Prediction of an Outcome Using NETwork Clusters (NET-C)

Abstract

Birth weight is a key consequence of environmental exposures and metabolic alterations and can influence lifelong health. While a number of methods have been used to examine associations of trace element (including essential nutrients and toxic metals) concentrations or metabolite concentrations with a health outcome, birth weight, studies evaluating how the coexistence of these factors impacts birth weight are extremely limited. Here, we present a novel algorithm NETwork Clusters (NET-C), to improve the prediction of outcome by considering the interactions of features in the network and then apply this method to predict birth weight by jointly modelling trace element and cord blood metabolite data. Specifically, by using trace element and/or metabolite subnetworks as groups, we apply group lasso to estimate birth weight. We conducted statistical simulation studies to examine how both sample size and correlations between grouped features and the outcome affect prediction performance. We showed that in terms of prediction error, our proposed method outperformed other methods such as a) group lasso with groups defined by hierarchical clustering, b) random forest regression and c) neural networks. We applied our method to data ascertained as part of the New Hampshire Birth Cohort Study on trace elements, metabolites and birth outcomes, adjusting for other covariates such as maternal body mass index (BMI) and enrollment age. Our proposed method can be applied to a variety of similarly structured high-dimensional datasets to predict health outcomes.

1. Introduction

Detecting associations between inter-related high-dimensional exposures and intermediates with health outcomes is an emerging challenge in biomedical re-search. As the complexity of interactions of high-dimensional factors increases, predicting the response variable can be difficult. This is because the network structures cannot be easily measured and the effect of high-dimensional data elements cannot be precisely estimated using traditional multi-linear regression techniques. In addition to prediction models using regression methods [1], approaches involving subgroups of highly correlated variables [2]; [3] with connectivity-based clustering methods, hierarchical clustering [4];[5], and Ran-dom forest regression utilizing tree bagging methods [6];[7] have been developed in literature. Hierarchical clustering fails to prioritize more important clusters over less significant groups of factors or disconnected factors due to its heuristic properties [8];[9]. Random forest regression requires more time to construct decision trees with high complexity [10] and does not consider highly correlated variables. while the network-based feature selection was used to predict prognosis of cancer [11], its application is limited to the classification of categorical outcomes rather than the prediction of a continuous outcome. To help overcome these limitations, we developed a statistical method to predict health outcomes using well-defined clusters that embrace complex relations between high dimensional biological factors of environmental exposures and metabolites.

Specifically, we present a method named NET-C (prediction of outcome using NETwork Clusters) designed to improve regression methods for predicting an outcome utilizing network analysis. We hypothesize that analyzing dense sub-networks of biological factors that are relevant to a particular health outcome can strengthen our prediction models. This method groups interacting biological factors using lasso and modularity maximization. Then it models their association with health outcome using regularized regression for grouped predictors. Model-based clustering implemented by lasso detects and prioritizes subgroups of strongly connected factors by shrinkage and selection. These grouped factors are used in regression models with covariates [12] to predict a health outcome. We tested the proposed method in simulation studies under various parameter settings (e.g. sample size and the association of features with the outcome) and compared its prediction performance with group lasso using groups determined by hierarchical clustering, or random forest regression. We chose the example of predicting birth weight as an outcome influenced by maternal exogenous and endogenous factors [13]. Specifically, we applied our method to data on placental trace element concentrations, cord-blood metabolite concentrations, birth weight and other covariates collected as part of the New Hampshire Birth Cohort Study (NHBCS).

Given the intricate biological pathway from environmental exposures such as trace elements and cord-blood metabolites to human health, we assume that our proposed method can be applied to various health outcomes. Applying our proposed method to birth weight is one significant approach to illustrate how high-dimensional exposures and metabolites can affect a specific outcome.

2. Methods

2.1. Overall Procedure

NET-C can be summarized in the following three main steps:

Step1: Applying lasso to all features to generate a network.

Step2: Identifying densely connected subnetworks in the estimated network and classify those subnetworks as distinct groups using modularity maximization algorithm such as Louvain.

Step3: Fitting a group lasso model for outcome and feature groups defined in Step 2.

We also compared the predictive performance of NET-C with alternative outcome prediction methods, including random forest regression, neural networks, and hierarchical clustering based group-lasso.

