BHMnet: Bayesian high-dimensional mediation analysis with network information integration for correlated mediators

Abstract

We consider identifying a small yet meaningful set of active mediators from a high-dimensional pool of potential mediators, commonly derived from “-omics” or imaging data. In these contexts, mediators are often correlated or exist network structures, which present unique opportunities to improve efficacy by using this valuable information. To this aim, we develop a Bayesian method that accommodates both high dimensionality and correlations among the mediators. Our approach flexibly learns the interconnection between the mediators while improving estimation accuracy by incorporating external knowledge about these relationships. Simulation studies demonstrate the effectiveness of the proposed method compared with alternative approaches. The analysis of the environmental toxicity data provides new insights into the intermediate effects of molecular-level traits.

Introduction

Mediation analysis is widely utilized across various disciplines, including epidemiology, biomedical research, and social sciences. The goal of mediation analysis is to understand how exposure affects the outcome, particularly in the presence of intermediate variables, known as mediators, that may form a pathway for transmitting the exposure’s impact. Recent advancements have led to the establishment of the counterfactual framework, which delineates natural direct and indirect effects by contrasting potential outcomes for the same individual, thus allowing for causal interpretation [1, 2]. Under some assumptions, the natural direct and indirect effects of counterfactual framework can be equivalently derived under conditional models of the linear structure equation model [3]. Briefly, this set of models includes the outcome model, which models the outcome variable conditioned on exposure, mediator, and confounding variables; and the mediator model, which models the mediator conditioned on exposure and confounding variables. This paradigm has been adopted in many high-dimensional mediation analyses, such as the studies by Song et al. [4], Guo et al. [5], and Cai et al. [6], among many others, and will be followed in our approach.

Recently, there has been rapid development in high-dimensional mediation analysis methods, largely driven by the need to manage a vast number of potential mediators, such as genetic biomarkers and brain imaging traits [7–9]. In these situations, techniques designed for high-dimensional variable selection can be applied to identify active mediators. A particular line of interest is by using the penalization methods [10] that can be adopted in studying the outcome model along [6, 11, 12] or for both the outcome and mediator models [4, 6, 13]. Multiple testing procedures can also be employed using a joint testing approach that assumes the null hypothesis, either that the mediator’s association in the outcome model is null, or that the exposure’s association in the mediator model is null. These procedures can be applied following marginal analysis where each mediator is modeled individually [14], or after a penalization method is used to select a subset of mediators [11], or as a built-in mixture model [4, 15]. For a comprehensive review of methods for high-dimensional mediation analysis, refer to [16].

However, most existing studies have examined mediators without accounting for the correlation structure among them. Typically, the mediator model is analyzed by focusing on each mediator individually [11] or by employing a multivariate normal distribution that uses an independence structure for convenience [6, 13]. The graphical representation of correlated mediators is shown in Fig. 1b, compared with the independent assumption as a commonly adopted priori in Fig. 1a. The correlation, frequently analyzed within network structure, is often observed among molecular biomarkers in practice. For instance, in high-throughput gene expression data, genes typically work through a group of interacting genes [7, 17, 18]. Another example is found in brain imaging data where a single pixel may not present strong effect but region as a whole may reveal a significant impact [19, 20]. On the other hand, the independence assumption simplifies the true relationship between the mediators, which may lead to loss of power in mediator selection [4, 13].

Figure 1.

Casual graphs of the multiple mediators: is the exposure, is the outcome variable, and are the multiple mediators. (a) Mediation model with independent mediators; (b) mediation model with connected mediators, represented by dashed lines.

Incorporating networks of the mediators means understanding how mediators interact with each other, where these interactions are represented as a network. In this network, the vertices represent the mediators, and the edges between the mediators indicate the relationships between them. Hence, estimating the relationships, represented by edges, can be integrated into the mediator selection process and further contribute to forming the mediator network. However, this essentially increases the number of parameters to be estimated, which can lead to unreliable estimation and selection, especially with a limited sample size. One potential solution to address this challenge is to borrow strength by integrating external biological knowledge about established network structures. Leveraging external information from previous studies is not a new idea; considerable efforts have demonstrated that utilizing such knowledge enhances both prediction accuracy and the interpretability of results [17, 18]. Such efforts are made possible in part due to several public databases, such as the Kyoto Encyclopedia of Gene Genomes [21] and Gene Ontology [22], which make external biological knowledge readily available.

