined the effects of moderate‐heavy alcohol consumption on brain connectivity in older adults. We used resting‐state fMRI scans from 41 participants that consumed light (<2 drinks/week and ≥ 1 drink/month, n = 20, age [years] = 71.1 ± 3.4, sex [M/F] = 12/8) or moderate‐heavy (7–21 drinks/week, nonbingers, n = 21, age [years] = 70.1 ± 4.2, sex [M/F] = 10/11) amounts of alcohol. These data came from a previous study (Mayhugh et al., 2016). More detail about demographics and applied preprocessing steps can be found in the referenced article.

A functional brain network for each participant was generated using mean time‐series of 268 ROIs from a functional parcellation (Shen, Tokoglu, Papademetris, & Constable, 2013). We used the functional parcellation in the native space of each participant (obtained using the corresponding inverse registration matrices) for extraction of each ROI's mean time series. Also, the spatial distances between brain ROIs of each participant were obtained using the transformed parcellations in the native space. We assessed the effects of moderate‐heavy alcohol use on the default mode network (DMN) as changes in this subnetwork have been associated with alcohol consumption in several studies (Muller‐Oehring, Jung, Pfefferbaum, Sullivan, & Schulte, 2015; Shokri‐Kojori, Tomasi, Wiers, Wang, & Volkow, 2017; Weber, Soreni, & Noseworthy, 2014; Zhu, Cortes, Mathur, Tomasi, & Momenan, 2017).

3. RESULTS

Using simulated and empirical data, we demonstrate that analyses of subnetworks within the context of their whole‐brain networks yield different outcomes when compared to analyzing them as independent networks. We also show that the proposed methodology better disentangles the differences between global and local levels when compared to standard t‐test analyses.

In all subsequent simulations, Equation (A5) was utilized but with different covariates (fixed effects). In simulations that only included the connection strength, only a random intercept was employed to model the within‐subject correlations of the connection strength. In all subsequent simulations, an additive noise (noise~N[μ = 0, σ = 0.02]) was added to the applied differences.

3.1. Population differences in connectivity at global and local levels

We made known population differences that were: (a) present at the global level but not in a subnetwork, (b) restricted to a subnetwork, and (c) restricted to only one of two subnetworks with common connections, (i.e., restricted to one of two overlapping subnetworks). We used a combination of interconnected nodes (regions) located in the frontal lobe as our subnetwork of interest and named it sub_a (you can simply think of sub_a as your subnetwork of interest such as the default mode network). In sections 3.1.1 and 3.1.2, simple models with four fixed effects, including intercept, population grouping covariate (COI), subnetwork covariate, and interaction of subnetwork covariate with COI were used (${\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}{\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}\right]}^{\prime },{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$). Applied differences and results are presented below. All relevant estimates and p‐values are bolded in the corresponding tables. Note that when a subnetwork is analyzed as an independent network, θ_{s, COI} itself shows the difference in the connectivity of that subnetwork between the two groups because all connections being analyzed are in that subnetwork.

3.1.1. Difference at the global level but not in a subnetwork

We changed the whole‐brain connection strength excluding sub_a in the first group by a factor of 0.8 to produce a data set with reduced strength for the second group (Figure 1a):

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=0.8\times {\mathbf{C}}_{\mathit{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections not located within}}{\mathit{sub}}_a,i=1,\dots ,40$$

Figure 1.

Models of brain networks that illustrate the global and local level differences. Regions located within sub_a are depicted with a slightly larger node size simply for identification. Edge color and thickness differentiate the connections that are different between the two groups (connections that are not different between the two groups are depicted with the blue color and thinner lines). (a) Connection strength difference between the two groups is present in all networks edges except those located within sub_a. (b) Connection strength difference between groups is restricted to edges within sub_a. The color bar indicates the edge strength difference between groups in arbitrary units. Brain networks shown here were generated using the BrainNet Viewer software (Xia, Wang, & He, 2013) [Color figure can be viewed at http://wileyonlinelibrary.com]

As Tables 1 presents, modeling sub_a within its whole‐brain network correctly identifies both the global difference and lack of a local difference in a single model. Note that the estimate for the parameter (${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}\mathit{+}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}$) provides the inference for connection strength difference between groups in sub_a (see Appendix D for more detail). Modeling sub_a as an independent network (Table 2) and modeling the whole‐brain network without regional subnetwork covariates (Table 3) correctly identify the differences as well for this simulation. Modeling without regional subnetworks and modeling sub_a as an independent network gave the same results as the standard t‐test analysis here since these models only contained the grouping covariate (i.e., we did not control for other effects in these simple models). This is shown in Table 4.

Table 1.

Modeling with regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3314	0.005535	<.0001
θs, COI	−0.06459	0.007828	<.0001
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}$	0.03503	0.002495	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.05883	0.003546	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	−0.00576	0.008424	.4941

The bold values indicate the relevant parameter estimates.

Note: Modeling results with ${\theta }_{\mathrm{s}}\mathbf{=}\left[{\theta }_{\mathrm{s},0}{\theta }_{\mathrm{s},\mathrm{COI}}{\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}{\theta }_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}\right]\prime$.

Table 2.

Modeling sub_a as an independent network

Parameter	Estimate	SE	p‐value
θs, 0	0.3656	0.007991	<.0001
θs, COI	−0.00652	0.01131	.5640

The bold values indicate the relevant parameter estimates.

Note: Modeling results with θ_s  = [θ_{s, 0} θ_{s, COI}]′ and sub_a connections.

Table 3.

Modeling without regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3354	0.005495	<.0001
θs, COI	−0.05777	0.007773	<.0001

The bold values indicate the relevant parameter estimates.

Note: Modeling results with θ_s  = [θ_{s, 0} θ_{s, COI}]′ and whole‐brain connections.

Table 4.

Standard t‐test analysis

Parameter	Average ‐ difference	p‐value
Whole − brain	−0.05775	1.14e‐10
suba	−0.00652	.562616

The bold values indicate the relevant parameter estimates.

3.1.2. Difference restricted to a subnetwork

We changed the connection strength in sub_a in the first group (while keeping the connection strength of all other brain regions the same) by a factor of 0.55 to produce a data set with reduced strength in sub_a for the second group (Figure 1.B):

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=0.55\times {\mathbf{C}}_{\mathit{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{within}}{\mathit{sub}}_a,i=1,\dots ,40$$

As Table 5 presents, modeling sub_a within its whole‐brain network correctly shows that the difference between the two groups is restricted to sub_a, and accounting for this subnetwork, connections of other regions of the brain (quantified by θ_{s, COI} in this model) are not different between the two groups. While analyzing sub_a as an independent network correctly shows the difference (Table 6), removing subnetwork covariates (i.e., ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}$ and ${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}$) from the model results in a significant whole‐brain difference between the two groups (Table 7) due only to the local sub_a difference (i.e., this model does not specify whether the observed global difference is actually present across all regions or is localized to sub_a). The t‐test analysis fails to make this distinction between the whole‐brain and sub_a differences between the two groups as well, and shows significant p‐values for both the whole‐brain and sub_a connections (Table 8). For the t‐test analysis and the model without regional subnetwork covariates to show the correct differences, additional tests on brain networks with sub_a connections excluded would have to be conducted. Although this could be done for this simple case, for more complex situations with multiple interconnected (overlapping) subnetworks being analyzed, the t‐test analysis and models without regional subnetwork covariates would fail to disentangle such differences. As shown by a more complex simulation in the next section, such interconnected subnetworks can be simply included as covariates within the proposed framework to disentangle the observed difference, and identify the specific brain networks responsible for differences identified between study populations.

Table 5.

Modeling with regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3314	0.005761	<.0001
θs, COI	−0.00471	0.008149	.5629
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}$	0.03503	0.002639	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	−0.1609	0.003762	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	−0.1656	0.008794	<.0001

The bold values indicate the relevant parameter estimates.

Note: Modeling results with ${\theta }_{\mathrm{s}}\mathrm{=}\left[{\theta }_{\mathrm{s},0}{\theta }_{\mathrm{s},\mathrm{COI}}{\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}{\theta }_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}\right]$.

Table 6.

Modeling sub_a as an independent network

Parameter	Estimate	SE	p‐value
θs, 0	0.3656	0.006537	<.0001
θs, COI	−0.1633	0.009253	<.0001

The bold values indicate the relevant parameter estimates.

Note: Modeling results with θ_s  = [θ_{s, 0} θ_{s, COI}] and sub_a connections.

Table 7.