2.2. Subnetwork generation and group assignment: Lasso and Louvain

We began by identifying the network of interacting features using lasso. Let N × p data matrix X represents N observations of a p-dimensional random vectors X= (X₁,X₂, …X_p) which are conventionally called features in literature, let x₁, x₂, …, x_N denote the rows of matrix X. We assume that X ~ N_p (0, Σ), with Σ being the positive definite covariance matrix while V={1,2,…,p] is the index set of the nodes corresponding to the p features. We further define the precision matrix of X as Ω = Σ⁻¹ which characterizes the conditional dependence between the features. The network G = (V,E) among the features can then be defined based on Ω, i.e., if ωij ≠ 6= 0, then we define an edge between node i and node j; otherwise, there is no edge between the node i and node j. A penalized linear regression [14] using each feature as outcome and the rest as predictors is utilized to determine if there is an edge between two given features. Since the normality is assumed for X, the penalized log-likelihood is defined as follows,

$$\frac{1}{N}{\sum }_{i=1}^N{\left(x_{ij}-{\beta }_0-x_i^T{\beta }_j\right)}^2+\lambda {\sum }_{k=1}^p\left|{\beta }_{jk}\right|$$ (1)

with β_j = (β_j1, …, β_j(j−1), 0, β_j(j+1), …, β_jp). Here we utilize L₁ penalty of β_j for a sparse network estimation. It is known that such L₁ penalty can remove the zero or very weak conditional correlation among the features so that we can focus on the significant relationship among X₁,…,X_p. After applying lasso to detect features significantly associated with each column of data, a p × p symmetrical matrix representing partial correlations of p features is formed based on βj, j=1,…,p. This p × p matrix corresponds to a network representing significant interactions of p elements. The tuning parameter λ involved in (1) is obtained by minimizing Akaike’s Information Criterion (AIC) [15] which is defined as,

$$AIC(\Omega )=-l_N(\widehat{\Omega }(G))+2k,$$ (2)

where $l_N(\widehat{\Omega }(G))$ represents the log-likelihood function corresponding to a given network G, $\widehat{\Omega }(G)$ is the maximum likelihood estimation of Ω given the network G. Because we have assumed the normal distribution for the features, we have the following expression for l_N(·),

$$l_N(\Omega )=N/2\left\{\text{log}det((\Omega ))-\mathit{\text{trace}}(S\Omega )\right\}$$ (3)

where S is the empirical covariance matrix.

Negative Gaussian log-likelihood in the formula (2) measures how well the model fits and indicates the difference in the negative Gaussian log-likelihood of the current model from a null model [16], and k is the number of estimated parameters which are non-zero. We choose the λ which minimizes AIC as the value for the tuning parameter. Given such choice of λ, we then minimize objective function (1) with respect to β_j, j = 1·, p. After this step, the set of estimated non-zero parameters which indicate interactions of features is obtained.

To determine the final groups of features consisting of dense networks, a community detection algorithm, Louvain, using maximal modularity score [17], is applied to the estimated network of features by lasso to detect and classify highly dense subnetworks of features.

Given a network with N nodes, the estimated interactions of nodes by lasso and estimated weights on edges between nodes, each node of the network is initially assigned to a distinct community. Each node is removed from its community and placed into another community to maximize the gain of modularity in the network. Given a network with correlations between variables, the modularity, M, is computed as follows [18]:

$$M=\frac{1}{2E}{\sum }_{ij}\left[W_{ij}-\frac{d_i{d}_j}{2E}f\left(g_i,g_j\right)\right]$$ (4)

where g_i is the community to which vertex i belongs, the function f(x, y) equals to 1 if x = y and 0 otherwise, d_i is the degree of vertex i, E is the number of edges in the network, and W_ij is the partial correlation of the edge between i and j. We assume that the gain of modularity acquired by moving a node i into a different community C is improved when the sum of edge weights between node i and the node members of C increases. This process continues until the positive gain modularity in the network is not possible and then each node stays in its finally assigned community. It is required that each node should always belong to one community throughout the whole process. Each finalized community is considered to be a subnetwork of node features [17]. After applying Louvain algorithm to the estimated interactions of features, the final dense subnetworks of interacting features are identified.