A significant advancement addressing this issue has been proposed [4], which developed a coordinated selection technique for identifying correlated mediators, building on an earlier approach [15]. In the work of Song et al. [15], two approaches for mediator selection are introduced: one utilizing a Gaussian mixture model (GMM) to categorize mediators into four categories based on the presence or the absence of exposure and outcome effects, and another using a Product Threshold Gaussian (PTG) prior to impose joint thresholding on the marginal effects and their product, which corresponds to the individual direct effect. However, neither method accounts for the correlations among mediators. In the later work [4], the GMM approach is extended by incorporating mediator correlations into the mixing probabilities to promote the joint selection of correlated mediators. Although not a primary focus of the paper, their method can also incorporate external information by allowing only mediators within the same biological pathway to have non-zero pairwise correlations, as demonstrated in the real data analysis of the LIFECODES birth cohort. Our work aligns with that of Song et al. [4] in aiming to account for correlations among mediators. However, there are a few key technical differences that highlight the contribution of our works. First, we model dependencies directly within the mediator model using a Gaussian graphical model (GGM). This improves effect estimation by accounting for their dependencies, which in turn enhances selection accuracy. In contrast, the approach of Song et al. [4] encodes correlation into the variable selection indicators via structured priors favoring inclusion of correlated mediators. Second, our framework incorporates external network data as a prior on the graphical structure, providing a principled way to integrate biological knowledge into the mediator model.

Methods

Model

Suppose there are individuals. For individual , we denote the outcome and exposure by and , respectively, and let and denote -dimensional potential mediators and -dimensional confounders, respectively. In this study, we focus on continuous variables for both and .

For inferring causal effects, we follow the counterfactual framework. For the th subject, we define as the potential outcome for mediators if the subject receives exposure . In addition, we define as the potential outcome for the subject given exposure and mediation level . Using these definitions, the total effect of exposure can be defined as , which is the overall effect of changing both the exposure and the mediators and can be further decomposed to the natural direct effect (NDE) and natural indirect effect (NIE). NDE is defined as , which characterizes the effect of changing the exposure from a reference level to , while keeping the mediators fixed at the level they would have naturally been under . NIE is defined as , which captures the effect of changing the mediators from the level associated with exposure to the level under exposure , while keeping the exposure fixed at .

The above causal effects are defined using counterfactual variables, which may not be directly observable in practice. Following the sequential ignorability assumption [2], and additional two assumptions commonly assumed in high-dimensional mediation analysis [5], the no-interaction assumption and linearity assumption, it has been shown that NDE and NIE can be identified by modeling and based on the observed data [2]. We therefore define the outcome model as

(1)

where is the effects of potential mediators, is the exposure effect, and is the effects of confounders, and . We set the first entries of to one for the intercept.

For the mediator model, a multivariate regression model is assumed as

(2)

where is the effect of exposure, is the effect of confounders with each being a vector, and with , which captures the conditional dependence structure between mediators.

Estimating mediation mechanisms

In high-dimensional mediation analysis, it is often assumed that only a small portion of the potential mediators are actual mediators, i.e. have non-zero mediation effects in the path through exposure to the outcome such that . To identify such mediators and account for the network structure, we introduce binary variable selection indicators at two levels, the group and the individual levels, to induce sparsity. Specifically, we let denote the binary indicator of whether the coefficients for the mediator effects in the th group are all zero, performing the mediator selection at the group level. Here we refer to network, pathway, or neighboring information as the group. We also let and denote the binary indicators for selecting individual elements in the th group corresponding to and , respectively. This performs within-group selection at the individual level. We then take the parameterization of mediator effects as follows: , and assume the following priors:

for , where and are prior inclusion probabilities for mediator selection variables. In order for the th potential mediator to have mediating effects, all related binary indicators should be one (i.e. ).

To complete the setup, we specify the following priors on the remaining parameters except for those on the precision matrix . For the prior inclusion probabilities, we assign Beta hyperpriors: and . The normal priors are assumed for the coefficients: , , and . Lastly, we assume .