Modeling without regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3354	0.005772	<.0001
θs, COI	−0.02297	0.008165	.0049

The bold values indicate the relevant parameter estimates.

Note: Modeling results with θ_s  = [θ_{s, 0} θ_{s, COI}] and whole‐brain connections.

Table 8.

Standard t‐test analysis

Parameter	Average ‐ difference	p‐value
Whole − brain	−0.02296	.006166
suba	−0.16305	3.70e‐29

The bold values indicate the relevant parameter estimates.

3.1.3. Difference restricted to one of two overlapping subnetworks

We made known population differences in one of two subnetworks with overlapping connections. The same sub_a from above served as our first subnetwork, and a combination of nodes located within the parietal lobe served as our second subnetwork (denoted by sub_b). To model subnetworks with overlapping connections, instead of modeling the connections within sub_a and sub_b, we modeled connections from sub_a to all other brain regions (denoted by sub_{a_out}) which included connections to sub_b as well, and connections from sub_b to all other brain regions (denoted by sub_{b_out}) which included connections to sub_a as well (Figure 2). Therefore, both sub_{a_out} and sub_{b_out} included the connections from sub_a to sub_b. The following fixed and random effects parameters were used in this simulation:

$${\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{{\mathrm{a}}_{\text{out}}}}{\mathrm{\theta }}_{\mathrm{s},{{\mathrm{sub}}_{\mathrm{b}}}_{\text{out}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{{\mathrm{a}}_{\text{out}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{{\mathrm{b}}_{\text{out}}}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$$

Figure 2.

Model of brain network that illustrates the difference between the two groups when the difference is restricted to one of two overlapping subnetworks. A simple 2D image that only contains the connections from sub_a and sub_b to other regions (not showing other connections) is shown here to better illustrate the difference. Red and blue circles in this figure depict the regions located within sub_a and sub_b, respectively. As shown here, only connections from sub_b to other brain regions (depicted by thicker lines), including the connections to sub_a (highlighted by a purple color) are different between the two groups [Color figure can be viewed at http://wileyonlinelibrary.com]

We changed the connection strength in sub_b to other brain regions (connections represented by sub_{b_out}) in the first group by a factor of 1.5 to produce a data set with increased strength in sub_{b_out} for the second group:

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=1.5\times {\mathbf{C}}_{\mathit{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections represented}}\mathit{by}{\mathit{sub}}_{b\_\mathit{\text{out}}},i=1,\dots ,40$$

As Table 9 presents, modeling subnetworks within the whole‐brain network correctly shows that the difference between the two groups is restricted to the connections from sub_b to other brain regions (i.e., sub_{b_out}) while an analysis of sub_{a_out} as an independent network (Table 10) and whole‐brain network without subnetwork covariates (Table 11) as well as the t‐test analysis (Table 12) fail to identify the correct difference. Modeling sub_{b_out} as an independent network gave the correct result (this is not shown here). Note that both subnetwork covariates must be included in the extended mixed models in order to identify the actual difference (this is shown in Appendix E (Table A1)).

Table 9.

Modeling with regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3595	0.006515	<.0001
θs, COI	0.000462	0.009214	.9600
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.03483	0.001891	<.0001
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.04746	0.002175	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.00259	0.002679	.3330
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	0.1547	0.003086	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.00213	0.009290	.8185
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	0.1552	0.009460	<.0001

The bold values indicate the relevant parameter estimates.

Table 10.

Modeling Sub_{a_out} as an independent network k

Parameter	Estimate	SE	p‐value
θs, 0	0.3169	0.006014	<.0001
θs, COI	0.02607	0.008506	.0022

The bold values indicate the relevant parameter estimates.Note: Modeling results with θ_s  = [θ_{s, 0} θ_{s, COI}] and sub_{a_out} connections.

Table 11.

Modeling without regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3354	0.006447	<.0001
θs, COI	0.03493	0.009118	.0001

The bold values indicate the relevant parameter estimates.

Note: Modeling results with θ_s  = [θ_{s, 0} θ_{s, COI}] and whole‐brain connections.

Table 12.

Standard t‐test analysis

Parameter	Average–difference	p‐value
Whole − brain	0.034913	.000255
suba_out	0.026026	.002935
subb_out	0.151816	8.67e‐24

The bold values indicate the relevant parameter estimates.

Additionally, we simulated more complex data sets with overlapping subnetworks having differences in different directions that counterbalance each other (i.e., sub_{a_out} had stronger connections in the second group while sub_{b_out} had weaker connections in this group). Modeling subnetworks independently as well as standard t‐test analyses indicated no significant differences in both sub_{a_out} and sub_{b_out} due to the counterbalancing effects of the overlapping connections while the extended mixed models correctly showed that both subnetworks were significantly different (this is shown in Appendix F (Tables A2‐A4)). Also, data sets with complex patterns of global and local connectivity differences (Appendix G (Table A5)), and data sets with exogenous confounding variables (gender) were modeled (Appendix H (Table A6)). Again, modeling subnetworks independently and t‐test analyses failed to disentangle all differences correctly.

3.2. Network features at local levels

We studied the difference of global efficiency in sub_a between the two groups by first modeling sub_a within the whole and then modeling sub_a as an independent network after producing a data set with increased strength in sub_a for the second group:

$${\mathit{C}}_{\mathit{ijk},{\mathit{gr}}_2}=1.3\times {\mathit{C}}_{\mathit{ijk},{\mathit{gr}}_1}+\mathit{\text{noise for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{within}}{\mathit{sub}}_a,i=1,\dots ,40$$

The following fixed and random effects (with a variance component covariance structure) parameters were used in the first model.

$${\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}{\mathrm{\theta }}_{\mathrm{s},\text{Geffi}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times \text{Geffi}}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\times \text{Geffi}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}\times \text{Geffi}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}\left[{\mathrm{b}}_{\mathrm{si},0}{\mathrm{b}}_{\mathrm{si},\text{Geffi}}\right]\prime$$

In which θ_{s, Geffi}, θ_{s, COI × Geffi}, ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\times \text{Geffi}}$, and ${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}\times \text{Geffi}}$ are parameters for the global efficiency and its interactions with other covariates. Other parameters are the same as the ones described in the previous sections. For computing the global efficiency, first we applied an inverse FZT on connectivity values as we had considered them to be simulated FZT‐connection strength (i.e., we used $\frac{\exp \left(2{\mathrm{C}}_{\mathrm{ijk}}\right)-1}{\exp (2{\mathrm{C}}_{\mathrm{ijk})}+1}$ values in computing the global efficiency).

Modeling sub_a within the whole (Table 13) indicates that the difference between the two groups is restricted to the connection strength in sub_a, and accounting for the connectivity, the global efficiency (as a measure of topology) in sub_a is not significantly different between the two groups (note that ${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times \text{Geffi}}+{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}\times \text{Geffi}}$ quantifies the slope difference of global efficiency/connections strength in sub_a). However, modeling sub_a as an independent network (Table 14) indicates that the slope of the global efficiency/connection strength relationship in sub_a is significantly higher in the second group, and thus, global efficiency explains more of the sub_a connection strength for the second group. This simulation clearly demonstrates the difference of analyzing global efficiency in sub_a within the whole and as an independent network. We used weighted networks in computing the global efficiency here and in the next section.

Table 13.

Modeling with regional subnetwork

Parameter	Estimate	SE	p‐value
θs, 0	0.3575	0.009728	<.0001
θs, COI	−0.01649	0.01375	.2305
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}$	0.04790	0.002796	<.0001
θs, Geffi	0.1158	0.002231	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.06298	0.004035	<.0001
θs, COI × Geffi	−0.01638	0.003089	<.0001
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\mathrm{\times }\mathrm{\text{Geffi}}}$	0.008885	0.002821	.0016
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}\mathrm{\times }\mathrm{\text{Geffi}}}$	0.02177	0.003922	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.04649	0.01421	.0011
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }\mathrm{\text{Geffi}}}+{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}\mathrm{\times }\mathrm{\text{Geffi}}}$	0.005393	0.004699	.2510

The bold values indicate the relevant parameter estimates.

Table 14.

Modeling sub_a as an independent network

Parameter	Estimate	SE	p‐value
θs, 0	0.4729	0.01521	<.0001
θs, COI	−0.08150	0.02122	.0001
θs, Geffi	0.1797	0.007420	<.0001
θb, COI × Geffi	0.02668	0.009942	.0073

The bold values indicate the relevant parameter estimates.

