Summary
Functional connectivity is defined as the undirected association between two or more functional magnetic resonance imaging (fMRI) time series. Increasingly, subject-level functional connectivity data have been used to predict and classify clinical outcomes and subject attributes. We propose a single-index model wherein response variables and sparse functional connectivity network valued predictors are linked by an unspecified smooth function in order to accommodate potentially nonlinear relationships. We exploit the network structure of functional connectivity by imposing meaningful sparsity constraints, which lead not only to the identification of association of interactions between regions with the response but also the assessment of whether or not the functional connectivity associated with a brain region is related to the response variable. We demonstrate the effectiveness of the proposed model in simulation studies and in an application to a resting-state fMRI data set from the Human Connectome Project to model fluid intelligence and sex and to identify predictive links between brain regions.
Keywords: fMRI, Networks, Nonparametric regression, Penalized splines, Sparsity
1. Introduction
A central goal in neuroimaging research is to relate interindividual variability in phenotypes to properties of brain functionality. In functional magnetic resonance imaging (fMRI) studies, blood oxygen-level dependent (BOLD) signals are used as a proxy for neural activity. Functional connectivity is defined as the undirected association between the measured BOLD signal between two or more anatomically distinct brain regions. Analysis of functional connectivity has been shown to be useful in the assessment of mental illnesses and disorders including autism (Nair and others, 2013), Alzheimer’s disease (Greicius and others, 2004), and Schizophrenia (Lynall and others, 2010).
Functional connectivity can be represented as a network, with nodes as brain regions and links or edges as relations or interactions between regions. Marginal correlation matrices, while computationally easy to estimate, quantify the indirect interactions between regions and hence are not quite interpretable. In contrast, partial correlation matrices quantify the direct interactions between regions and are therefore scientifically more meaningful (Marrelec and others, 2006; Varoquaux and others, 2010; Smith and others, 2011; Solo and others, 2018). However, they are computationally more challenging to estimate, especially in large dimensional settings, as their estimation often involves the inversion of the covariance matrix. Thus, regularization methods are required, with the most common approach being sparse network estimation (Friedman and others, 2008; Mazumder and Hastie, 2012a,b; Ryali and others, 2012; Liu and Luo, 2012). Sparsity is appealing as it implies no direct relationship between brain regions. The joint estimation of multisubject inverse covariance matrices for sharing information about the strength of connections and sparsity patterns has also been studied and applied to measuring brain region connectivity (Danaher and others, 2014; Liang and others, 2016; Qiu and others, 2016; Mejia and others, 2018; Colclough and others, 2018).
Recent literature has focused on identifying the association between brain networks and a response variable using functional connectome-based predictive methods (Woo and others, 2017; Dadi and others, 2019). Standard statistical models have previously been used to predict and classify clinical outcomes and patient attributes based on functional connectivity. For example, random forest classifiers have been used to classify schizophrenic patients (Shen and others, 2010) and support vector machines have been used to discriminate between young and old subjects (Meier and others, 2012) and to identify depressed subjects (Zeng and others, 2012). Despite the promise of these high-performing predictive models, they often can be difficult to interpret, which impedes the analysis of brain pattern associations. Models that exploit the network structure of functional connectivity could lead to increased statistical efficiency and improved interpretation. Some recent literature has focused on developing classification models that incorporate network structure. Wang and others (2019, 2020), and Vogelstein and others (2012) propose linear classification models that incorporate subgraph structure in predictors, Guha and Rodriguez (2020) propose a class of network shrinkage priors for Bayesian learning of undirected network predictor coefficients, and Relión and others (2019) propose a linear classification model that encourages sparsity in the number of nodes and edges of the coefficient matrix, a pattern of sparsity we also seek to incorporate in our proposed model.
A key assumption common to the majority of the previously discussed models is the linear relationship between response variables and predictors, which can be seriously violated in practice. Nonparametric regression methods can be used to accommodate potential nonlinear relationships present in the data. In fully nonparametric regression, the response variable is generally assumed to satisfy some smoothness assumptions, but no assumptions are made on the form of dependence on the predictors. While this offers flexibility in modeling, results are often undesirable due to the curse of dimensionality. A useful alternative to a fully nonparametric regression approach is the single-index model, in which the response variable is modeled as a nonparametric function of a linear combination of the predictors called an index (Härdle and Stoker, 1989; Ichimura, 1993; Carroll and others, 1997; Wang and Yang, 2009).