Utilizing structural information between mediators

In this subsection, we discuss the prior assignment on the precision matrix to account for mediator network structures. Meanwhile, we assume that the external network information between mediators is available a priori and aim to improve the estimation by incorporating it into the model. First, to learn the dependencies among the mediators, we adopt the GGM for in the mediator model, where defines the conditional independence structure among mediators. Here, each element of represents the probabilistic relationship between the two variables. Achieving sparsity in the estimation of has been a key objective in the application of GGM, which naturally leads to the use of regularization techniques. In the Bayesian framework, priors that can induce sparsity in estimating are assigned [23, 24]. This sparsity pattern of can be formally represented by an undirected graph , where is the set of mediators (nodes) and the set of edges. The graph can be indexed by a set of binary variables, where denotes a binary indicator for the presence of an edge between variables and in the graph . Specifically, if and only if the edge between variables and is missing from the graph , meaning conditionally independence given all the other variables.

In this paper, we follow the proposal of Wang [23] and use the continuous spike-slab prior, which allows for sparsity estimation by augmenting latent binary indicators for edge inclusion and offers efficient and scalable computation. The basic idea of the continuous spike–slab prior is that, given edge inclusion indicators, it allows elements to come from either a normal distribution with a large variance or a normal distribution with a small variance. Specifically, for the off-diagonal elements, if , the corresponding off-diagonal element follows a normal distribution with a large variance , representing the slab part of the continuous spike–slab prior; conversely, if , comes the spike part represented by a normal distribution with a small variance . For the diagonal entries, an exponential prior is assigned. The continuous spike–slab prior is then defined as

(3)

where is small, is large, and is an indicator function that is a symmetric positive definite matrix.

We next discuss our prior choice for the edge selection indicators. Although an independent Bernoulli distribution is a popular prior for them, it assumes independence a priori and hence does not conform to our goal of incorporating prior knowledge about the network between the mediators. As such, we propose a prior that favors the inclusion of the edges that are connected a priori, providing additional information about network structure. To this end, we assume a Markov Random Field (MRF) prior on edge selection indicators, with a prior network knowledge as a hyperparameter of the MRF prior. Specifically, let denotes a binary indicator matrix containing external network information. For illustration, we take an example in the right panel of Fig. 2, where blue and white squares represent 1 and 0, respectively. In this example, mediator 1 (M1) is connected with mediators 2 and 3 (M2 and M3), and the corresponding edges, and , take a value of 1. This suggests that and should be selected together. On the other hand, mediators 1 and 4 are not connected, and hence, the selection of or is unrelated to the selection of . Building upon this observation, we transform this matrix into a matrix by taking a pairwise multiplication of the edges as illustrated in the right panel of Fig. 2. Here, we note that any alternative method of constructing can be used, provided it results in a binary matrix that contains external network information. Let denote the vectorized binary indicators corresponding to the off-diagonal entries of . The prior for conditional on then is defined as follows:

Figure 2.

Overview of the proposed method. Left panel is a summary of the study models and prior for parameters of mediator selection indicators and precision matrix. Right panel illustrates the use of external network information matrix and its re-construction to matrix.

where regulates the sparsity of , with a smaller value favoring sparser models, and controls the extent to which the inclusion of the edge indicators is influenced by the inclusion of connected edges, with a higher value indicating stronger influence from the selection of those connected edges. The values of should be carefully chosen as increasing the values of may result in a sharp increase in the number of variables selected, leading to a phenomenon called transition phase [7]. In this paper, we follow the guidance of Li and Zhang [7] and Peterson et al. [25] when selecting a value of .

Remark. In our model, the binary indicators , , and together determine whether the th mediator is an active mediator or not. When , the corresponding mediator is excluded from the model, regardless of the values of and , leading to an identifiability issue for inactive mediators (i.e. ). Some existing studies address this issue by imposing conditional priors or additional constraints to eliminate parameter settings that lead to non-identifiability [26, 27]. However, this identifiability issue does not affect the selection or estimation of active mediators. Hence, in this study, we choose not to impose additional constraints to resolve this issue, following previous approaches that prioritize computational efficiency [28]. We consider this a reasonable trade-off that facilitates computational efficiency.