3.3. Moderate‐heavy alcohol consumption lifestyle in older adults

To further demonstrate the utility of the developed models and how analyzing brain subnetworks within the context of their whole‐brain networks can provide a deeper understanding of the brain, we analyzed the effects of moderate‐heavy alcohol consumption on the default mode network in older adults. It is important to note that the main goal of this analysis is to demonstrate the utility of the developed methodology and difference between analyzing subnetworks within their whole‐brain networks and analyzing them as independent networks in real studies, not to perform a thorough investigation of the effects of alcohol consumption on default mode network functionality in older adults.

3.3.1. Modeling DMN within the whole‐brain network

The fixed effects covariates included: (a) COI: covariate of interest (alcohol group—a binary variable separating the light from moderate‐heavy drinkers); (b) DMN: subnetwork covariate (a binary variable that separates the connections located within DMN from other brain connections). Thirty seven ROIs (of 268) of the utilized atlas were identified as regions within DMN, and connections between these 37 ROIs were coded as one (1) while assigning zero (0) to all other brain connections (the mask is shown in Figure A1 in Appendix I); (c) Geffi: the average of global efficiency in each dyad (we only used one network feature to keep the model simple); (d) Int: Interaction covariates: COI/DMN, COI/Geffi, DMN/Geffi, and COI/DMN/Geffi interactions; (e) Conf: Confounders: spatial distance and square of spatial distance between brain ROIs, age, sex, and cognitive test performance. We used random effects (with a variance component covariance structure) for the connection strength, global efficiency, spatial distance, and square of spatial distance.

Table 15 presents the results for the probability and strength models. Estimates of (θ_{r, COI × geffi} + θ_{r, COI × DMN × geffi}) and (θ_{s, COI × geffi} + θ_{s, COI × DMN × geffi}) represent the slope differences between the moderate‐heavy and light drinkers in the relationships between connection probability and global efficiency in the DMN, and connection strength and global efficiency in the DMN, respectively. Global efficiency in the DMN is more negatively related to brain connection probability in moderate‐heavy drinkers when compared to light drinkers (the slope difference is significantly negative). This indicates that in the DMN, moderate‐heavy drinkers are less likely to have connections between nodes with higher global efficiency than light drinkers. Similarly, global efficiency in the DMN is more negatively (less positively) related to brain connection strength in moderate‐heavy drinkers when compared to light drinkers (the slope difference is significantly negative). Again, this indicates that in the DMN, moderate‐heavy drinkers have weaker connections between nodes with higher global efficiency than light drinkers. Interestingly, the slope for the relationship between whole‐brain connectivity and global efficiency is not different between the two groups. This difference is restricted to the DMN. This indicates that moderate‐heavy drinkers tend to lose the integration between the DMN regions as the connectivity increases but whole brain integration is not affected.

Table 15.

Results for modeling DMN within the whole‐brain network in old moderate‐heavy alcohol drinkers

Probability model			Strength model
Parameter	Estimate	p‐value	Parameter	Estimate	p‐value
θr, 0	−0.1030	.0046	θs, 0	0.2289	<.0001
θr, COI	−0.0479	.2105	θs, COI	−0.0172	.1825
θr, DMN	0.5281	<.0001	θs, DMN	0.0738	<.0001
θr, Geffi	0.1001	<.0001	θr, Geffi	0.1022	<.0001
θr, COI × DMN	0.0425	.1082	θs, COI × DMN	0.0083	.0048
θr, COI × Geffi	−0.0014	.9647	θs, COI × Geffi	−0.0041	.4886
θr, DMN × Geffi	−0.1224	<.0001	θs, DMN × Geffi	0.0124	<.0001
θr, COI × DMN × geffi	−0.1199	<.0001	θs, COI × DMN × geffi	−0.0090	.0041
θr, dist	−0.5829	<.0001	θs, dist	−0.0657	<.0001
${\theta }_{\mathrm{r},{\mathit{\text{dist}}}^2}$	0.1248	<.0001	${\theta }_{\mathrm{s},{\mathit{\text{dist}}}^2}$	0.0276	<.0001
θr, Sex	−0.0163	.6797	θs, Sex	−0.0011	.9313
θr, Age	−0.0052	.7843	θs, Age	−0.0029	.6487
θr, Cog	−0.0008	.9661	θs, Cog	0.0078	.2520
θr, COI + θr, COI × DMN	−0.0053	.9079	θs, COI + θs, COI × DMN	−0.0089	.5014
θr, COI × Geffi + θr, COI × DMN × Geffi	−0.1214	.0039	θs, COI × Geffi + θs, COI × DMN × Geffi	−0.0132	.0502

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

3.3.2. Modeling DMN as an independent network

We modeled DMN as an independent network as well to show how such an analysis could be different from analyzing it within the whole‐brain network. We used the same fixed‐effects covariates except the subnetwork covariate and its interactions because in this analysis COI itself separates the DMN connections between the two groups (all connections being analyzed are in the DMN). Also, the same random effects were used. As presented in Table 16, moderate‐heavy and light drinkers do not have a different relationship between global efficiency and the connection probability or strength. This clearly demonstrates the importance of analyzing brain subnetworks within the context of their global networks.

Table 16.

Results for modeling DMN as an independent network in old moderate‐heavy alcohol drinkers

Probability model			Strength model
Parameter	Estimate	p‐value	Parameter	Estimate	p‐value
θr, 0	0.2198	.0138	θr, 0	0.2501	<.0001
θr, COI	−0.0320	.7337	θr, COI	0.0067	.5945
θr, Geffi	0.4972	<.0001	θr, Geffi	0.1108	<.0001
θr, COI × Geffi	0.1060	.3853	θr, COI × Geffi	0.0070	.5397
θr, dist	−0.5225	<.0001	θs, dist	−0.0945	<.0001
${\theta }_{\mathrm{r},{\mathit{\text{dist}}}^2}$	0.2159	<.0001	${\theta }_{\mathrm{s},{\mathit{\text{dist}}}^2}$	0.0539	<.0001
θr, Sex	0.0341	.7265	θs, Sex	0.0137	.2935
θr, Age	−0.0349	.4625	θs, Age	0.0156	.0130
θr, Cog	0.0284	.5656	θs, Cog	0.0037	.5779

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

4. DISCUSSION

Analysis of the human brain from a network perspective has increased our understanding of brain function and its abnormalities. However, despite the exponential increase in brain network studies with a major focus on regional analyses, statistical methods for such analyses are yet to be developed. Most current studies extract subnetworks of interest and analyze them as independent networks. This approach is in contrast to the fact that brain subnetworks, as components of a complex system, are constantly modified through interactions with other subnetworks. Thus, analyses of subnetworks extracted from the whole brain should be interpreted with great caution (Bullmore & Sporns, 2009; Simpson et al., 2011). Considering the function of brain subnetworks within the context of their global (i.e., the whole‐brain) networks is a promising approach toward understanding the coordination of brain activity and how it facilitates complex behavioral and cognitive tasks (McIntosh, 2000). An increasing number of studies suggest that complex psychiatric and neurological disorders such as Alzheimer's disease, Schizophrenia, ADHD, depression, and autism are characterized by structural and functional abnormalities in local and global circuits of the brain (Menon, 2011). Such disorders usually affect multiple systems of the brain with a complex pattern. For example, Alzheimer's disease has been associated with decreased connectivity in the DMN (Jacobs, Radua, Luckmann, & Sack, 2013; Li et al., 2015) and increased connectivity in the salience network (Wang et al., 2016). It is quite challenging to understand the implications of such subnetwork connectivity increases/decreases when the subnetworks are analyzed in isolation.

In this study, using simulated and empirical data, we showed how analyzing brain subnetworks as independent networks can produce misleading results. Furthermore, we introduced a promising multivariate framework for analyses of brain subnetworks within the context of their whole‐brain networks. Development of multivariate frameworks for investigating phenotypic associations with brain subnetworks has lagged behind other methodologies (Simpson & Laurienti, 2016). We devised a procedure within a promising whole‐brain multivariate analysis framework (Simpson & Laurienti, 2015) that enables analyzing brain subnetworks, while maintaining the capabilities of the original framework for whole‐brain studies. This new framework not only allows controlling for endogenous and exogenous confounding effects and assessing the effects of multiple covariates on brain networks at the same time, but also allows analyzing brain subnetworks within the context of their global networks and thus accounting for the extremely relevant and complex interactions between a given subnetwork and all other regions or subnetworks that make up the remainder of the brain (Bullmore & Sporns, 2009; Zhou, Zemanova, Zamora, Hilgetag, & Kurths, 2006). It also allows predicting an outcome from the brain connectivity structure and vice versa in global and local networks, and simulating global or local brain networks. Additionally, our framework could be used to fit multiple models to multiple parcellations of a subnetwork and do model comparisons employing information criteria (e.g., BIC) to see which model fits best for the given context, and thus which network resolution or scale is the most appropriate for characterizing dysfunction in a given context for a given behavioral or health outcome. Although the original and extended frameworks in this study are developed for nondirectional (weighted) networks, they could be used for directional networks as well. However, this would double the data size due to the networks being nonsymmetric, and thus increase the computational cost. It would also necessitate a change in the way the dyadic covariates are defined and interpreted. Future work will further examine the possibility of employing our approach with directed networks.