In this article, we propose a single-index model, where response variables and sparse functional connectivity network valued predictors are linked by an unspecified smooth function, which accommodates potential nonlinear relationships between the response and the predictors. The parameter space can be thought of as a network of parameters, and we impose two types of sparsity in the effects of the elements of the networks, similar to Relión and others (2019). The first type of sparsity, denoted edge sparsity, places constraints on individual edges in the parameter network. This allows for the pruning of the connection between two nodes while simultaneously allowing the two nodes to remain connected to other nodes in the network. The second type of sparsity, denoted node sparsity, places constraints of the nodes themselves. Hence, in this model, any node is either in or out of the network. In the latter case, all connections (edges) between the node and all other nodes are removed from the network. The proposed model is estimated via an iterative algorithm, which involves a subalgorithm based on the alternating direction method of multipliers (ADMM). The proposed model and estimation method can be easily generalized for general matrix predictors with other types of sparsity penalties; see Section 4 for more details. Additionally, we consider various regularization approaches to estimating individual functional connectivity matrices as inputs to this model and evaluate their performances for predicting the responses.
The remainder of the article is organized as follows: Section 2 describes a motivating data set, Section 3 discusses the estimation of functional connectivity networks, Section 4 introduces the proposed single-index model and our estimation methods, and Section 5 assesses the numerical performance of our model with a simulation study. In Section 6, we apply our prediction method to the motivating data set. Finally, Section 7 concludes the article with a discussion.
2. Materials
Resting-state fMRI (rs-fMRI) has become increasingly popular for evaluating regional interactions between brain regions that occur when a subject is not performing an explicit task. It has been shown that fluctuations in BOLD signal show strong and reliable correlations at rest even in distant brain regions. In addition, the spatial patterns of the correlations have been shown to be reliably observed across subjects and scanning sessions. Because of the lack of task demands, rs-fMRI removes the burden of subject compliance and training, making it an attractive option in many studies of development and clinical populations.
We analyzed data from 820 subjects with complete rs-fMRI scans from the 900-subject public data release from the Human Connectome Project (HCP). The HCP provides the required ethics and consent needed for study and dissemination. The sample consists of healthy adults (22–35 years old, 453 females) scanned on a 3-T Siemens connectome-Skyra. For each subject, a total of four 15 min fMRI scans with a temporal resolution of 0.73 s and a spatial resolution of 2-mm isotropic were available. The preprocessing pipeline followed the procedure outlined in Smith and others (2013a,b). Spatial preprocessing was applied using the procedure described by Glasser and others (2013). ICA, followed by FMRIBs ICA-based X-noisefier from the FMRIB Software Library (Griffanti and others, 2014), was used for structured artifact removal, removing more than 99% of the artifactual ICA components in the data set. Group spatial ICA was used to obtain a parcellation of 100 components that cover both the cortical surfaces and the subcortical areas.
The parcellation was used to project the fMRI data into 100-dimensional data corresponding to 100 brain regions. Data from the four scans, each measuring the BOLD signal at 1200 time points, were concatenated into a single run. Hence, the final rs-fMRI data used in the analysis consisted of a 
matrix for each of 820 subjects. In addition, fluid intelligence and sex were extracted for each subject and were used as response variables in our model.
3. Estimation of functional connectivity networks
Functional connectivity networks describe how different regions of the brain interact. The strength of interaction is often characterized by partial correlations. Let 
be the 
th subject’s inverse covariance matrix. That subject’s partial correlation matrix is then 
with 
for 
and 
. Thus, estimation of partial correlations can be derived from that of the inverse covariance matrix.
To estimate a subject’s inverse covariance matrix, a simple method is to invert the sample covariance matrix, i.e., to let 
, where 
is the 
th subject’s sample covariance matrix. For example, for the rs-fMRI data, the 
data matrix for each subject leads to a 
sample covariance matrix. Although not an issue for our data application, a drawback of this approach is that the sample covariance is not invertible when the number of regions exceeds the sample size (the number of time points for the rs-fMRI data), which could be the case if a high resolution of brain regions is desired. One solution is to instead invert the Ledoit–Wolf estimator (Ledoit and Wolf, 2004), which takes the form 
, where 
is a scalar between 0 and 1. In application, we will use the closed-form optimal value of 
as derived by Ledoit and Wolf (2004). This optimal value minimizes the expected squared Frobenious norm of 
using theoretical results. Each subject will have a different value of 
. Then 
is the resulting inverse covariance for subject 
.
Alternatively, one may consider the graphical lasso (Friedman and others, 2008), which gives 
, where 
is the trace operator and 
is a regularization parameter controlling the level of regularization in the estimate. The estimate may be element-wise sparse, which, in the context of functional connectivity networks, implies conditional independence between the signals of two brain regions given the signals of all other regions. We will use 5-fold cross-validation (Bien and Tibshirani, 2011) for selecting the value of 
, the value of which could vary across subjects. The glasso estimates for three subjects of the HCP study are shown in Figure 1.