Posterior inference

The posterior distribution of the proposed model is analytically intractable, and we rely on a Markov chain Monte Carlo algorithm to simulate samples from the posterior. All parameters are updated using Gibbs sampling, exploiting conjugacies. The basic structure of the proposed Gibbs sampler is as follows:

• Update the mediator selection indicators and by drawing from the conditional posterior distributions, which follow Bernoulli distributions, where , , and

• Update the graph and the precision matrix using a block Gibbs sampler proposed by Wang [23].

• Update (, , , , , , ) by sampling from their respective full conditional posterior distributions, where and .

Full details of the algorithm are provided in the Supplementary data. The code for implementing the proposed method is publicly available at https://github.com/yunjuim/BHMnet.

Our primary focus for the posterior inference is on the identification of the active mediators. Following Bayesian variable selection literature, we estimate the posterior inclusion probabilities (PIPs), which in this study are defined as , for . We then select the mediators with PIPs >0.5. The model that retains these variables is referred to as the median probability model [29].

Simulation

In this section, the performance of the proposed method, BHMnet, is assessed using the simulated data and compared against the alternative methods. For the alternative methods, we consider the following: (i) HIMA by Zhang et al. [12] that uses a sure independence screening and a minimax concave penalty for identifying and estimating the effects of mediators, and conducts a joint significance test to determine active mediators with control on false discovery rate (FDR). We present final results by setting two commonly used FDR thresholds 0.05 and 0.1, producing HIMA0.05 and HIMA0.1. (ii) HDBM by Song et al. [30], a Bayesian approach that employs separate shrinkage priors on outcome and mediator models for each mediator. Due to its observed limitation in selection performance during simulation, we also introduce an alternative version named the HDBM-outcome model. In HDBM-outcome, variables selected in the outcome model are identified as active mediators and estimation of mediation effects follows. (iii) PTG by Song et al. [15], that assigns the product threshold prior, which thresholds not only the marginal effects but also the indirect effects (i.e. the product of the marginal effects) for each mediator. Note that HIMA, HDBM, and PTG do not account for the network structure of the mediators. (iv) The proposed model but without prior information (BHMnet-woPI), which assigns the independent Bernoulli prior on edge selection indicators. This method is adopted to show benefit by incorporating prior knowledge. Due to the lack of available functional codes for Song et al. [4], we exclude it for the comparison despite its capability to account for correlation. The simulation is conducted by Juila version 1.7 for the BHMnet and BHMnet-woPI and R version 4.2 for all the other methods. HIMA is performed by the R package HIMA. HDBM and PTG are performed by the R package hdbm.

We consider independent individuals with and generate the data from and , where and . ’s are generated from a standard normal distribution. In all scenarios, there are 20 non-zero elements of and with 10 being mediators (i.e. ). We consider two different sets of values for the non-zero effects of and : and , corresponding to moderate and smaller effects, respectively. The signs of the non-zero effects are randomly assigned as positive or negative. Lastly, we set . For the precision matrix , we consider a block diagonal matrix, where each block is a size of but has different structures. In setting 1, consists of 10 dense modules with elements defined as and for ; otherwise . In setting 2, the first seven blocks of follow the same dense structure as in setting 1 while the last three blocks have a circle module such that elements defined as ; for ; = and , otherwise. In setting 3, consists of 10 blocks having the autoregressive structure, such that each element within a block is defined as . We assess the performance under varying levels of correlation by adjusting the values of to 0.4 and 0.6. The diagonal elements are adjusted as needed to keep positive definite.

For hyperparameters for the proposed method, we set their values either reflecting a vague prior belief on them or following literature: . For Beta hyperpriors in the outcome model, we set . For hyperparameters in continuous spike-and-slab priors in GGM, we use (, , ) = (1.0, 0.02, 1.0) following the suggestion of Wang [23]. In addition, when applying the proposed method, we need to specify the prior knowledge about the network structure between the mediators, which is summarized in . Assuming that the prior knowledge is fairly accurate but may contain some false information, we generate based on the true network structures with a random 10% false negatives and a 1% false positives. To examine the impact of the accuracy of the external knowledge on BHMnet, we conduct a small set of sensitivity analyses on setting 3. Specifically, we consider varying proportions of false negatives (20% and 50%) and false positives (5%, 10%, and 20%) in the external knowledge.