Algorithm 1:

Generating Clusters with Lasso and Louvain

Input: An N-by-P input data matrix Output: Community (Group) index for all feature variables in the data matrix: for p = 1 to P do find optimal λ by minimizing AIC in Equation (2) using Lasso A network of P nodes with correlations between nodes is generated While ΔM > 0 do Compute ΔM using equation (3) by plugging values into E, Wij, di and dj Move nodes into different communities Output final community assignment for all feature variables

2.3. Alternative Methods

We have three alternative methods, group lasso using all distinct groups, group lasso using groups determined by hierarchical clustering and random forest regression. The simplest way to group features is to consider that all features belong to all distinct groups. Hierarchical clustering classifies objects into a dendrogram whose branches are the target clusters using dissimilarities over samples by implementing branch cutting or pruning [19]. To generate the dendrogram, distance between features was measured as follows using Euclidean distance measurement [20]:

$$\mathit{\text{Dist}}(x,y)=\sqrt{{\sum }_{i=1}^f{\left(x_i-y_i\right)}^2}$$ (5)

where f is the attribute size. We used complete linkage rule for hierarchical clustering which maximizes the distance between two features in distinct clusters. The number of highly connected subnetworks or clusters generated by hierarchical clustering is identically set to the number of groups detected byNET-C.

Random forest regression can compute the importance of each feature and predict a continuous outcome with the training dataset to fit the models and the testing datasets [10];[21]. In the random forest regression algorithm, with the fixed number of trees, each regression sub-tree is generated in a bootstrap sample from the initial training data. Features which can minimize the residual sum of squares along down left and right branches of each node in the tree are selected [22]. In our statistical simulation, the number of trees is 500, the number of features randomly sampled as candidates at each split on the node is set to 40 and the minimum size of terminal nodes is 5. Variable importance is assigned to each feature after testing how the assignments of bigger importance to specific features increase prediction error as the features are permuted.

Input feature variables and the target outcome for the supervised training were used to implement a single-hidden layer neural network with 120 hidden neurons. Input weight values for each pair of input and hidden neurons and output weight values for each pair of hidden and output neurons are computed [23].

2.4. Parameter estimation by group lasso

After features are grouped using lasso and modularity maximization, or other grouping methods, we utilized group lasso to associate outcome with all features. Group indexes are assigned to features belonging in each group. We use group lasso formula as follows:

$${\text{min}}_{\beta \in R^p}\frac{1}{n}{\sum }_{i=1}^n{\left(y_i-x_i^T\beta \right)}^2+\lambda {\sum }_j\sqrt{K_j}{\Vert {\beta }_j\Vert }_1$$ (6)

A lasso penalty is applied to the Euclidean (L2) norm of the coefficients in each group, leading group-specific variable selection. If one group is selected, all elements in the group would have non-zero coefficient estimates. Otherwise, the effect of all elements in the group would shrink to zero. Our proposed method, NET-C, groups features with lasso and modularity maximization and predicts an outcome with these grouped features.

2.5. Datasets and additional pre-processing and validation steps for the real data application

We applied NET-C to training and testing datasets from a large human subjects study, the NHBCS, to predict birth weight using placental trace elements and cord blood metabolites. The NHBCS a longitudinal pregnancy cohort of mother-infant pairs that has been described previously [24]. Briefly, pregnant women who seek prenatal care at a participating study clinic were eligible for enrollment if they were 18 to 45 years old, used a private, unregulated well for drinking water, and were not planning to move prior to delivery. All study participants provided written informed consent prior to participation according to the guidelines of the Committee for the Protection of Human Subjects at Dartmouth.

Placental biopsies were collected from infants born to NHBCS-enrolled mothers at the time of delivery. Following removal of any decidua tissue and avoiding calcium deposits and connective tissue, 1 cm deep and 1–2 cm wide biopsies are taken at the base of the cord insertion of the fetal side of the placenta to minimize heterogeneity. Placental biopsies were stored in trace element-free tubes, which were labeled with a sample barcode ID and stored at −80°C until analysis. Cord-blood was collected at the time of delivery by clinical staff and processed and stored at −80°C until analysis.

Mass spectrometry (MS) was used to quantify the metallome of nutrient and toxic elements in placental biopsies, as well as the metabolome in cord blood plasma [25];[26]. A total of 220 cord blood metabolites and computed ratios of metabolites were measured using the Biocrates AbsoluteIDQTM p180 kit (Biocrates Life Sciences AG, Innsbruck, Austria). In placental biopsies, 24 trace elements were quantified by Inductively Coupled Plasma Mass Spectrometry (7700x Agilent, Santa Clara, CA), with analysis following the quality control procedures described in EPA 6020a [27]. The NHBCS dataset also included relevant covariate information such as maternal enrollment age and body mass index (BMI). Covariate selection followed the standard described in this paper [27]. When we generated highly dense subnetworks of grouped trace elements or metabolites, log-transformed (base e) was also considered to improve model fit and normalize residuals.