Fig. 1.
Partial correlation estimates via the glasso for three subjects of the HCP study. The estimate for each subject is computed for 100 brain regions over 4800 time points recorded over one hour. Red elements indicate positive partial correlations, while blue elements indicate negative partial correlations. Nodes are grouped by network and the corresponding networks are denoted in the rows of each matrix. V, visual network; S, somatomotor; DA, dorsal attention network; VA, ventral attention network; FPN, frontoparietal network; DMN, default mode network; BG, basal ganglia; C, cerebellum; BS, brain stem.
The glasso estimate usually gives different sparsity patterns between subjects. It may sometimes be desirable to instead obtain the same sparsity pattern across subjects. This can be achieved by joint estimation with group penalties. The set of joint glasso estimates 
is the solution to
 |
where 
is a tuning parameter. We will use 5-fold cross-validation for selecting the value of 
. We refer to this approach as joint glasso as it is a special case of the joint graphical lasso (Danaher and others, 2014).
The three methods provide three different sparsity patterns: the Ledoit–Wolf estimator has no sparsity, the glasso gives sparsity at the individual level, and the joint glasso ensures sparsity at the group level. As a zero value of partial correlation implies conditional independence between the two associated brain regions, the Ledoit–Wolf estimator is the least interpretable while the joint glasso gives the most parsimonious interpretation as the same sparsity pattern is enforced across subjects. However, the glasso is more flexible than the joint glasso as it accommodates potentially varying sparsity patterns across subjects. We will estimate individual functional connectivity matrices using these methods and consider the estimates as inputs to our regression model. We shall investigate how these estimates compare when they are used as predictors to relate to response variables. Specifically, we are interested in their predictability of the response variable and the interpretability of the resulting model estimation for identifying meaningful associations of interactions between brain regions with the response variable.
4. Single-index model
Suppose that 
is the response variable and 
is the functional connectivity network for subject 
. Assuming that the response variable is from a distribution in the exponential family, we propose a single-index model in which the conditional mean of 
given 
is
 |
(4.1) |
where 
is a known monotonic function, 
is an unknown and unspecified index function, 
is an unknown coefficient matrix, and the inner product 
is 
. As 
is a partial correlation matrix with 1s on the diagonal, it is reasonable to assume that the coefficient matrix 
is also symmetric and has 0 values on the diagonal. Let 
be a matrix operator that stacks the lower triangle (excluding the diagonal) of a square matrix into a vector. Then, 
, where 
and 
.
As the number of coefficients in the coefficient matrix 
is 
, which could well exceed the sample size 
, regularization is required for model estimation. More importantly, it is desirable to assume sparsity in the coefficient matrix for better interpretation. Indeed, if an edge is not associated with the response, then the corresponding coefficient should be zero, and if a node is not associated with the response, then the coefficients corresponding to the edges that are associated with the node should all be zero. The former is called “edge sparsity” and can be achieved using the lasso, while the latter is called “node sparsity” and can be achieved by imposing a carefully constructed group lasso. Therefore, we consider the following optimization problem:
 |
(4.2) |
subject to the identifiability constraints 
, 
, and 
for 
, in addition to the symmetry constraint 
. Here, 
is the negative log-likelihood function that depends on both 
and 
and 
is a penalty function. To impose node & edge sparsity, we let
 |
The term associated with the tuning parameter 
is the node penalty, as it penalizes as a group all the coefficients associated with all edges connecting each node. The term associated with 
is the edge penalty as it penalizes separately each coefficient associated with individual edges. The proposed method is called a “node & edge” model. When 
, it becomes an “edge” only model and when 
, it becomes a “node” only model. The node & edge regularization term has been used previously in Relión and others (2019) and is suitable for the model with functional connectivity networks as predictors. However, model (4.1) allows for more general matrix predictors, and other regularization terms may be useful. Some popular matrix penalties include the row penalty, column penalty, and the rank penalty, which penalizes the nuclear norm of the coefficient matrix. Zhou and others (2013) and Zhou and Li (2014) discuss regularization techniques in regression models with matrix-valued covariates and Feng and others (2020) make use of general matrix regularization penalties in fitting single-index models. In Section 5, we will compare the node & edge penalty to the lasso (which is equivalent to the edge only model) and the elastic net, which linearly combines the 
and squared 
norms of the elements of 
.