We used AAL (Tzourio‐Mazoyer et al., 2002) and Shen (Shen et al., 2013) parcellations as they have been extensively used for analyses of structural/functional brain networks. Although using the same parcellation for the simulation and empirical data analyses would produce more aligned results, we intentionally used different parcellations to show that our results (i.e., the demonstrated difference between analyzing the subnetworks within their whole‐brain networks and analyzing them as independent networks) can be generalized to networks obtained from any desired parcellation as it is not the network/subnetwork size that matters, but the failure to account for dependencies between subnetworks that engenders misleading results. However, the parcellation choice will affect the computational costs, which could limit the utility of the provided framework for parcellations with a large number of ROIs (e.g., 1,000 ROIs) or voxel‐wise networks. Nevertheless, almost all currently‐used parcellations—including MMP (360 ROIs) (Glasser et al., 2016), the Shen parcellation (Shen et al., 2013), AAL (Tzourio‐Mazoyer et al., 2002), and the Craddock parcellation (600–1000 ROIs) (Craddock, James, Holtzheimer, Hu, & Mayberg, 2012)—have less than 1,000 ROIs. Additionally, parcellations with less than 1,000 ROIs provide an optimal representation of underlying voxel‐wise networks in analyzing brain connectivity and its association with phenotypic characteristics (Shehzad et al., 2014). Thus, we believe that parcellation choice will not have a substantial impact on the utility of this method.

Inconsistencies in the literature make characterizing the effects of brain disorders on network connectivity difficult (Badhwar et al., 2017). One of the major reasons for such inconsistencies is the lack of modeling frameworks that allow accounting for the complex interactions between the brain subnetworks. Thus, multivariate methods that allow analyzing brain subnetworks within the context of their global circuitries can provide a deeper understanding of the complex patterns of systemic and local brain network differences and potential compensatory mechanisms. The introduced framework in this article allows performing such analyses. As shown by simulation studies, a variety of system‐level and regional hypotheses can be tested through the new models that allow identifying sophisticated patterns of brain network differences among groups in global and local circuits.

Analysis of subnetworks within the context of their global networks allows including the complex interplay between brain subnetworks. This is especially important in the study of network features such as path length or global efficiency since changes in a brain region not comprised by the subnetwork can have a systemic effect on all other brain regions and subnetworks. Such effects cannot be examined when subnetworks are analyzed independently or when simple connections between subnetworks are considered. For example, information flow between specific regions in a brain subnetwork is not exclusively determined by connectivity within that subnetwork. Rather, information flow between such regions may be more efficiently exchanged through connections of that subnetwork to other brain subnetworks (Rubinov & Sporns, 2010; Sporns, 2018). Considering global efficiency as a measure of information flow, the alcohol consumption study in section 3.3 demonstrated that analyzing the DMN within the context of the whole network captures the exchanged information flow with other regions of the brain. It also demonstrated how analyzing the brain subnetworks within the context of their global networks could result in different conclusions when compared to analyzing them as independent networks. While investigations of the effects of moderate‐heavy alcohol use on brain networks does not lie within the focus of this work, decreased efficiency in the brain networks of alcoholics has been reported in several studies (Jacobus et al., 2009; McQueeny et al., 2009; Spear, 2018) as we found here in the DMN of moderate‐heavy alcohol users.

In addition to modeling the interactions within and between brain subsystems at a single scale, the introduced modeling framework can also be used for modeling the complex interactions of hierarchical brain subsystems at multiple resolutions. This is important because the brain, as a complex multiscale system, is characterized by similar organization at multiple resolutions (Bassett & Gazzaniga, 2011). For example, while the brain can be decomposed into highly inter‐connected and weakly intra‐connected communities or subnetworks at a single scale (Bullmore & Sporns, 2009), each subnetwork also displays this modular structure at the next level, such that it can be decomposed into further communities, and this hierarchical modular structure is present at subsequent (sub) levels as well (Meunier, Lambiotte, & Bullmore, 2010). Development of modeling frameworks that allow accounting for such complex hierarchical interactions among brain subsystems at different scales is a major step toward understanding brain function and structure (Bassett & Gazzaniga, 2011). Hierarchical subsystems of the brain can be analyzed within our framework as subnetwork covariates. However, the nature of such relationships at a specific resolution with phenotypic characteristics should be predefined as our framework only reveals the presence/absence of linear relationships. A recently introduced approach in (Vogelstein et al., 2019) not only allows identifying the existence of relationships with different natures, but also provides an adaptive framework to identify the most informative components (subsystems) that contribute to such relationships. However, unlike our framework, it may not be able to disentangle the relationships with phenotypic characteristics when (multiple overlapping) subsystems with complex interplays at different resolutions are included.

The simulated and empirical data analyses presented demonstrate that there are important differences between brain subnetwork analyses when the subnetwork is isolated compared to when it is analyzed within the context of the whole‐brain. With regional subnetwork analyses dominating the recent literature, this study could attract much attention, and the developed multivariate methodology for analyses of brain subnetworks within the whole brain could have great appeal and wide applicability. Nevertheless, this study is not without limitations. Simulated networks are inherently limited as they do not account for the full complexity of the human brain. The simulations used here were devised to demonstrate important outcomes that could change the biological interpretation of a brain network analysis. There are certainly many alterations that could be made to the simulations to test other outcomes but an exhaustive assessment of such possibilities is beyond the scope of this project. In addition, the empirical data from a prior study were chosen to simply demonstrate utility. Considerable research will need to be done on both smaller focused data sets such as the one we used as well as on large databases like the Human Connectome Project. This future work will be able to test how this new subnetwork analysis methodology alters our biological interpretation of brain network connectivity.

The provided framework in this manuscript is a model‐based method that allows testing a variety of hypotheses about the relationship between brain network connectivity and (phenotypic) covariates of interest while controlling for endogenous and exogenous confounding effects. Data‐driven methods such as independent component analysis (ICA), on the other hand, are generative models, which cannot be used for testing such hypotheses. However, data‐driven methods could be used as complementary methods in identifying useful features and nuisance components from the data before using them in our framework. For example, local subnetworks could be the first identified or population‐based parcellations could be the first generated using ICA, and subsequently be used in our framework. Data‐driven methods are especially useful when used in combination with a statistical modeling framework (Calhoun & Adali, 2012; Erhardt, Allen, Damaraju, & Calhoun, 2011), including our framework.

Supporting information

Appendix S1. Supporting Information.

TWO‐PART MIXED‐EFFECTS MODELING FRAMEWORK FOR WHOLE‐BRAIN NETWORKS

1.

Most network studies of the brain use either binary or positively‐weighted networks because extracting network features such as clustering coefficient is not straight‐forward and these features remain poorly understood in networks that include negative weights (edges) (Telesford, Simpson et al., 2011, Friedman, Landsberg, Owen, Li, & Mukherjee, 2014). Thus, as weighted networks are used within the mixed‐effects modeling framework, negative weights (e.g., correlation values) are the first set to zero.