3. Simulation Study

3.1. Statistical Simulation

Consider the following regression equation, Y (outcome) = β₀ + X * β + ε, in which N × p matrix X is the N observations of p features; ε = (ε₁, …, ε_N) is the noise vector drawn from the uniform distribution U[−20,20] with independent components; and β is the coefficient vector for p features. The features were considered as continuous variables with normal distribution. The sample size N considered include 150, 180, 210 and 240. The number of features is set to be 120. We assume there exist 9 clusters of highly connected features, which consist of 3 big clusters of size 9, 3 medium clusters of size 6 and 3 small clusters of size 3. There were 54 features involving in these clusters. The correlation between these 120 features can be depicted by a correlation matrix C. For the first cluster, we have C[i,j] with integers i, j ∈ S₁ × T₁ where S₁ and T₁ are integers from 1 to 9; for second cluster, we have i, j ∈ S₂ × T₂ where S₂ and T₂ are integers from 10 to 18; for the third cluster, we have i, j ∈ S₃ × T₃ where S₃ and T₃ are integers from 19 to 27 indicates 3 large clusters; for the cluster 4, we have C[i,j] with i, j ∈ S₄ × T₄ where S₄ and T₄ are integers from 28 to 33; for the cluster 5, we have i, j ∈ S6 × T₅ where S₅ and T₅ are integers from 34 to 39; for cluster 6, we have i, j ∈ S₆ × T₆ where S₆ and T₆ are integers from 40 to 45 indicates 3 medium clusters; for cluster 7, we have C[I,j] with integers i, jS₇ × T₇ where S₇ and T₇ are integers from 46 to 48; for cluster 8, we have i, j ∈ S₈ × T₈ where S₈ and T₈ are integers from 49 to 51; for cluster 9, we have i, j ∈ S₉ × T₉ where S₉ and T₉ are integers from 52 to 54 indicates 3 small clusters (Figure 1). The strength of feature interactions in each square submatrix, C[i,j], were randomly selected between 0.1 and 0.7 or between −0.7 and −0.1 and the strength of other feature interactions outside submatrices of clusters was selected between 0.01 and 0.012 or between −0.01 and −0.012. The elements of such matrix C below the diagonal are then set to be equal to the corresponding elements above the diagonal. In order to ensure C can be used as precision matrix, the diagonal entries of C is further set to be 1. We employed mvrnorm function in R package, “MASS”, to generate an N × p continuous matrix X based on C

Figure 1:

The matrix representation of three large clusters, three medium clusters and three small clusters. The strength of feature interactions in each square submatrix, C[i,j], in the matrix is randomly chosen between 0.1 and 0.7 or between −0.7 and −0.1, and the strength of other feature interactions outside submatrices of clusters is chosen between 0.01 and 0.012 or between −0.012 and −0.01. C[i,j] with integers i, j ∈ S₁ × T₁ where S₁ and T₁ are integers from 1 to 9, i, j ∈ S₂ × T₂ where S₂ and T₂ are integers from 10 to 18, and i, j ∈ S₃ × T₃ where S₃ and T₃ are integers from 19 to 27 represents 3 large clusters, C[i,j] with integers i, j ∈ S₄ × T₄ where S₄ and T₄ are integers from 28 to 33, i, j ∈ S₅ × T₅ where S₅ and T₅ are integers from 34 to 39, and i, j ∈ S₆ × T₆ where S₆ and T₆ are integers from 40 to 45 represents 3 medium clusters and C[i,j] with integers i, j ∈ S₇ × T₇ where S₇ and T₇ are integers from 46 to 48, i, j ∈ S₈ × T₈ where S₈ and T₈ are integers from 49 to 51, and i, j ∈ S₉ × T₉ where S₉ and T₉ are integers from 52 to 54 represents 3 small clusters

Since we have 120 features, a vector of coefficients, β is a vector of positive or negative values. The first 54 values in β indicate β_c, a vector of coefficients of features in any of 9 defined clusters and the other 66 values in β indicate β_n, a vector of coefficients of features not belonging to any clusters. We define three different ranges of coefficients; L, M, and S which are randomly chosen from uniform distribution of [10,20], [3,9], and [0,2] respectively. Using, L, M, and S, β_c is defined as follows.