4.1. Model estimation
Simultaneously estimating the unknown smooth function 
and the coefficient matrix 
is computationally challenging; sequentially estimating 
given 
and estimating 
given 
is computationally more efficient and a more commonly used strategy in fitting single-index models. We will use this strategy for estimating the model parameters. Specifically, the first step will be to estimate 
given 
and the second step will be to estimate 
given 
. The solutions to both steps are as follows.
Given the coefficient matrix 
, we estimate 
using penalized splines (Eilers and Marx, 1996). Let 
be the index of the 
th subject so that 
. The penalized splines approximate 
by a linear combination of B-splines; that is, 
where 
is a set of cubic B-spline basis functions defined in the range of the 
s, and 
is a vector of associated coefficients. Specifically, we estimate the coefficients by minimizing
 |
where 
is the second-order difference operator. The second term is a roughness penalty controlled by the smoothing parameter 
. Existing theoretic works on penalized splines show that as long as the number of basic functions is sufficiently large, penalized splines work well (Xiao, 2019). In our simulation studies and real data analysis, we use 10 basis functions. We select 
via generalized cross-validation (GCV; Ruppert, 2002).
Given 
, we estimate the coefficient matrix 
. To deal with the overlapping group lasso in the objective function (4.2), we adopt the union of groups method (Jacob and others, 2009), which allows variables to be selected as long as they belong to at least one group with positive coefficients. Then, we solve the optimization using the alternating direction method of multipliers (ADMM; Boyd and others, 2011). We reformulate (4.2) as
 |
where 
denotes the group of indices of coefficients in 
that are associated with the 
th node, 
is a local variable, and 
is the indicator function of the constraint set given by
 |
The above reformulation leads to the following scaled form ADMM updates:
 |
where 
and 
are the stacked vectors of 
’s and 
’s, respectively, and 
is an indicator matrix with 1s in row 
indicating the index of two “copies” of the 
th element of 
in 
and 0s otherwise. Essentially, the matrix multiplication 
creates a vector of averages of the “copies” of the elements of 
, so that each element of 
is regularized towards the average of its two corresponding elements in 
.
The update of 
is solved via the method of proximal gradient descent. Letting 
denote the current iterate 
, we make the iterative update
 |
until convergence of 
with learning rate 
. The derivation of the gradients in the case of normally distributed and binary responses is provided in Section A of the Supplementary material available at Biostatistics online. In practice, we use backtracking line search for determining the value of 
for each step. We refer to Boyd and others (2004) for further details. The update of 
is solved by
 |
where 
is the projection of 
onto the surface of a unit ball with 
. The update of 
is solved by the proximal operator of the 
norm. Letting 
,
 |
The estimation of 
is solved by computing these iterative updates until convergence of the global variable 
. The overall minimization problem (4.2) is solved by iterating over the two-step algorithm until convergence. Each step of the two-step algorithm is guaranteed convergence. If the tuning parameter 
in the estimation of 
were fixed, the overall two-step algorithm is guaranteed convergence by properties of block coordinate descent. However, as we propose to select 
via GCV at each iteration, the overall algorithm does not have a guarantee of convergence. In our experiments, we have found that the selected value of 
stabilizes quickly, which gives empirical evidence that the algorithm is well-behaved in practice.
For ease of selecting values for the tuning parameters 
and 
via cross-validation, it is convenient to reparameterize the tuning parameters 
and 
as 
and 
respectively, so that 
is the overall level of regularization and 
controls the balance between the two sparsity inducing penalties. Optimal values of 
and 
are obtained through grid search in a 2D sequence.
5. Simulation study
We conduct a simulation study to evaluate the performance of the proposed regression model in terms of variable selection and out-of-sample prediction. We consider three link functions, 
, 
, and 
and simulate responses from model (4.1). The predictors 
s are the 820 functional connectivity matrices estimated from the HCP data with the glasso estimator. We consider two settings for the number of nodes in the covariate matrices. In the first setting, the 
s will consist of only the estimated edges between the 25 nodes in either the visual and default mode network. In the second, the 
s will consist of the estimated edges between the all 100 nodes. We will refer to these two settings as 
and 
respectively.
Continuous responses are simulated from a normal distribution with mean 
and variance 
, with 
in high signal-to-noise (SNR) settings and 
in low SNR settings. Binary responses are simulated from a 
distribution with 
, with 
in high signal settings and 
in low signal settings. We consider three cases for the coefficient matrix 
. The first case is the adjacency matrix of a circle network (Wang and others, 2012), where 
with 
for 
and the upper triangle elements are matched with the lower triangle elements for symmetry. The second case is a random symmetric matrix where each node (column/row) has a probability of 0.5 of being included in the network, and each edge has a probability of 0.5 being included, conditional on node inclusion. The value of each included element is 1. The third case is the adjacency matrix of a small-world network (Watts and Strogatz, 1998), a graph with a high clustering coefficient and a short average path length. The small-world network is generated according to the Watts–Strogatz model as implemented in the R package igraph (Csardi and others, 2006). The random symmetric matrix demonstrates node & edge sparsity, while the circle network adjacency matrix demonstrates only edge sparsity.
