ERS-Based Mediation Analysis

• The paper introduces ERS-MA as a novel mediation method that compresses multivariable exposure mixtures into a single scalar risk score to estimate global indirect effects.
• ERS-MA constructs the environmental risk score using elastic net regression with a training-analysis data split, reducing overfitting and capturing mixture-level effects.
• ERS-MA facilitates mediation analysis by estimating natural direct and indirect effects, though its outcome-dependent score may overlook exposure-specific details.

Environmental risk score-based mediation analysis (ERS-MA) is a mediation approach for exposure mixtures in which a multivariable environmental mixture is compressed into a single scalar score and standard mediation analysis is then conducted using that scalar score as the exposure. In this formulation, the environmental risk score (ERS) is a weighted linear summary learned from training data, typically in a supervised fashion using the outcome together with confounder adjustment, and the principal target is the ERS-level natural indirect effect for a specified contrast in the score rather than the mediated effect of any one chemical (Wang et al., 13 Sep 2025).

1. Conceptual definition and position among mixture methods

ERS-MA is presented as one of four mediation approaches for exposure mixtures. Its defining move is to replace the original exposure vector by a single learned scalar summary and then use that scalar in an otherwise standard mediation analysis. This distinguishes it from approaches that analyze one exposure at a time, from unsupervised dimension reduction, and from fully nonparametric mixture modeling (Wang et al., 13 Sep 2025).

Approach	Exposure representation	Main characterization
SE-MA	One exposure at a time	Baseline that analyzes each exposure individually
PC-MA	Principal components	Compresses exposures into unsupervised principal components
ERS-MA	Environmental risk score	Supervised summary approach
BKMR-CMA	Full mixture	Models the entire mixture flexibly and nonparametrically

The ERS is described as supervised, mixture-level, and not a measure of exposure amount per se. A higher ERS means higher predicted outcome risk from the mixture, not necessarily higher pollution exposure. This distinction is central to interpretation: ERS-MA addresses whether a learned mixture-risk index is transmitted through a mediator, not whether a physical increase in any specific exposure is mediated. A plausible implication is that ERS-MA is best aligned with mixture-level etiologic questions rather than chemical-specific attribution.

2. Construction of the environmental risk score

The paper uses elastic net regression as a transparent example for building the ERS. In that construction, exposures are penalized, confounders are left unpenalized, and cross-validation is used to tune $\lambda_1$ and $\lambda_2$. The data are randomly split into a training set, used to fit elastic net and learn coefficients, and an analysis set, used to compute the ERS and perform mediation analysis. This split is intended to reduce overfitting and to make downstream inference more defensible (Wang et al., 13 Sep 2025).

With main effects only, the score in the analysis set is

$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$

If squared terms or pairwise interactions are used, the score becomes

$$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$

The tutorial workflow is explicit. First, the sample is split into training and analysis sets. Second, elastic net is fit in the training data using gcdnet, with exposures penalized and confounders unpenalized. Third, the ERS is constructed in the analysis data. Fourth, mediation analysis is carried out with CMAverse::cmest, using the ERS “as the exposure variable in mediation analysis,” model = "rb", linear mediator and outcome models, no exposure-mediator interaction in the tutorial example, and inference by delta method or bootstrap. The paper also notes a practical safeguard: if elastic net selects too few exposures, the penalty is adjusted so that at least three exposures remain in the score.

This construction makes ERS-MA attractive when exposures are highly correlated and a compact mixture summary is desirable. At the same time, the score is outcome-dependent, so its meaning is tied to the specific outcome used to learn it.