The two‐part mixed‐effects models use positively weighted networks, with negative weights set to zero as discussed above. Let R_ijk denote the binary variable specifying whether a connection is present between node j and node k of the ith subject's network and Y_ijk denote the strength (e.g., correlation value) of this connection if it exists. Thus, R_ijk = 0 if Y_ijk = 0 and R_ijk = 1 if Y_ijk > 0 with the following conditional probability:

$$\left\{\begin{array}{l}P\left(R_{\mathit{ijk}}=1|{\mathit{\beta }}_r;{\mathit{b}}_{\mathit{ri}}\right)=p_{\mathit{ijk}}\left({\mathit{\beta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\\ P\left(R_{\mathit{ijk}}=0|{\mathit{\beta }}_r;{\mathit{b}}_{\mathit{ri}}\right)=1-p_{\mathit{ijk}}\left({\mathit{\beta }}_r;{\mathit{b}}_{r\mathrm{i}}\right)\end{array}\right.$$ (A1)

where p_ijk is the probability of having a connection between node j and node k for the ith subject's network, β_r is the vector of fixed effects (population) parameters, and b_ri is the vector of random effects parameters for subject i. The fixed effects vector (β_r) represents the population estimates of the relationship between the connection probability of each nodal pair (dyad) and sets of covariates. The random effects vector (b_ri) represents the subject‐specific parameters that capture the correlation between repeated measurements (i.e., dyadic network features or connectivity values). The random effects capture how the relationships between the connection probability and sets of covariates vary around β_r by subject and node.

Given this and assuming S_ijk = [Y_ijk| R_ijk = 1], the two‐part mixed‐effects modeling framework for the probability and strength of brain connections can be defined by the following equations:

$$\left\{\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathit{\beta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\mathit{X}}_{\mathit{ijk}}^{\prime }{\mathit{\beta }}_r+{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\left(A2\right)\\ \\ \mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathit{\beta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\mathit{X}}_{\mathit{ijk}}^{\prime }{\mathit{\beta }}_s+{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+{\mathit{e}}_{\mathit{ijk}}\left(A3\right)\end{array}\right.$$

where similar to β_r and b_ri, β_s, and b_si are vectors of fixed and random effects parameters, respectively, that relate the strength of brain connections to sets of covariates. X_ijk is the known design matrix for the fixed effects vectors (i.e., β_r and β_s), Z_ijk is the known design matrix for the random effects that is analogous to X_ijk for the fixed effects, and e_ijk captures the random noise (not captured by random effects) in the connection strength between node j and node k of the ith subject's network. Equation (A2) is a logistic mixed model (from the GLMMs family) quantifying the relationship between the connection probability and covariates. Equation (A3) quantifies the relationship between the strength of brain connections and the same sets of covariates. FZT is the Fisher's Z‐transform applied to ensure that normality assumption is met. The fixed effects vectors can be decomposed into: β_r  = [β_{r, 0}β_{r, net} β_{r, COI}β_{r, int}β_{r, conf}]′ and β_s  = [β_{s, 0}β_{s, net} β_{s, COI}β_{s, int}β_{s, conf}]′, and the random effects vectors can be decomposed into: b_ri  = [b_{ri, 0}b_{ri, net}b_{ri, dist}δ_{ri, j}δ_{ri, k}]′ and b_si  = [b_{si, 0}b_{si, net}b_{si, dist}δ_{si, j}δ_{si, k}]′. β_{r, 0} and β_{s, 0} are population intercepts in the probability and strength models, respectively, and b_{ri, 0} and b_{si, 0} are subject intercept deviations from β_{r, 0} and β_{s, 0}. β_{r, net} and β_{s, net} quantify the population relationships between the dyadic and global network features and probability and strength of brain connections, respectively, and b_{ri, net} and b_{si, net} capture the deviation of these relationships by subject in the probability and strength models, respectively. β_{r, COI} and β_{s, COI} quantify the relationships between the (main) covariate of interest (e.g., task performance, age group or disease state) and probability and strength of brain connections, respectively. β_{r, int} and β_{s, int} quantify the relationships between the specified interactions, such as interactions of network features with the covariate of interest, and probability and strength of brain connections, respectively. β_{r, conf} and β_{s, conf} quantify the relationships between the confounders (e.g., sex, years of education) and the probability and strength of brain connections, respectively. The confounders also include spatial distance and square of spatial distance between brain regions as important geometric confounders (Friedman et al., 2014; Simpson & Laurienti, 2015), and b_{ri, dist} and b_{si, dist} capture the deviation from distance relationships with the probability and strength of brain connections by subject, respectively. δ_{ri, j} and δ_{ri, k} (in jk^th dyad) represent the propensities of node j and node k to be connected to other nodes in the network, and δ_{si, j} and δ_{si, k} represent the magnitudes of their connections. The random effects, b_ri and b_si, are assumed to be normally distributed and mutually independent.

TWO‐PART MIXED‐EFFECTS MODELING FRAMEWORK FOR GLOBAL AND LOCAL NETWORKS

1.

To make the whole‐brain modeling framework described in section 2.1 useable for analyzing local brain networks (subnetworks) within the context of the global network, we extend the models by adding binary subnetwork covariates as additional fixed effects. These covariates distinguish the subnetworks of interest from other regions of the brain. More detail is provided below.

Let T_ijk denote the new design matrix that includes additional columns for subnetworks of interest and their interactions with other covariates (of interest), and γ_r and γ_s denote the new fixed effects parameters that quantify the relationships between sets of covariates and the probability and strength of brain connections, respectively. Thus, we can write:

$$\left\{\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathit{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\mathit{T}}_{\mathit{ijk}}^{\prime }{\mathit{\theta }}_r\mathit{+}{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\left(A4\right)\\ \mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathit{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\mathit{T}}_{\mathit{ijk}}^{\prime }{\mathit{\theta }}_s\mathit{+}{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+{\mathit{e}}_{\mathit{ijk}}\left(A5\right)\end{array}\right.$$

where the new design matrix and fixed effects parameters can be decomposed into: ${\mathbf{T}}_{\mathrm{ijk}}^{\prime }=\left[{\mathbf{T}}_{\mathrm{ijk},\text{global}}^{\prime }{\mathbf{T}}_{\mathrm{ijk},\mathrm{soi}}^{\prime }{\mathbf{T}}_{\mathrm{ijk},\mathrm{int}\text{\_}\mathrm{soi}}^{\prime }\right]$, θ_r = [θ_{r, global}θ_{r, soi}θ_{r, int_soi}]′, and θ_s = [θ_{s, global}θ_{s, soi}θ_{s, int_soi}]′, in which T_{ijk, global}  = X_ijk, θ_{r, global}  = β_r, θ_{s, global}  = β_s, and X_ijk, Z_ijk, β_r, β_s, b_ri, b_si, and e_ijk are the same whole‐brain parameters described above. T_{ijk, soi} is the design matrix for subnetworks of interest (soi) with binary columns distinguishing the subnetwork regions from other regions of the brain, and T_{ijk, int_soi} is the design matrix for interactions of subnetwork covariates with other covariates of interest. T_{ijk, soi} can be further decomposed into: ${\mathbf{T}}_{\text{ijik},\mathrm{soi}}^{\prime }=\left[{\mathbf{T}}_{\text{ijik},{\mathrm{soi}}_1}^{\prime }{\mathbf{T}}_{\text{ijik},{\mathrm{soi}}_2}^{\prime }\dots {\mathbf{T}}_{\text{ijik},{\mathrm{soi}}_{\mathrm{n}}}^{\prime }\right]$, in which ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}}$ is one (1) if the connection between node j and node k of the ith subject's network is located within the qth subnetwork and zero (0) otherwise, and n is the number of focal subnetworks being analyzed. Similarly, θ_{r, soi} and θ_{s, soi} can be further decomposed into: ${\mathbf{\theta }}_{\mathrm{r},\mathrm{soi}}=\left[{\mathrm{\theta }}_{\mathrm{r},{\mathrm{soi}}_1}{\mathrm{\theta }}_{\mathrm{r},{\mathrm{soi}}_2}\dots {\mathrm{\theta }}_{\mathrm{r},{\mathrm{soi}}_{\mathrm{n}}}\right]\prime$ and ${\mathbf{\theta }}_{\mathrm{s},\mathrm{soi}}=\left[{\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_1}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_2}\dots {\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_{\mathrm{n}}}\right]\prime$, in which ${\mathrm{\theta }}_{\mathrm{r},{\mathrm{soi}}_{\mathrm{q}}}$ and ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_{\mathrm{q}}}$ quantify the relationships between the qth subnetwork covariate and the probability and strength of brain connections, respectively.