While we have proposed the node & edge sparse regularization term for the estimation of 
, other regularization terms may also be used to estimate the coefficient matrix in the second step of the two-step algorithm described in Section 4. Specifically, we may employ the lasso penalty 
and the elastic net penalty 
(Zou and Hastie, 2005), where 
is a weight parameter. A simple modification to the proposed estimation procedure of 
can be used and hence the details are omitted. Five-fold cross-validation is used for tuning the parameter(s). Note that 50 simulations are conducted under each setting and we compare the node & edge penalty with the above two. In addition, we also fit linear models by setting the estimate of the smooth function 
to the identity function and running only the second step of the two-step algorithm.
The simulation results are summarized for the random network coefficient matrix in Tables 1 and 2, and the simulation results for the circle network and small-world network coefficient matrices are summarized in Section B of the Supplementary material available at Biostatistics online. Predictive performance is measured by the mean squared errors of the predictions in the case of continuous responses, and by area under the receiver operating characteristic curve in the case of the binary responses. True positive rate is defined as the proportion of elements of 
that are correctly estimated as zero. Similarly, false positive rate is defined as the proportion of elements of 
that are incorrectly estimated as zero. We do not include a measure of estimation error of the coefficient matrix as 
is only identifiable up to a constant of proportionality.
Table 1.
Mean squared error (standard error) of the normal response simulations and AUC (standard error) of the binary response simulations with the random network coefficient matrix for the proposed model and linear model with three penalties.
| Nonlinear | Linear | |||||||
|---|---|---|---|---|---|---|---|---|
|
|
SNR | Node & edge | Elastic net | Lasso | Node & edge | Elastic net | Lasso |
| Normal response | ||||||||
|
25 | Low | 0.175 (0.119) | 0.177 (0.121) | 0.181 (0.124) | 0.170 (0.111) | 0.174 (0.113) | 0.168 (0.109) |
|
25 | High | 0.079 (0.032) | 0.079 (0.033) | 0.080 (0.033) | 0.080 (0.034) | 0.081 (0.034) | 0.079 (0.034) |
|
100 | Low | 0.896 (0.200) | 0.930 (0.208) | 0.965 (0.216) | 0.997 (0.204) | 1.033 (0.212) | 0.963 (0.197) |
|
100 | High | 0.480 (0.163) | 0.509 (0.173) | 0.540 (0.182) | 0.510 (0.165) | 0.541 (0.174) | 0.481 (0.155) |
|
25 | Low | 0.021 (0.007) | 0.021 (0.007) | 0.021 (0.007) | 0.150 (0.061) | 0.153 (0.062) | 0.149 (0.060) |
|
25 | High | 0.009 (0.005) | 0.009 (0.005) | 0.009 (0.005) | 0.072 (0.038) | 0.073 (0.038) | 0.071 (0.037) |
|
100 | Low | 0.045 (0.029) | 0.046 (0.031) | 0.048 (0.032) | 0.942 (0.226) | 0.976 (0.233) | 0.910 (0.217) |
|
100 | High | 0.030 (0.019) | 0.031 (0.020) | 0.033 (0.021) | 0.564 (0.164) | 0.594 (0.172) | 0.536 (0.156) |
|
25 | Low | 0.146 (0.087) | 0.147 (0.087) | 0.150 (0.088) | 0.523 (0.204) | 0.532 (0.206) | 0.514 (0.201) |
|
25 | High | 0.088 (0.044) | 0.089 (0.044) | 0.091 (0.045) | 0.494 (0.111) | 0.503 (0.109) | 0.484 (0.112) |
|
100 | Low | 0.756 (0.041) | 0.761 (0.040) | 0.764 (0.040) | 0.962 (0.225) | 0.998 (0.234) | 0.928 (0.216) |
|
100 | High | 0.592 (0.024) | 0.598 (0.025) | 0.601 (0.025) | 0.504 (0.124) | 0.533 (0.130) | 0.477 (0.119) |
| Binary response | ||||||||
|
25 | Low | 0.700 (0.041) | 0.697 (0.041) | 0.697 (0.041) | 0.770 (0.074) | 0.746 (0.096) | 0.786 (0.073) |