Case 1: β_c = a vector of 54 {S}

Case 2: β_c = a vector of 54 {M}

Case 3: β_c = a vector of 18 {L, M, S}

Case 4: β_c = a vector of 27 {L} for clusters of size 9 and of 9 {L, M, S} for clusters size of 6 and 3

Case 5: β_c = a vector of 45 {L} for clusters of size 9 and of 3 {L, M, S} for clusters size of 3

β_n is a vector of 66 randomly chosen numbers ranging from 0 to 20. Positive and negative signs are randomly assigned to βc and β_n. With βc and β_n, β = [βc, β_n]. In the regression formula, we set β₀ to 20 and b to a real number randomly chosen from Uniform distribution of [−20, 20]. The outcome is generated using 20 + Xβ + U(−20, 20).

3.2. Validation – definition of prediction error

To evaluate how our proposed model will perform in new data, we compared the prediction error by four prediction methods: NET-C, group lasso with groups defined by hierarchical clustering, group lasso with all distinct groups, neural networks and random forest regression. In each generated dataset, samples were randomly reordered with replacement and 70% of reordered samples were allotted to the training set and 30% of reordered samples were allotted to the testing set; In the real data application, 267 samples were as-signed to the training set and 114 samples were assigned to the testing set. The overall prediction error was computed by averaging the prediction error over distinct training and testing datasets for as follows:

$$\text{Prediction error}=\frac{1}{N}{\sum }_{i=1}^N{\left(Y_{\mathit{\text{test}}}^i-X_{\mathit{\text{test}}}^i{\beta }_{\mathit{\text{train}}}\right)}^2,$$ (7)

where N is the number of distinct datasets, $Y_{\mathit{\text{test}}}^i$ is observed outcome values in the testing set and $X_{\mathit{\text{test}}}^i{\beta }_{\mathit{\text{train}}}$ is predicted outcome values in the testing set; This estimation of prediction error can be considered as a “hold-out” validation process. Distinct training and testing datasets were generated 100 times for simulation and 1000 times for real data application. In total, 20 examples were generated with four different sample sizes and five different cases of coefficients of features indicating effect of features on the outcome.

In our real data application, to estimate positive or negative coefficients of trace elements, metabolites, or covariates affecting birth weight, we implemented a 95% bootstrap confidence interval method. Over 1000 datasets, we get 1000 coefficient values for trace elements, metabolites, and covariates respectively. For each variable, after sorting the 1000 relevant coefficients in increasing order, we obtained a lower bound of the confidence interval, the 25^th smallest coefficient, and an upper bound of the confidence interval, the 975^th smallest coefficient.

4. Results

4.1. Comparisons of clustering approaches over different parameters for simulated data

In 20 cases with different pairs of sample size and coefficients of features, NET-C con-sistently outperforms the prediction method with all distinct groups by at least 6% as shown in Table 1. Increasing sample size decreases prediction error overall. For all five cases, as the sample size increased from 150 to 240, the overall error in estimating the outcome by four prediction methods decreased. Error difference between the prediction method with all distinct groups and our proposed method is smaller with sample size 240 than sample size 150. This trend in prediction error by different sample size occurs in all cases except case 5.

Table 1:

Sample size of 150, 180, 210 and 240 with 120 features and 5 cases of diverse correlations between features and outcome. Prediction errors were estimated using Mean Squared Error. The values in brackets are 95% confidence intervals and those in parentheses indicate error difference between one alternative and the proposed methods. Error difference = ((error_alternative − error_proposed)/error_proposed) * 100%