To evaluate the performance, we report the following measures based on 100 simulation replicates: (i) metrics to describe the mediator selection performances including the mean and standard deviation (SD) of the true positive rate (TPR), false positive rate (FPR), and the Matthews correlation coefficient (MCC), which provides an overall summary of classification accuracy. MCC is defined as where and are the numbers of true and false positives, respectively, and and are the numbers of true and false negatives, respectively. For Bayesian methods, mediators are selected if their PIPs exceed 0.5. (ii) mean squared error of the indirect effect for the truly active mediators (MSE). (iii) coverage rates of credible intervals of all true active mediators (calculated as the average coverage rates among those true mediators), the direct effect, and the indirect effect. This coverage comparison focuses on the Bayesian methods, including the proposed methods and PTG. HDBM is excluded because it fails to identify active mediators in our settings, and HIMA is excluded as its confidence intervals are not readily available. Results for Bayesian methods are based on 8000 post burn-in samples after the burn-in period of 2000 iterations.

Simulation results are illustrated in Figs 3–5. A table of simulation results can be found in the Supplementary Table B.1. In general, BHMnet and BHMnet-woPI outperform alternative methods by improved mediator selection as well as accuracy of estimation for the mediation effects. Comparing BHMnet and BHMnet-woPI, the larger differences between the two are observed when a weaker mediation effect is used (), indicating the benefit of considering external information in those cases. Additionally, higher correlation among mediators appears to slightly enhance performance in these cases. Furthermore, BHMnet(-woPI) achieves higher coverage rates for the credible intervals of active mediators and mediation effects, suggesting greater accuracy estimation and reliable credible intervals. Nonetheless, these rates do not consistently meet the nominal coverage level. Specifically, we observe that some intervals can exhibit under-coverage, which is a well recognized and complex challenge in high-dimensional inference [31]. Finally, we observe that the performance of BHMnet(-woPI) is stable across settings.

Figure 3.

Mediator selection accuracy based on 100 replicates. In each figure, the y-axis represents MCC, and the x-axis represents TPR. Each bubble’s center corresponds to the average TPR and MCC. The bubble size represents variability, with horizontal and vertical dimensions indicating the standard errors for TPR and MCC, respectively.

Figure 5.

Top: Boxplots of the coverage of the true positives: Bottom: Coverage rates of the direct effect and indirect effect are shown on x-axis and y-axis, respectively. Each bubble’s center corresponds to the average coverage rates of the direct and indirect effects. The bubble size represents variability, with horizontal and vertical dimensions indicating the standard errors for direct and indirect effects, respectively. All results are based on 100 replicates.

Figure 4.

Boxplot of mean squared error of indirect effects, MSE, based on 100 replicates.

In cases with weak correlation , HIMA may represent the strongest-performing alternative following BHMnet(-woPI). Compared with the other Bayesian methods, HIMA often exhibits a lower TPR, but this is accompanied by a lower FPR, resulting in a generally higher MCC. However, when the correlation strength increases with , HIMA has reduced performance, which is previously noted by Ashburner et al. [32]. As a result, its performance may decline when the mediation effect is weaker (). However, it should be noted that HIMA applies more rigorous control over variable selection compared with other methods that do not incorporate hypothesis testing following selection. Therefore, the reduced TFR is reasonable and expected. Within the HIMA method, a threshold of 0.1 generally yields slightly more favorable results compared with a threshold of 0.05 in our settings. Finally, as shown in Supplementary Table C.1, the results of BHMnet are relatively robust to changes in the accuracy of the external knowledge.

Among the other Bayesian methods, PTG demonstrates strong performance when the signal strength is high (). However, its performance deteriorates markedly under conditions of weakened signal strength (). For HDBM, we consistently observe a failure to identify any mediators. A closer examination reveals that HDBM can identify outcome-mediator correctly but fail to identify mediator-exposure effects. This results in low TPRs for detecting active mediators. As a result, we include HDBM-outcome in comparison in which we may consider those selected by the outcome model as mediators. It is clear that HDBM-outcome has improved performance in both selection and estimation. In some cases, the goal is to cast a wide net to identify as many potential mediators as possible. In such scenarios, greater emphasis may be placed on TPR relative to FPR, which can influence the choice of method.