|
25 | High | 0.851 (0.036) | 0.850 (0.036) | 0.850 (0.036) | 0.910 (0.042) | 0.900 (0.045) | 0.919 (0.040) |
|
100 | Low | 0.701 (0.036) | 0.692 (0.037) | 0.692 (0.037) | 0.806 (0.085) | 0.777 (0.098) | 0.831 (0.082) |
|
100 | High | 0.839 (0.048) | 0.826 (0.054) | 0.826 (0.054) | 0.892 (0.059) | 0.873 (0.063) | 0.910 (0.055) |
|
25 | Low | 0.804 (0.061) | 0.789 (0.063) | 0.772 (0.065) | 0.542 (0.033) | 0.542 (0.033) | 0.545 (0.033) |
|
25 | High | 0.923 (0.028) | 0.914 (0.030) | 0.905 (0.032) | 0.655 (0.080) | 0.655 (0.080) | 0.664 (0.072) |
|
100 | Low | 0.843 (0.075) | 0.819 (0.078) | 0.794 (0.080) | 0.540 (0.056) | 0.540 (0.056) | 0.542 (0.043) |
|
100 | High | 0.929 (0.050) | 0.912 (0.055) | 0.894 (0.059) | 0.693 (0.165) | 0.693 (0.165) | 0.678 (0.161) |
|
25 | Low | 0.801 (0.054) | 0.784 (0.058) | 0.767 (0.061) | 0.568 (0.045) | 0.568 (0.045) | 0.576 (0.046) |
|
25 | High | 0.910 (0.036) | 0.901 (0.037) | 0.919 (0.034) | 0.674 (0.133) | 0.674 (0.133) | 0.686 (0.127) |
|
100 | Low | 0.795 (0.097) | 0.763 (0.116) | 0.826 (0.083) | 0.574 (0.099) | 0.574 (0.099) | 0.586 (0.099) |
|
100 | High | 0.910 (0.046) | 0.889 (0.052) | 0.929 (0.040) | 0.602 (0.073) | 0.602 (0.073) | 0.612 (0.083) |
Table 2.
True positive rate (false positive rate) of the normal response and binary response simulations with the random network coefficient matrix for the proposed model and linear model with three penalties.
| Nonlinear | Linear | |||||||
|---|---|---|---|---|---|---|---|---|
|
|
SNR | Node & edge | Elastic net | Lasso | Node & edge | Elastic net | Lasso |
| Normal response | ||||||||
|
25 | Low | 0.844 (0.028) | 0.805 (0.019) | 0.758 (0.011) | 0.864 (0.028) | 0.821 (0.020) | 0.772 (0.012) |
|
25 | High | 0.827 (0.000) | 0.780 (0.000) | 0.727 (0.000) | 0.828 (0.000) | 0.785 (0.000) | 0.731 (0.000) |
|
100 | Low | 0.985 (0.943) | 0.982 (0.934) | 0.979 (0.924) | 0.987 (0.950) | 0.984 (0.942) | 0.980 (0.932) |
|
100 | High | 0.974 (0.895) | 0.970 (0.881) | 0.966 (0.869) | 0.973 (0.893) | 0.968 (0.880) | 0.964 (0.867) |
|
25 | Low | 0.887 (0.004) | 0.848 (0.003) | 0.799 (0.003) | 0.879 (0.022) | 0.842 (0.011) | 0.798 (0.008) |
|
25 | High | 0.862 (0.001) | 0.822 (0.001) | 0.768 (0.001) | 0.870 (0.012) | 0.831 (0.009) | 0.783 (0.008) |
|
100 | Low | 0.985 (0.943) | 0.982 (0.932) | 0.978 (0.922) | 0.992 (0.973) | 0.990 (0.968) | 0.988 (0.961) |
|
100 | High | 0.973 (0.897) | 0.969 (0.888) | 0.965 (0.876) | 0.988 (0.954) | 0.986 (0.948) | 0.984 (0.941) |
|
25 | Low | 0.878 (0.013) | 0.840 (0.006) | 0.796 (0.005) | 0.974 (0.662) | 0.956 (0.612) | 0.933 (0.575) |
|
25 | High | 0.833 (0.001) | 0.791 (0.001) | 0.749 (0.000) | 0.974 (0.734) | 0.959 (0.690) | 0.940 (0.638) |
|
100 | Low | 0.981 (0.938) | 0.978 (0.928) | 0.974 (0.916) | 1.000 (1.000) | 0.999 (0.999) | 0.998 (0.998) |
|
100 | High | 0.970 (0.889) | 0.966 (0.877) | 0.961 (0.866) | 1.000 (1.000) | 0.999 (0.999) | 0.998 (0.998) |
| Binary response | ||||||||
|
25 | Low | 0.940 (0.611) | 0.915 (0.546) | 0.880 (0.483) | 0.932 (0.562) | 0.901 (0.509) | 0.864 (0.437) |
|
25 | High | 0.867 (0.090) | 0.829 (0.058) | 0.780 (0.036) | 0.860 (0.090) | 0.818 (0.059) | 0.776 (0.038) |
|
100 | Low | 0.992 (0.980) | 0.990 (0.975) | 0.987 (0.969) | 0.993 (0.984) | 0.991 (0.978) | 0.988 (0.973) |
|
100 | High | 0.986 (0.958) | 0.983 (0.950) | 0.980 (0.941) | 0.985 (0.953) | 0.982 (0.944) | 0.978 (0.934) |
|
25 | Low | 0.926 (0.590) | 0.898 (0.521) | 0.857 (0.457) | 0.992 (0.980) | 0.979 (0.944) | 0.962 (0.912) |
|
25 | High | 0.862 (0.111) | 0.821 (0.076) | 0.772 (0.064) | 0.949 (0.696) | 0.922 (0.631) | 0.895 (0.575) |