Estimates of the subnetwork covariate parameters (θ_{r, soi} and θ_{s, soi}) and their interaction covariate parameters (θ_{r, int_soi} and θ_{s, int_soi}) allow analyzing subnetworks. For example, estimate of ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_{\mathrm{q}}}$ shows if connection strength in subnetwork q is, on average, different than connection strength of other brain regions, or assuming a cognitive task performance measurement as a continuous variable, estimate of ${\mathrm{\theta }}_{\mathrm{r},\mathrm{cog}\times {\mathrm{soi}}_{\mathrm{q}}}$ shows if the cognitive score (denoted by cog here) is differentially related to connection probability in subnetwork q compared to connection probability in other brain regions. The estimates provide the quantified relationships (magnitude and direction) and statistical inference (p‐values). For more complex analyses or comparisons such as comparing subnetworks among groups (i.e., for assessing the effects of binary or categorical covariates of interest on subnetworks), estimates of linear combinations of parameters (i.e., contrast statements), including combinations of subnetwork covariate parameters (θ_{r, soi} and θ_{s, soi}), their interaction covariate parameters (θ_{r, int_soi} and θ_{s, int_soi}), and parameters of other covariates of interest should be used. (Estimation and inferential procedures for linear combinations of parameters in mixed models are briefly described in Appendix C). More detail about deriving linear combinations of parameters (linear contrast statements) for comparing subnetworks of interest among groups is provided in Appendix B.

Estimation of linear combinations of parameters in mixed models

1.

For linear combinations of parameters, statistical inference can be obtained by testing the following hypothesis:

$$H:\mathit{L}\left[\begin{array}{c}\mathbf{\theta }\\ \mathit{b}\end{array}\right]=\mathbf{0}\left(A6\right)$$

where θ (θ _r and θ _s in our case) and b are vectors of fixed and random effects parameters, and L is a numeric matrix with rows specifying the desired combinations of parameters. For inference, the following F‐statistics can be used:

$$F=\frac{\left[\begin{array}{c}\widehat{\mathbf{\theta }}\\ \widehat{\mathit{b}}\end{array}\right]{\mathit{L}}^{\prime }{\left(\mathit{L}\widehat{\mathit{C}}{\mathit{L}}^{\prime }\right)}^{-1}\mathit{L}\left[\begin{array}{c}\widehat{\mathbf{\theta }}\\ \widehat{\mathit{b}}\end{array}\right]}{\mathit{rank}\left(\mathit{L}\right)}\left(A7\right)$$

in which $\widehat{\mathbf{\theta }}$ and $\widehat{\mathbf{b}}$ are the fixed and random effects parameters estimates, and $\widehat{\mathbf{C}}$ is an approximation of the covariance matrix of ($\widehat{\mathbf{\theta }},\widehat{\mathbf{b}}\mathbf{-}\mathbf{b}$). F has an approximate F‐distribution, with rank (L) numerator degrees of freedom, and the denominator degrees of freedom are taken from the tests of fixed effects parameters. For more detail see (McLean and Sanders 1988).

CONTRAST STATEMENTS FOR COMPARING REGIONAL SUBNETWORKS

1.

Connection Probability And Strength In Subnetwork q

Let θ_{r, COI} and θ_{s, COI} denote the parameters for a (main) covariate of interest (i.e., grouping covariate denoted by COI), ${\mathrm{\theta }}_{\mathrm{r},{\mathrm{soi}}_{\mathrm{q}}}$ and ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_{\mathrm{q}}}$ denote the parameters for subnetwork q (a binary variable denoted by soi_q that distinguishes the connections within subnetwork q from all other network connections), ${\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}}$ and ${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}}$ denote the parameters for interactions of the covariate of interest (COI) with subnetwork q (soi_q), and θ_{r, rem} and θ_{s, rem} denote the parameters for the remaining covariates. Parameters with subscript r relate the covariates to the probability of brain connections, and those with subscript s relate the covariates to the connection strength. Using the defined parameters, we can expand Equation (A4) and Equation (A5) as follows:

$$\left\{\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathbf{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\mathit{T}}_{\mathit{ijk},\mathit{COI}}^{\prime }{\gamma }_{r,\mathit{COI}}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{soi}}_q}^{\prime }{\theta }_{r,{\mathit{soi}}_q}\mathbf{+}{{\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{soi}}_q}^{\prime }{\theta }_{r,\mathit{COI}\times {\mathit{soi}}_q}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{r,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\left(A8\right)\\ \mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathbf{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\mathit{T}}_{\mathit{ijk},\mathit{COI}}^{\prime }{\theta }_{s,\mathit{COI}}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{soi}}_q}^{\prime }{\theta }_{s,{\mathit{soi}}_q}\mathbf{+}{{\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{soi}}_q}^{\prime }{\theta }_{s,\mathit{COI}\times {\mathit{soi}}_q}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{s,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+e_{\mathit{ijk}}\left(A9\right)\end{array}\right.$$

in which Z_ijk is the design matrix for the random effects, b_ri and b_si are the random effects parameters, and ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI},}{\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}},{\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}},$ and T_{ijk, rem} are design matrices (or vectors) for the corresponding fixed effects covariates obtained from decomposing T_ijk (Note that ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}}\mathbf{=}{\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}}\mathbf{\times }{\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}}$). Thus, we will have the following equations for the two groups:

Group 1: ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}}\mathbf{=}\mathbf{1}$, T_{ijk, COI}  = 0, and ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}}\mathbf{=}\mathbf{0}$

$$\to \left\{\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathbf{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\theta }_{r,{\mathit{soi}}_q}{\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{r,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\left(A10\right)\\ \mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathbf{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\theta }_{s,{\mathit{soi}}_q}\mathbf{+}{{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{s,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+e_{\mathit{ijk}}\left(A11\right)\end{array}\right.$$

Group 2: ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{roi}}_{\mathrm{q}}}\mathbf{=}\mathbf{1}$, T_{ijk, COI}  = 1, and ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{roi}}_{\mathrm{q}}}\mathbf{=}\mathbf{1}$

$$\to \left\{\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathbf{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\theta }_{r,\mathit{COI}}+{\theta }_{r,{\mathit{soi}}_q}{\mathbf{+}{\theta }_{r,\mathit{COI}\times {\mathit{soi}}_q}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{r,\mathit{rem}}\mathbf{+}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\left(A12\right)\\ \mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathbf{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\theta }_{s,\mathit{COI}}+{\theta }_{s,{\mathit{soi}}_q}\mathbf{+}{\theta }_{s,\mathit{COI}\times {\mathit{soi}}_q}\mathbf{+}{{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{s,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+e_{\mathit{ijk}}\left(A13\right)\end{array}\right.$$

Thus, estimates of $\left({\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}}+{\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}}\right)$ and $\left({\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}\mathbf{+}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}}\right)$ quantify the differences of the connection probability and strength, respectively, between the two groups (Group2–Group1) in subnetwork q. This procedure can be repeated by deriving and estimating the appropriate linear combinations of parameters to analyze the effects of any desired binary or categorical (with any number of categories) covariate on the probability and strength of brain connections in subnetworks.

Network Features In Subnetwork q

In the original models (Equation A2 and Equation A3), the effect of any covariate of interest on brain network features (i.e., brain topology) is assessed through estimates of the interaction of that covariate with the network features. For example, considering COI to be a binary grouping covariate, estimate of θ_{s, COI × Geffi} shows if the global efficiency (denoted by Geffi) is differently associated with whole‐brain connection strength between the two groups. This provides inference about the differences of the global efficiency between the two groups. More specifically, θ_{s, Geffi} and θ_{s, Geffi}  + θ_{s, COI × Geffi} represent the slopes of the relationships between the global efficiency and connection strength in the two groups, respectively, and thus θ_{s, COI × Geffi} specifies the slope difference of the relationships which in turn can provide inference about the global efficiency difference between the two groups (see [Simpson & Laurienti, 2015] and [Bahrami et al., 2017] for more detail). Thus, to compare a network feature g in subnetwork q between the two groups, we should estimate the slope difference for the relationships of network feature g and connection probability and strength in subnetwork q between the two groups.

$$\left\{\begin{array}{l}\begin{array}{c}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathbf{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{r,{\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{net}}_g}^{\prime }{\theta }_{r,\mathit{COI}\times {\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{soi}}_q\times {\mathit{net}}_g}^{\prime }{\theta }_{r,{\mathit{soi}}_q\times {\mathit{net}}_g}\dots \\ {\mathbf{+}\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}^{\prime }{\theta }_{r,\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{r,\mathit{rem}}\mathbf{+}{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\end{array}\left(A14\right)\\ \begin{array}{c}\\ \end{array}\\ \begin{array}{c}\mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathbf{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{s,{\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{net}}_g}^{\prime }{\theta }_{s,\mathit{COI}\times {\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{soi}}_q\times {\mathit{net}}_g}^{\prime }{\theta }_{s,{\mathit{soi}}_q\times {\mathit{net}}_g}\cdots \\ \mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}^{\prime }{\theta }_{s,\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}{\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{s,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+e_{\mathit{ijk}}\end{array}\left(A15\right)\end{array}\right.$$