3. Mediation estimands and identification

Once the ERS is constructed, it replaces the original exposure vector in a standard mediation model. The paper uses the usual product-method decomposition to define mediation effects for the score. The natural direct effect, natural indirect effect, and total effect are written as (Wang et al., 13 Sep 2025)

$$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$

$$\widehat{\text{NIE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\alpha}_x^{\text{ERS}}\hat{\beta}_m^{\text{ERS}},$$

$$\widehat{\text{TE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\left(\hat{\beta}_x^{\text{ERS}}+\hat{\alpha}_x^{\text{ERS}}\hat{\beta}_m^{\text{ERS}}\right).$$

The key estimand is therefore the ERS-level natural indirect effect for a specified contrast in the score. The paper also frames this as a global indirect effect for the mixture, because the ERS is a single summary measure of the whole exposure mixture. In the PROTECT application, this contrast was operationalized as shifting the ERS from the 25th to the 75th percentile.

For causal interpretation, the usual mediation assumptions are required: Consistency, Positivity, SUTVA, No unmeasured confounding for exposure-outcome, exposure-mediator, and mediator-outcome relations, and Cross-world independence for natural direct and indirect effects. The sequential ignorability-style conditions are stated as

$$Y(x,m) \perp X \mid C,\quad M(x) \perp X \mid C,\quad Y(x,m) \perp M \mid X,C,$$

together with the cross-world condition

$Y(x,m) \perp M(x) \mid C.$

ERS-MA adds an identification requirement specific to score-based representation: the ERS must be a sufficiently rich summary of the original exposures. The paper explicitly states that ERS-MA relies on the assumption that the score captures the relevant information in the original exposures. This is conceptually similar to the sufficiency assumption discussed for PCA. A plausible implication is that causal interpretation weakens when mediation-relevant mixture structure is discarded during score construction.

4. Operating characteristics, applications, and limitations

The simulation study evaluates ERS-MA for global NIE estimation under varying sample sizes, mediation strengths, and correlated mixture structures. ERS-MA performed substantially better than unadjusted SE-MA and generally better than PC-MA. Reported relative bias for the global NIE was about 18% at $n=1000$ and 11% at $\lambda_2$0 under strong mediation, and about 29% at $\lambda_2$1 and 18% at $\lambda_2$2 under weak mediation. The method was described as reasonably robust, improved with larger sample size, and performed somewhat worse when the mediated signal was weak (Wang et al., 13 Sep 2025).

The same simulation section cautions that summary-based methods, including ERS-MA, may underperform when important mediation-relevant patterns are not captured by the score, can be sensitive to how the score is constructed, and may lose information when the exposure-mediator-outcome relation is more complex than the chosen ERS model can encode. It also does not evaluate ERS-MA for variable selection performance, because ERS-MA is not designed to identify individual active exposures.

The PROTECT birth cohort example illustrates the method in a concrete environmental-health setting. The analysis examined prenatal exposure to phthalate mixtures and neonatal head circumference Z-score, with leukotriene E4 as a mediator. ERS-MA was run with main effects only and with main effects + higher-order terms, and the ERS contrast was again taken from the 25th to 75th percentile. The reported global indirect effects were:

• Main effects only: NIE $\lambda_2$3, 95% CI $\lambda_2$4
• With high-order terms: NIE $\lambda_2$5, 95% CI $\lambda_2$6

In that application, ERS-MA suggested little evidence of mediation through LTE4 at the mixture level.

Several limitations are emphasized. ERS-MA provides no exposure-specific mediation, depends on score quality, uses an outcome-dependent score, may incur potential loss of power because splitting data into training and analysis halves reduces effective sample size, and raises issues of interpretability of the score, since the ERS is not a physical exposure quantity but a learned risk index. These are not peripheral caveats; they define the boundary between valid mixture-level interpretation and over-interpretation.

5. Hypothesis testing for score-style mediation pathways

A separate inferential problem arises when mediation is tested through a product-of-coefficients null. Although not specifically about environmental risk scores, the subsampling-based framework of "Subsampling-based Tests in Mediation Analysis" is directly relevant to settings in which mediator or outcome models are applied to many predictors or scores and the null of no mediation is composite (Roy et al., 2024).