|
100 | Low | 0.991 (0.976) | 0.989 (0.973) | 0.987 (0.966) | 0.999 (0.999) | 0.999 (0.999) | 0.998 (0.997) |
|
100 | High | 0.984 (0.948) | 0.981 (0.939) | 0.977 (0.930) | 0.996 (0.991) | 0.994 (0.989) | 0.992 (0.985) |
|
25 | Low | 0.932 (0.571) | 0.906 (0.503) | 0.868 (0.415) | 0.988 (0.926) | 0.976 (0.898) | 0.953 (0.856) |
|
25 | High | 0.878 (0.100) | 0.831 (0.067) | 0.786 (0.043) | 0.955 (0.707) | 0.940 (0.660) | 0.911 (0.624) |
|
100 | Low | 0.992 (0.981) | 0.990 (0.977) | 0.987 (0.970) | 0.999 (0.999) | 0.999 (0.999) | 0.997 (0.998) |
|
100 | High | 0.984 (0.949) | 0.980 (0.941) | 0.977 (0.932) | 0.999 (0.999) | 0.999 (0.999) | 0.997 (0.997) |
As expected, there is no significant difference in the performances of the linear and nonlinear models when the index function is linear. However, when the index function is nonlinear, the nonlinear models clearly outperform the linear models in prediction error, true positive rate, and false positive rate. Further, the lasso and elastic net penalties both result in node sparsity less frequently than the node–edge penalty. While the results of the three penalties are often similar, the node & edge model results in the lowest prediction errors and best sparsity classifications more often than not. This, along with the desirable interpretation of the node & edge sparsity pattern, demonstrates the value of the proposed model.
6. Application to HCP data
We apply the proposed methods to the HCP data discussed in Section 2. The predictors, i.e., the functional connectivity matrices, can be estimated using either the inverted Ledoit–Wolf estimator, the glasso, or the joint glasso as described in Section 3. To assess their resulting performances, we compute the three estimators for all subjects in the HCP study. The estimates for 80% of the subjects are used to fit the proposed single-index models of sex and fluid intelligence and the estimated model components are then used to generate out-of-sample predictions for the remaining subjects.
The ROC curves for the fitted models of sex using the three estimators of functional connectivity are compared in Figure 2(a). We see no apparent difference between the estimator imposing individual level sparsity (glasso) and the estimator that does not impose sparsity (Ledoit–Wolf). The results suggest that while imposing sparsity in the estimation of functional connectivity leads to clearer interpretation of model results, the predictive performance of the node & edge model is not affected by the inclusion of nonzero estimates of the node weights between conditionally independent nodes. We also see that the joint glasso leads to a lower area under the ROC curve (AUC), which might be due to individual differences in functional connectivity being dampened by the joint regularization. The mean squared errors (MSE) of the models of fluid intelligence are plotted against the level of regularization in Figure 2(b). The conclusions regarding the three estimators of functional connectivity are the same as in the model for the sex variable. Based on these results, we suggest the use of the glasso in application.
Fig. 2.
(a) ROC curves of the single-index model predictions using the Ledoit-Wolf, glasso, and joint glasso as input for performing sex discrimination. AUC values are reported along with legend labels. (b) MSE of the single-index model predictions by level of regularization using the Ledoit-Wolf, glasso, and joint glasso as input for performing fluid intelligence prediction.