Let ${\mathrm{\theta }}_{\mathrm{r},{\mathrm{net}}_{\mathrm{g}}}$ and ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{net}}_{\mathrm{g}}}$ denote the parameters for network feature g, and ${\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}\times {\mathrm{net}}_{\mathrm{g}}}$, ${\mathrm{\theta }}_{\mathrm{r},{\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}$, ${\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}$, ${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{net}}_{\mathrm{g}}}$, ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}$, and ${\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}$ denote the interaction parameters in the probability and strength models, in which COI and soi_q are the grouping and subnetwork covariates, respectively. Using the defined parameters, we can expand Equations (A4) and (A5) as follows:

in which θ_{r, rem} and θ_{s, rem} are parameters for the remaining covariates. ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}$is the design vector for network feature g (i.e., ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}$ is the average network feature g of node j and node k of the i^th subject's network). Thus, we will have the following equations for the two groups:

Group 1: ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}\mathbf{=}\mathbf{1}\mathbf{\times }{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}\mathbf{=}{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}$, ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{net}}_{\mathrm{g}}}\mathbf{=}\mathbf{0}\mathbf{\times }{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}\mathbf{=}\mathbf{0}$, and ${\mathit{T}}_{\mathit{ijk},\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}\mathit{=}\mathbf{0}\mathit{\times }{\mathit{T}}_{\mathit{ijk},{\mathit{soi}}_q\times {\mathit{net}}_g}\mathit{=}\mathbf{0}$ →

$$\left\{\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathbf{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{r,{\mathit{net}}_g}\mathit{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{{\theta }_{r,{\mathit{soi}}_q\times {\mathit{net}}_g}}_{\mathit{ijk}}\mathit{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{r,\mathit{rem}}\mathit{+}{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\left(A16\right)\\ \mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathbf{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{s,{\mathit{net}}_g}\mathit{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{{\theta }_{s,{\mathit{soi}}_q\times {\mathit{net}}_g}}_{\mathit{ijk}}\mathit{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{s,\mathit{rem}}\mathit{+}{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+e_{\mathit{ijk}}\left(A17\right)\end{array}\right.$$

Group 2: ${\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}\mathbf{=}\mathbf{1}\mathbf{\times }{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}\mathbf{=}{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}$, ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{net}}_{\mathrm{g}}}\mathbf{=}\mathbf{1}\mathbf{\times }{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}\mathbf{=}{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}$, and ${\mathbf{T}}_{\mathrm{ijk},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}\mathbf{=}\mathbf{1}\mathbf{\times }{\mathbf{T}}_{\mathrm{ijk},{\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}\mathbf{=}{\mathbf{T}}_{\mathrm{ijk},{\mathrm{net}}_{\mathrm{g}}}$

$$\to \left\{\begin{array}{l}\begin{array}{l}\mathit{\text{logit}}\left(p_{\mathit{ijk}}\left({\mathbf{\theta }}_r;{\mathit{b}}_{\mathit{ri}}\right)\right)={\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{r,{\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{r,\mathit{COI}\times {\mathit{net}}_g\cdots }\\ \mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{{\theta }_{r,{\mathit{soi}}_q\times {\mathit{net}}_g}}_{\mathit{ijk}}{\mathbf{+}\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{r,\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{r,\mathit{rem}}\mathbf{+}{\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{ri}}\end{array}\left(A18\right)\\ \begin{array}{c}\\ \end{array}\\ \begin{array}{l}\mathit{FZT}\left(S_{\mathit{ijk}}\left({\mathbf{\theta }}_s;{\mathit{b}}_{\mathit{si}}\right)\right)={\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{s,{\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{s,\mathit{COI}\times {\mathit{net}}_g}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{{\theta }_{r,{\mathit{soi}}_q\times {\mathit{net}}_g}}_{\mathit{ijk}}\cdots \\ \mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{{\theta }_{s,{\mathit{soi}}_q\times {\mathit{net}}_g}}_{\mathit{ijk}}\mathbf{+}{\mathit{T}}_{\mathit{ijk},{\mathit{net}}_g}^{\prime }{\theta }_{s,\mathit{COI}\times {\mathit{soi}}_q\times {\mathit{net}}_g}{\mathbf{+}{\mathit{T}}_{\mathit{ijk},\mathit{rem}}^{\prime }{\mathbf{\theta }}_{s,\mathit{rem}}\mathbf{+}\mathit{Z}}_{\mathit{ijk}}^{\prime }{\mathit{b}}_{\mathit{si}}+e_{\mathit{ijk}}\end{array}\left(A19\right)\end{array}\right.$$

Thus, the estimates of $\left({\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}\times {\mathrm{net}}_{\mathrm{g}}}+{\mathrm{\theta }}_{\mathrm{r},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}\right)$ and $\left({\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{net}}_{\mathrm{g}}}+{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{soi}}_{\mathrm{q}}\times {\mathrm{net}}_{\mathrm{g}}}\right)$ quantify the slope differences of the relationships of network feature g with connection probability and strength in subnetwork q between the two groups (Group 2–Group 1). This approach can be repeated to analyze the effects of any desired binary or categorical (with any number categories) covariate on any network feature in subnetworks.

MODELING RESULTS FOR OVERLAPPING SUBNETWORKS WITH INCORRECT θ_s

1.

For overlapping regions, sub_{a_out} and sub_{b_out}, with the following fixed and random effect parameters, the modeling results change to the ones shown in Table A1. (Since modeling without regional subnetworks and modeling subnetworks as independent networks yield the same results as the standard t‐test analysis for our simulations [for the reasons mentioned in the article], we only show the results of the t‐test analysis in this sections and the following sections.)

$${\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{a_{\mathit{\text{out}}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{a_{\mathit{\text{out}}}}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$$

Table A1.

Modeling and t‐test results for known difference in only one of two overlapping regions with incorrect γ_s

	Modeling results				T‐test results
Parameter	Estimate	SE	*p‐value	Parameter	Avg–Diff	*p‐value
θs, 0	0.472	0.006538	<.0001
θs, COI	0.04049	0.009247	<.0001	Whole‐brain	0.034913	.000255
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.03111	0.001900	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.01469	0.002691	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	0.02580	0.009342	.0058	suba_out	0.026026	.002935

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

As Table A1 presents, removing the subnetwork covariates for sub_{b_out} results in incorrect results (see Table 9).

POPULATION DIFFERENCES IN DIFFERENT DIRECTIONS IN TWO OVERLAPPING SUBNETWORKS

1.

Modeling results for simulated data sets with different connection strength in overlapping regions, sub_{a_out} (higher in the second group) and sub_{b_out} (lower in the second group), are shown below for different choices of the fixed effects parameters.

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=1.048\times {\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{in}}{\mathit{sub}}_{a\_\mathit{\text{out}}},i=1,\dots ,40$$

${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=0.935\times {\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_1}+\text{noise}$

for jk^thconnections locatedinsub_{b_out}, i = 1, …, 40

$$1.{\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{a_{\mathit{\text{out}}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{a_{\mathit{\text{out}}}}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$$

$$2.{\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{b_{\mathit{\text{out}}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{b_{\mathit{\text{out}}}}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$$

$$3.{\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{a_{\mathit{\text{out}}}}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{b_{\mathit{\text{out}}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{a_{\mathit{\text{out}}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{b_{\mathit{\text{out}}}}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0.}$$

Table A2.

Modeling and t‐test results for known differences in different directions in two overlapping subnetworks

	Modeling results				T‐test results
Parameter	Estimate	SE	*p‐value	Parameter	Avg‐diff	*p‐value
θs, 0	0.3472	0.006083	<.0001
θs, COI	−0.00489	0.008604	.5699	Whole‐brain	0.00254	.765847
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.03111	0.001786	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	0.01948	0.002540	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	0.01459	0.008696	.0934	suba_out	0.014454	.07843

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

As Table A2 presents, when covariates for sub_{b_out} are not included, the model fails to find the actual simulated differences.

Table A3.

Modeling and t‐test results for known differences in different directions in two overlapping subnetworks

	Modeling results				T‐test results
Parameter	Estimate	SE	*p‐value	Parameter	Avg–diff	*p‐value
θs, 0	0.3455	0.006100	<.0001
θs, COI	0.007141	0.008627	.4078	Whole‐brain	0.00254	.765847
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.04385	0.002047	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.02111	0.002916	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.01397	0.008891	.1162	subb_out	−0.01338	.100654

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

As Table A3 presents, when covariates for sub_{a_out} are not included, the model fails to find the actual simulated differences.