		Case 1	Case 2	Case 3	Case 4	Case 5
Sample Size 150	All distinct groups	18568.68 [5974.73, 41126.96] (25.39%)	62833.2 [19056.68,115757.3] (19.75%)	66290.04 [17788.26, 126248.01] (16.31%)	85915.94 [22199.13,188909.02] (23.15%)	109567.9 [29177.38, 220182.34] (13.98%)
	Random forest	80527.18 [77439.97,82922.59] (443.77%)	146612.4 [142971.27,150083.17] (179.42%)	176946.9 [172482.82,182477.75] (210.47%)	320221.8 [307611.43,331080.32] (359.01%)	462729.4 [443418.79, 482350.36] (381.38%)
	Hierarchical Clustering	23115.31 [5924.6, 58431.84] (56.09%)	67426.33 [23289.28,141161.24] (28.51%)	69058.95 [20304.65, 127986.74] (21.17%)	83990.03 [26727.68, 156791.92] (20.39%)	111273.5 [40909.36, 204437.1] (15.76%)
	Neural Network	82671.22 [65224.87, 98158.46] (458.25%)	126666.3 [96673.71,152119.01] (141.41%)	132216.4 [88408.62, 164300.7] (131.99%)	225135.3 [144572.34, 290927.45] (222.71%)	212898.7 [153445.03, 290296.66] (121.48%)
	NET-C	14809.08 [4253.95, 26546.37] (−)	52469.51 [16635.74, 96948.04] (−)	56993.2 [17144.97, 100261.95] (−)	69763.29 [22408.49, 153802.14] (−)	96125.39 [29092.24, 199140.78] (−)
Sample Size 180	All distinct groups	14400.58 [4263.61, 28411.74] (13.01%)	44194.68 [15944.62,90938.87] (26.14%)	47357.77 [12020.73, 92841.86] (24.92%)	56164.97 [20072.7, 116324.99] (21.58%)	67699.63 [20912.07, 138292.09] (15.48%)
	Random forest	89945.03 [86842.7, 92954.87] (605.87%)	160179.1 [156310.96, 163649.34] (357.17%)	240391.9 [233776.42, 246768.04] (534.10%)	656608.1 [638741.55, 673103.46] (1321.32%)	808929.4 [783126.61, 831468.29] (1279.88%)
	Hierarchical Clustering	17565.72 [7402.82, 40751.04] (37.85%)	48398.8 [16635.2, 87704.28] (38.13%)	50935.02 [12385.62, 97952.97] (34.36%)	61213.54 [20435.06, 119424.77] (32.51%)	75531.55 [22399.73, 144443.37] (28.84%)
	Neural Network	70776.80 [57653.27, 80329.09] (455.44%)	97271.28 [78847.95, 111702.15] (177.62%)	133749.2 [113283.54, 153510.06] (252.80%)	407915.9 [341324.53, 456098.73] (782.99%)	426748.5 [358249.97, 500890.16] (627.95%)
	NET-C	12742.49 [3649.06, 28832.63] (−)	35037.4 [9660.61, 68941.24] (−)	37910.71 [8131.88, 79338.32] (−)	46196.19 [14446.62, 105897.32] (−)	58623.18 [19284.89, 134004.98] (−)
Sample Size 210	All distinct groups	13576.77 [7159.25, 24482.09] (10.74%)	30503.26 [11458.59,58262.57] (24.93%)	31200.36 [12715.6, 70858.52] (17.27%)	35472.79 [10617.84, 76525.08] (30.65%)	43164.89 [9488.41, 102281.12] (39.14%)
	Random forest	107248 [104697.1,109720.89] (774.78%)	188861.7 [183232.04,191897.38] (673.50%)	265437.5 [257855.88,270791.96] (897.65%)	498784.9 [483101.93,512250.22] (1737.13%)	611100.4 [590475.07,625845.84] (1869.85%)
	Hierarchical Clustering	15746.16 [8567.33, 25181.28] (28.44%)	33881.52 [13252.88,64691.01] (38.77%)	35271.82 [13488.29, 65607.85] (32.57%)	38694.58 [12699.72, 80869.13] (42.52%)	46643.94 [12338.08, 107048.6] (50.35%)
	Neural Network	84586.79 [68602.39, 102087.15] (589.94%)	97091.62 [79892.35, 112219.07] (297.65%)	99468.98 [81846.43, 115955.34] (273.86%)	163776.1 [125533.18, 204886.74] (503.22%)	186608.2 [145036.57, 227575.46] (501.52%)
	NET-C	12260.00 [6348.56, 20002.31] (−)	24416.41 [11265.1, 48331.15] (−)	26606.17 [9878.37, 48504.34] (−)	27150.23 [7922.28, 57580.08] (−)	31022.72 [8777.89, 65977.65] (−)
Sample Size 240	All distinct groups	14130.16 [7023.12, 22189.39] (6.09%)	24787.49 [11499.41,48052.27] (14.32%)	23559.76 [11217.06, 40356.36] (8.31%)	27884.84 [12605.56, 49574.59] (14.09%)	30942.51 [12917.27, 51819.06] (18.92%)
	Random forest	131145.7 [127364.01, 135141.95] (884.66%)	212061.5 [207104.7, 216528.84] (878.07%)	300151.2 [292243.98, 308762.77] (1279.86%)	610733.3 [588138.18, 626159.55] (2398.87%)	829610 [803871.9, 858056.09] (3088.44%)
	Hierarchical Clustering	16069.76 [7600.94, 26281.03] (20.65%)	27315.92 [11796.44,48638.82] (25.99%)	26142.85 [11500.4, 47778.4] (20.18%)	30165.19 [15083.98, 52850.44] (23.42%)	33547.22 [13671.76,58795.88] (28.93%)
	Neural Network	83712.84 [71489.45,96757.48] (528.53%)	87518.12 [69688.57,104204.54] (303.65%)	103910.4 [84451.76, 127396.82] (377.70%)	170889.4 [118336.48,212808.13] (599.21%)	238134.7 [173360.46,320509.51] (815.22%)
	NET-C	13318.82 [6068.15, 22066.34] (−)	21681.7 [9951.26, 42505.94] (−)	21752.24 [11038.68, 36481.85] (−)	24440.43 [12845.34, 39898.85] (−)	26019.32 [12621.36, 46810.1] (−)