In sum, the proposed method outperforms alternative approaches in both the identification of active mediators and estimation accuracy. This reinforces the substantial advantages of incorporating mediators’ network structure and integration of external knowledge about their relationships.

Real data analysis

Phthalates are chemical compounds commonly used to make plasticizers more flexible and durable, and they can be found in a wide range of everyday products and materials [33]. Substantial evidence has shown that exposure to phthalates has positive associations with an increased risk of preterm birth and shorter gestational periods [34, 35]. Our study aims to illustrate how the proposed and alternative methods can be leveraged to deepen understanding of the molecular mechanisms connecting phthalate exposure to preterm birth, focusing on potential biomarker-mediated pathways. We will utilize a synthetic dataset curated by Aung et al. [36] which is available at https://github.com/umich-cphds/environmental_mediation_framework. This dataset was also analyzed in a recent mediation study by Cai et al. [37].

This dataset includes information on pregnant women, with measurements on environmental exposures, baseline clinical variables, and 61 endogenous biomarkers. The biomarkers consist of 51 eicosanoids, five oxidative stress markers, and five immunological markers spanning seven distinct biological pathways. The classification into seven biological pathways was predefined in the dataset, following established biochemical knowledge as described in Aung et al. [38]. In that study, eicosanoids were organized into three enzymatic pathways (cyclooxygenase, cytochrome P450, and lipoxygenase) based on previous studies [39], and further classified additional biomarkers into other biological pathway groups (oxidative stress, protein damage, parent compound, and inflammatory markers). Each biomarker in the dataset is explicitly annotated with its respective pathway. We leverage these pathway annotations as external information in our model. These measurements were derived from urine and blood samples collected during clinical visits between 23.1 and 28.9 weeks of gestation. The outcome of interest is gestational age at delivery, with exposure focused on nine distinct phthalate toxicants. We adjusted for confounding variables including age, maternal BMI at the initial visit, race, education, urinary specific gravity, and type of health insurance. We applied both the proposed and alternative methods for the analysis. Specifically, for the proposed method, we run the Gibbs sampler for 50 000 iterations, collecting 30 000 samples after discarding the first 20 000 samples as burn-in. The mediators with a PIP >0.5 were identified as active mediators for Bayesian methods.

The mediation selection results for Bayesian methods are summarized in Fig. 6. Overall, BHMnet and BHMnet-woPI produce consistent mediator identification patterns. Nonetheless, the inclusion of pathway information in BHMnet typically leads to the identification of additional mediators. For example, with MEHP as the exposure, BHMnet identified PGD3 and 18-HETE as active mediators while BHMnet-woPI only identified PGD3 as the active mediator. In contrast, the other two Bayesian methods, PTG and HDBM, generally do not select mediators, as their PIPs remain below the 0.5 threshold for most exposures. An exception occurs with MECPP as the exposure, where PTG identifies IL-10 and TNF-alpha as significant mediators.

Figure 6.

PIPs for each exposure across the Bayesian methods for exploring the pathway of Phthalates–Biomarkers–Gestational age. The red dotted line indicates the PIP threshold of 0.5. Exposures include monoethyl phthalate (MEP), mono-N-butyl phthalate (MBP), mono-isobutyl phthalate (MiBP), monobenzyl phthalate (MBzP), mono(2-ethylhexyl) phthalate (MEHP), mono(2-ethyl-5-hydroxyhexyl) phthalate (MEHHP), mono(2-ethyl-5-carboxypentyl) phthalate (MECPP), mono(2-ethyl-5-oxohexyl) phthalate (MEOHP), and mono(3-carboxypropyl) phthalate (MCPP).

The data are also analyzed using HIMA with the threshold for FDR set at 0.1, which identifies 18-HETE and 5-oxoETE as active mediators when MEHP, MEHHP, and MEOHP are used as exposures. With MECPP as the exposure, HIMA identifies 18-HETE as the only active mediator. For all other exposures, HIMA does not identify any active mediators.