Table A4.

Modeling and t‐test results for known differences in different directions in two overlapping subnetworks

	Modeling results				T‐test results
Parameter	Estimate	SE	*p‐value	Parameter	Avg–diff	*p‐value
θs, 0	0.3595	0.006180	<.0001
θs, COI	−0.00014	0.008740	.9873	Whole‐brain	0.00254	.765847
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.03483	0.001785	<.0001
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.04746	0.002053	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	0.01789	0.002538	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.01929	0.002924	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	0.01775	0.008813	.0440	suba_out	0.014454	.07843
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.01943	0.008973	.0303	subb_out	−0.01338	.100654

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

As Table A4 presents, with both subnetwork covariates included in the model, the modeling framework correctly finds the differences in each of the subnetworks, and controls for the counterbalancing (systemic) effects of each subnetwork on the other one as well as their effects on other brain regions.

DIFFERENCE PRESENT AT THE GLOBAL LEVEL, BUT WITH A MORE COMPLEX PATTERN AT LOCAL LEVELS

1.

Here, we used four regional covariates for modeling connections within subnetworks and b (sub_a and sub_b), and for connections from sub_a and sub_b to other brain regions (sub_{a_out}, and sub_{b_out}). Thus, we had the following fixed and random effects parameters:

$${\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_a}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_b}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{a\_\mathit{\text{out}}}}{\mathrm{\theta }}_{\mathrm{s},{\mathit{sub}}_{b\_\mathit{\text{out}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_a}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_b}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{a\_\mathit{\text{out}}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathit{sub}}_{b\_\mathit{\text{out}}}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$$

Known differences were created such that for the second group, the strength of: (a) connections within sub_a were increased by a factor of 1.5, (b) connections within sub_b were reduced by a factor of 0.8, (c) connections from sub_a to other brain regions (i.e., connections represented by sub_{a_out}) were not changed, (d) connections from sub_b to other brain regions (i.e., connections represented by sub_{b_out}) were increased by a factor of 1.5, and finally (e) connections of all other brain regions were reduced by a factor 0.9:

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=1.5\times {\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{wihtin}}{\mathit{sub}}_a,i=1,\dots ,40$$

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=0.8\times {\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{wihtin}}{\mathit{sub}}_b,i=1,\dots ,40$$

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=1.5\times {\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{in}}{\mathit{sub}}_{b\_\mathit{\text{out}}},i=1,\dots ,40$$

$${\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_2}=0.9\times {\mathbf{C}}_{\mathrm{ijk},{\mathrm{gr}}_1}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{in}}\mathit{\text{other regions}},i=1,\dots ,40$$

As Table A5 presents, the modeling framework correctly identifies the known differences at both the global and local levels due to its multivariate nature and its inclusion of interactions between local regions. However, the t‐test analysis fails to disentangle the differences correctly. The t‐test results incorrectly indicate that the connections of sub_{a_out} (connections from sub_a to other brain regions) are significantly different between the two groups. Also, as shown by the positive “Avg–Diff,” the t‐test identified significantly stronger whole‐brain connections (connections of other brain regions not included through subnetwork covariates) in Group 2, which is entirely misleading because the data simulation included weaker global connections in Group 2. The modeling results, accounting for the subnetwork covariates, did identify that connections of all brain regions other than subnetworks a and b are significantly weaker in the second group.

Table A5.

Modeling and t‐test results for known difference at the global level, but with a more sophisticated pattern at local levels

	Modeling results				T‐test results
Parameter	Estimate	SE	*p‐value	Parameter	Avg–diff	*p‐value
θs, 0	0.3552	0.006629	<.0001
θs, COI	−0.03012	0.009377	.0013	Whole‐brain	0.041041	2.49e‐05
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}$	0.01109	0.003147	.0004
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{b}}}$	0.03066	0.005695	<.0001
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.03114	0.002042	<.0001
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	−0.04430	0.002283	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.2145	0.004466	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}}}$	−0.04985	0.008090	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	0.01795	0.002905	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	0.1784	0.003249	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.1843	0.009910	<.0001	suba	0.181719	1.31e‐20
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}}}$	−0.07997	0.01199	<.0001	subb	−0.07814	8.58e‐08
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}\text{\_}\mathrm{\text{out}}}}$	−0.01218	0.009382	.1943	suba_out	0.020381	.017487
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{b}\text{\_}\mathrm{\text{out}}}}$	0.1483	0.009557	<.0001	subb_out	0.151459	6.20e‐24

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

GENDER AS AN EXOGENOUS CONFOUNDING VARIABLE

1.

Finally, we evaluated the performance of the methodological extensions with respect to differences in an exogenous confounding effect, gender in this case. We labeled 14 subjects in the first group and 26 subjects in the second group as males, and the remaining subjects in the two groups as females. sub_a served as our subnetwork covariate. The following fixed and random effects parameters were used:

$${\mathbf{\theta }}_{\mathrm{s}}\mathbf{=}\left[{\mathrm{\theta }}_{\mathrm{s},0}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{sex}}{\mathrm{\theta }}_{\mathrm{s},\mathrm{COI}\times {\mathrm{sub}}_{\mathrm{a}}}{\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\times \mathrm{sex}}\right]\prime ,{\mathrm{b}}_{\mathrm{si}}\mathbf{=}{\mathrm{b}}_{\mathrm{si},0}$$

where θ_{s, sex} and ${\mathrm{\theta }}_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\times \mathrm{sex}}$ are parameters for gender (a binary variable with one for males and zero for females) and the interaction of gender with sub_a, and other parameters are the same as the ones described throughout the article. A difference was created such that for males, the strength of connections within sub_a was increased by a factor of 1.5:

$${\mathbf{C}}_{\mathit{ijk},\text{male}}=1.5\times {\mathbf{C}}_{\mathrm{ijk},\text{male}}+\text{noise}\mathit{\text{for}}{\mathit{jk}}^{\mathit{th}}\mathit{\text{connections located}}\mathit{\text{wihtin}}{\mathit{sub}}_a,i=\mathit{\text{male subjects}}$$

As Table A6 presents, unlike the t‐test analysis, our methodology controls for the confounding effect of gender and correctly shows that the difference is restricted to the sub_a between males and females.

Table A6.

Modeling and t‐test results for known local difference in gender as an exogenous confounding variable

Modeling results				T‐test results
Parameter	Estimate	SE	*p‐value	Parameter	Avg–diff	*p‐value
θs, 0	0.3317	0.007242	<.0001
θs, COI	0.000377	0.009533	.9688	Whole‐brain	0.009492	.316916
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}}$	0.03295	0.003251	<.0001
θs, sex	−0.00153	0.009533	.8728	Sex	0.029124	4.49e‐102
${\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.005089	0.004343	.2413
${\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\mathrm{\times }\mathrm{sex}}$	0.2716	0.004343	<.0001
${\theta }_{\mathrm{s},\mathrm{COI}}\mathrm{+}{\theta }_{\mathrm{s},\mathrm{COI}\mathrm{\times }{\mathrm{sub}}_{\mathrm{a}}}$	0.005466	0.01027	.5946	suba	0.085305	.011279
${\theta }_{\mathrm{s},\mathrm{sex}}\mathrm{+}{\theta }_{\mathrm{s},{\mathrm{sub}}_{\mathrm{a}}\mathrm{\times }\mathrm{sex}}$	0.3045	0.004101	<.0001	Sex in suba	0.272581	0

The bold values indicate the relevant parameter estimates.

Note: Modeling p‐values are adjusted using the adaptive FDR procedure detailed in (Benjamini & Hochberg, 2000).

DEFAULT MODE NETWORK MASK

1.

The DMN mask used for extracting the 37 ROIs of DMN is shown in Figure A1 below.

Figure A1.

Axial and sagittal images of the utilized default mode network mask [Color figure can be viewed at http://wileyonlinelibrary.com]

Bahrami M, Laurienti PJ, Simpson SL. Analysis of brain subnetworks within the context of their whole‐brain networks. Hum Brain Mapp. 2019;40:5123–5141. 10.1002/hbm.24762

DATA AVAILABILITY STATEMENT

Simulated datasets and implemented SAS scripts are provided as supplementary files accompanying this paper. Simulated datasets and codes are provided on the mendeley data repository (https://data.mendeley.com/datasets/wpxk9s6wbf/1).