The next step was to validate how the coefficients of features in the subnet-work affect accuracy in predicting the outcome. To compare prediction error by different prediction methods, we investigated difference between prediction error of NET-C and those of other methods in the two extreme cases: Case 1 290 and Case 5. From case 1 to case 5, there was no clear increasing or decreasing trend in error difference. However, it was obvious that our proposed method gave lowest prediction error over all situations, the prediction method with all distinct groups outperformed one with groups defined by hierarchical clustering. Random forest regression and neural networks gave much higher prediction errors than other prediction methods did. The accuracy results of finding true clusters using hierar-chical clustering and NET-C are shown in Appendix B-1

4.2. Application to placental metallomics and infant cord plasma metabolomics with covariates in NHBCS

We applied the proposed method to predict birth weight using 220 cord metabolites, 24 trace elements, and maternal characteristics (age and BMI prior to pregnancy) from 381 participants in NHBCS. The first step of NET-C, which involves grouping highly correlated variables, was applied to the placental trace element and cord-blood metabolite data to find the clusters of highly correlated trace elements and metabolites and removing weak or insignificant interactions. Trace elements clustered into 5 groups. Group 1 included Mg, P, and Zn. Group 2 included K, Ca, and Pb. Group 3 included Al, Cr, Co, Ni, As, Mo, Ag, Cd, Sb, and Hg. Group 4 included Sr and Ba. Group 5 included Na, Si, Mn, Cu, Se, Tl. Cord-blood metabolites clustered into 10 groups as shown in Appendix B-2.

The second step of proposed method used group lasso to predict birth weight with the clustered trace elements and cord-blood metabolites, along with maternal enrollment age and maternal BMI as covariates. In this real data application, NET-C and other three pre-diction methods such as group lasso with all distinct groups and group lasso with hierarchical clustering gave similar results in prediction accuracy. A summary of our findings appears in Table 2, Table 3 and Appendix B-2, and Figure 2. In Table 2 and Table 3, all the coefficients were modelled by group lasso method and 95% confidence intervals were calculated using replications of prediction procedures 1000 training and testing instances. Trace elements, metabolites, or covariates with consistently nonnegative coefficients were listed in Table 2, and those with consistently nonpositive coefficients were listed in Table 3. The meaning of each metabolite symbol in Tables 2 and 3 is explained in Appendix B-3 [28]. Lysophosphatidyl choline metabolites were consistently positively associated with birth weight and isoleucine (ile) and other metabolites were possibly positively associated with birth weight. As shown in Figure 2, five lysophosphatidyl choline metabolites which are positively associated with birth weight also formed a closely interconnected subnet-work. Table 3 shows a trace element, Cu, and metabolites including c5 which was consist-ently negatively associated with birth weight and formed a subnetwork consisting of c5 and c3. Cu, and other metabolites were possibly negatively associated with birth weight.

Table 2:

Covariates (e.g., maternal BMI), and metabolites nonnegatively correlated with birth weight: Metabolites with lower C.I. above zero are colored green

Variable Name	Mean	Lower C.I.	Upper C.I.
Maternal BMI	9.883713	0.000000	19.746845
Ile	0.001994	0.000000	0.005080
lysopc_a_c16_1	0.034728	0.006077	0.066917
lysopc_a_c18_1	0.015001	0.004081	0.026512
lysopc_a_c18_2	0.010608	0.000857	0.022875
lysopc_a_c20_3	0.051269	0.012737	0.091663
lysopc_a_c20_4	0.010309	0.000260	0.022651
pc_aa_c38_3	0.001703	0.000000	0.005539
pc_aa_c40_5	0.013531	0.000000	0.033373
pc_ae_c40_1	0.195712	0.000000	0.516420
pc_ae_c40_4	0.066743	0.000000	0.157386
pc_ae_c42_3	0.313017	0.000000	0.800545
sm_oh_c22_1	0.015206	0.000000	0.035723