While the Ledoit–Wolf and glasso estimates result in similar out-of-sample predictive performance, the resulting coefficient matrix estimates do show a slight difference in levels of sparsity. In the model of sex, the estimate of the coefficient matrix using the Ledoit–Wolf estimates of functional connectivity has 
edge-wise sparsity, while the estimate of the coefficient matrix using the glasso estimates has 
. In the model of fluid intelligence, the Ledoit–Wolf estimates and glasso estimates have edge-wise sparsities of 
and 
respectively. Defining node-wise sparsity as the proportion of nodes with more than 
of their associated edges estimated to be zero, the estimate of the coefficient matrix using the Ledoit-Wolf estimates of functional connectivity has 
node-wise sparsity, while the estimate of the coefficient matrix using the glasso estimates has 
. In the model of fluid intelligence, the Ledoit–Wolf estimates and glasso estimates have edge-wise sparsities of 
and 
respectively.
Figures 3(a) and (b) illustrate the estimated coefficient matrix and mean function of the single-index model of sex, respectively. The proposed model of sex includes significant interactions between the visual and frontoparietal networks, and within the default mode network. These results agree with existing studies. For example, Filippi and others (2013) found that the strength of connectivity between sensorimotor, visual, and rostral lateral prefrontal areas was higher in males than in females, and Ritchie and others (2018) found the strength of connectivity within the default mode network to be higher in females than males. Figures 3(c) and (d) illustrate the estimated coefficient matrix and mean function of the single-index model of fluid intelligence, respectively. The proposed model of fluid intelligence includes significant interactions between the default mode and frontoparietal networks. These results also agree with existing studies. For example, Hearne and others (2016) identified a relationship between fluid intelligence and functional connectivity between the default mode and frontoparietal networks, and Finn and others (2015) found that connectivity associated with the frontoparietal network was predictive of individuals’ fluid intelligence. The effective degrees of freedom of the estimate of 
is 4.8, indicating a nonlinear relationship between the response and covariates. Indeed, fitting a linear model to the index results in an MSE of 19.4, whereas the MSE of the single-index model is 16.2.
Fig. 3.
(a) Estimated coefficient matrix for the single-index model of sex. The grid lines in the matrix plot distinguish the nodes associated with each region. (b) Estimated mean function of the single-index model of sex in red and logistic fit in blue. (c) Estimated coefficient matrix for the single-index model of fluid intelligence. The grid lines in the matrix plot distinguish the nodes associated with each region. (d) Estimated mean function of the single-index model of fluid intelligence in red and linear fit in blue.
7. Discussion
We proposed a single-index model that relates brain functional connectivity networks to response variables to deal with potential nonlinear relationships. We employed two types of sparsity in model estimation motivated by the network structure of the functional connectivity matrix. The induced patterns of sparsity provided variable selection in both the edges and the nodes of the network associated with a sparse estimate of functional connectivity. The proposed model therefore could not only identify links between regions that relate to the response but also directly assess whether the functional connectivity associated with a brain region is related to the response variable.
We found in simulation studies that the proposed model has better prediction error and true positive and false positive rates on average in estimating network sparsity patterns than linear models and nonlinear models that do not exploit network structure. We also found via applications to the HCP data that sparse estimation of functional connectivity matrices led to more interpretable model estimates.
Finally, we applied the proposed approach to perform sex discrimination and fluid intelligence prediction in the HCP data set. The most discriminative connectivity between regions and their effects reported by our model were found to be consistent with existing results.
While the proposed model was developed using functional connectivity matrix data, it can be applied to any predictor with a form of symmetric matrices. In the article, the proposed model was applied to rs-fMRI data in HCP; however, we anticipate that it might be useful for task-based data as well.
8. Software
Software in the form of R code, together with a sample input data set and complete documentation, are available at https://github.com/clbwvr/FC-SIM.
Supplementary Material
Acknowledgments
We gratefully acknowledge the comments and suggestions made by the Associate Editor and two referees that led to a much improved paper. Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research; and by the McDonnell Center for Systems Neuroscience at Washington University.
Conflict of Interest: None declared.
Contributor Information
Caleb Weaver, Department of Statistics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27606, USA.
Luo Xiao, Department of Statistics, North Carolina State University, 2311 Stinson Drive, Raleigh, NC 27606, USA.
Martin A Lindquist, Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health, 615 N. Wolfe Street, Baltimore, MD 21205, USA.
Supplementary material
Supplementary material is available online at http://biostatistics.oxfordjournals.org.
Funding
National Institute of Health (NIH) (R01 NS112303, R56 AG064803, and R01 AG064803 to L.X., in part); National Institute of Health (NIH) (R01 EB016061 and R01 EB026549 to M.A.L.) from the National Institute of Biomedical Imaging and Bioengineering.
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