To illustrate the identified mediation effects, Table 1 presents the estimated indirect effects and their 95% credible intervals for mediators selected under MEHP exposure as an example. It is shown that PGD3 presents a negative indirect effect. For 18-HETE, a mediator selected by both BHMnet and HIMA, the estimated indirect effect direction is consistent across the two approaches. Although its 95% credible interval (−0.009, 0.322) slightly includes zero, potentially due to limited power with a small sample size or a weak signal, there is strong evidence for a positive effect, as the posterior probability that this mediator has a positive indirect effect, conditional on its inclusion, exceeds 95%.

Table 1.

Estimated indirect effects and their 95% credible intervals in parentheses for Bayesian approaches for exploring the pathway of MEPH–Biomarkers–Gestational age

Selected mediators	BHMnet	BHMnet-woPI	HIMA0.1
PGD3	−0.066 (−0.127, −0.021)	−0.070 (−0.133, −0.022)
18-HETE	0.146 (−0.009, 0.322)		0.133
5-oxoETE			−0.102

The most frequently selected mediators using BHMnet across all exposures are PGD3 and 18-HETE, a finding, i.e. well supported by existing literature. These biomarkers are associated with the cyclooxygenase pathway and the cytochrome p450 pathway, respectively. The cyclooxygenase pathway, which regulates various physiological and pathological processes such as inflammation, immune response, and homeostasis, is involved in preterm labor. Elevated inflammation, a key feature of the cyclooxygenase pathway, is a recognized risk factor for adverse preterm delivery [40, 41]. The cytochrome p450 pathway, a family of enzymes important for metabolizing drugs, environmental toxins, and endogenous molecules, has been previously identified as a target of metabolic disruption by phthalates [42, 43]. Consistent with our findings, Aung et al. [36] also reported PGD3 and 18-HETE as potential mediators linking phthalate exposure to adverse pregnancy outcomes. Taken together, these findings suggest that exposure to phthalate may have affected metabolite compounds through those two pathways, and in turn, potentially contributing to the risk of adverse preterm delivery. Our data analysis provides valuable insights into the molecular mechanisms linking environmental exposure to preterm birth through endogenous biomarkers, with potential implications for early intervention, prevention, and therapeutic development.

Discussion

We note a few areas where the method could be further developed. First, the proposed framework can be extended to incorporate non-continuous mediators such as binary and categorical variables. One potential approach is to adopt a latent Gaussian graphical model that introduces latent variables corresponding to the non-continuous mediators and models their correlations through the latent representation. Such extensions would broaden the applicability of the method to settings where the Gaussian assumption for mediators is not appropriate. Another possible extension is to introduce greater flexibility in how external network information is incorporated. The current approach leverages prior network information through a structured prior that encourages edge inclusion when supported by known external connections. While we assume that the external network information is reasonably accurate in this study, such information may be partially incorrect, uncertain, mis-specified, or vary in relevance across different parts of the network in practice. A more general formulation would treat external knowledge not as fixed input, but as uncertain information whose influence can be adaptively modulated based on its alignment with the observed data. This would allow the model to selectively borrow strength from external sources where appropriate, while remaining robust to potential misspecification and allowing for a more nuanced and data-driven incorporation of prior structure. A final avenue for future research is the calibration of the model’s credible intervals. We observed that our proposed methods, while demonstrating superior performance, can still exhibit under-coverage, which is commonly observed in high-dimensional models [31]. Future work could aim to produce credible intervals with more accurate nominal coverage, thereby strengthening the reliability of inference on the identified mediators.

Key Points

• We propose a novel Bayesian approach for high-dimensional mediator analysis to identify active mediators.

• By incorporating correlations among mediators, our approach improves selection accuracy and biological relevance.

• Through leveraging external network information, our approaches has boosted power especially under scenarios with weaker mediation effects.

Conflict of interest: None declared.

Funding

The project described is supported by the National Institute of General Medical Sciences, U54 GM115458, which funds the Great Plains IDeA-CTR Network. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.

Data availability

The code for BHMnet is accessible through the GitHub repository at https://github.com/yunjuim/BHMnet. The data are available at https://github.com/umich-cphds/environmental_mediation_framework.