Table 3:

Trace elements or metabolites nonpositively correlated with birth weight. Metabolites with upper C.I. below zero are colored red

Variable Name	Mean	Lower C.I.	Upper C.I.
Cu	−0.177953	−0.394365	0.000000
c5	−0.677693	−1.408347	−0.012290
c14_2	−6.780522	−16.859072	0.000000
pc_aa_c34_1	−0.000687	−0.001734	0.000000
pc_aa_c42_1	−0.307263	−0.823219	0.000000
pc_aa_c42_4	−0.408022	−1.064783	0.000000
pc_ae_c32_2	−0.229497	−0.673711	0.000000
sm_oh_c16_1	−0.047273	−0.118574	0.000000
sm_c22_3	−0.528129	−1.312262	0.000000
sm_c26_1	−0.475565	−1.117796	0.000000

Figure 2:

Metabolites in this subnetwork including lysophosphatidylcholines were nonnegatively associated with birth weight. The metabolite nodes such as lysophosphadidylcholines colored green were positively associated with birth weight

5. Discussion

We proposed the statistical method, NET-C, a regularized regression method with grouped features that were determined by lasso and network community detection by Louvain [29]. In simulations, NET-C outperformed other clustering strategies such as hierarchical clustering and random forest regression. Conditions such as varying sample size or correlations of features with the outcome variables influenced outcome prediction accuracy. Our proposed network approach considered partial correlations of all pairs of features in the data and selected significant interactions of features in the network. There are many statistical methods of computing correlations between two features for all possible interactions of the given data. The network approach extracts and visualizes “sub-networks” of interacting features rather than a single interaction between two features. Thus, a more efficient computation and representation of interactions is possible using this dimensionality reduction step compared to other regression methods.

In statistical simulations, we found that NET-C, the group lasso method based on lasso, outperformed group lasso predictions using other grouping strategies. Hierarchical clus-tering did not perform well because in defined clusters, features are either positively or negatively correlated. Thus, hierarchical clustering cannot always accurately group clus-ters of features. In the high-dimensional setting, random forest regression cannot efficiently estimate a continuous outcome since it cannot select the most important features among the vast set of features. However, as sample size increases, the error difference between NET-C and other methods decreases. A future strategy may be to use prior knowledge of the correlations between predictor features and the outcome to improve prediction. Generating subnetworks of features which are a priori known to be strongly correlated with an outcome would likely produce a better prediction accuracy.

In our real data application, NET-C slightly outperformed the prediction methods such as group lasso with all distinct groups or group lasso with groups defined by hierarchical clustering. As shown in our statistical simulations, the higher the sample size, the less the difference between the prediction error computed by NET-C and the prediction errors computed by alternative methods. In this real data application, the sample size of 380 ma-ternal-infant dyads is greater than the number of metabolites, 220. It implies that the outperformance of NET-C over alternative methods can be reduced. Further, trace elements and metabolites were mostly positively correlated, and hierarchical clustering tends to group positively correlated variables more accurately than equally positively and negatively correlated variables. Because of these properties of our real data, performance of group lasso methods with distinct groups or groups defined by hierarchical clustering was improved. Thus, in the real data application, our proposed method did not greatly outperform alternative methods in terms of prediction accuracy, but it was clear that our proposed method successfully identified sub-networks of metabolites strongly affecting birth weight given that our proposed method can predict birth weight as accurately as promising.

The associations found in our analysis were, for the most part aligned, with known biology [30];[31];[32];[33]. In a previous study, higher Cu concentrations in both maternal and cord blood was associated with fetal growth restriction and maternal Cu concentration was related to reduced birth weight [34]. The concentrations of certain maternal amino acids and proteins such as isoleucine in the third trimester of pregnancy also have been positively correlated with fetal growth and neurodevelopment [35]. Cord-blood lysophosphatidyl choline metabolites 16: 1 and 18:1 in particular were positively correlated with birth weight [36]. In contrast, concentrations of C3, C3DC, C5 and C5OH 385 were higher in low birth weight infants compared to normal birth weight infants [37]. While consistent with the available with literature, they will need to be confirmed or refuted by additional studies.

In summary, we found that NET-C identified a network of features that predict birth weight. while the proposed method may more accurately predict outcome compared to alternative clustering methods. Nonetheless, the generalizability of this method will need to be studied further to understand its ability to predict health outcomes under a broader set of features characteristic of real-world data.