In the standard single-mediator structural equation model,

$\lambda_2$7

$\lambda_2$8

the mediation target is the classical product-of-coefficients effect proportional to $\lambda_2$9. The null

$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$0

contains three cases: $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$1

The paper emphasizes that many existing tests are calibrated only under $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$2 and $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$3 and become conservative under $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$4, which is common in high-dimensional omics / risk-score applications. Under Assumptions 1–2, the classical Sobel statistic has different null variances: $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$5

To address this, the authors split the sample into $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$6 disjoint subsamples, compute a Sobel statistic $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$7 in each split, average them to $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$8, and studentize: $\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij}.$9 For fixed $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$0, Theorem 2 shows that under $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$1, regardless of whether $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$2, $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$3, or $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$4 holds,

$$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$5

The asymptotic null law is therefore pivotal and universal across the three null cases. The recommended practical choice is

$$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$6

To reduce dependence on a single random split, the method repeats the subsampling procedure $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$7 times, computes $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$8 for each partition, and combines them using the Cauchy method. The resulting procedure, CSMT (Cauchy-combined Studentized Mediation Test), showed best size control in simulations, whereas Sobel and MaxP were conservative and ABtest had inflated size. In score-based mediation analyses, this suggests that careful testing of the indirect pathway can matter as much as score construction itself.

6. Relation to broader mediation methodology

ERS-MA occupies one point in a broader methodological landscape. When the scalar-score reduction is scientifically acceptable and the goal is the overall mediated effect of an exposure mixture, ERS-MA offers a computationally simpler route than full mixture modeling. When that reduction is too restrictive, alternative frameworks become relevant.

For complex mixtures with nonlinear and interactive exposure-mediator-outcome relations, "Bayesian kernel machine regression-causal mediation analysis" retains the exposure mixture as a multivariate object and estimates $$\text{ERS}_i=\sum_{j=1}^p \hat{\beta}_j X_{ij} + \sum_{k=1}^{p}\sum_{l=1}^p \hat{\beta}_{kl} X_{ik}X_{il}.$$9, $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$0, $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$1, and $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$2 through posterior predictive simulation rather than compressing the mixture into a single score. Its simulation study showed that when exposure-mediator and exposure-mediator-outcome relationships are complex, BKMR-Causal Mediation Analysis performs better than current mediation methods (Devick et al., 2018). This suggests a practical boundary for ERS-MA: once higher-order structure becomes scientifically central rather than incidental, a scalar summary may be inadequate.

For settings with multiple exposures, multivariate mediators, and binary or survival outcomes, "A Framework for Mediation Analysis with Multiple Exposures, Multivariate Mediators, and Non-Linear Response Models" allows the exposure of interest to be one component of a joint exposure vector and estimates a single joint indirect effect through the set $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$3 without imposing a causal ordering among mediators. It uses a regression-based plug-in / g-computation estimator with Monte Carlo integration and supports mediation on the mean difference, odds ratio, and restricted mean survival scales (Long et al., 2020). In an ERS setting, the natural adaptation is to treat the ERS as one exposure $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$4 or as a component of the exposure vector $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$5.

For binary outcomes with a continuous mediator, "Omitting continuous covariates in binary regression models: implications for sensitivity and mediation analysis" provides analytic approximations for the marginal effect of $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$6, the natural direct effect (NDE), the natural indirect effect (NIE), and delta-method standard errors, including treatment-mediator interaction. The paper shows that the proposed estimators perform about as well as exact numerical integration and natural effect model approaches while preserving closed-form interpretability (Gasparin et al., 2023). This is relevant when an ERS is used as the exposure and the target is an interpretable direct-indirect decomposition on the log-odds scale.

For survival outcomes, "Comparison of Effect Size Measures for Mediation Analysis of Survival Outcomes with Application to the Framingham Heart Study" adapts an $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$7-based mediation effect size framework to Cox models and compares five survival $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$8 measures. The recommended measures are $$\widehat{\text{NDE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\beta}_x^{\text{ERS}},$$9 and $$\widehat{\text{NIE}}_{\text{ERS}}(\text{ERS},\text{ERS}^\ast \mid C) = (\text{ERS}-\text{ERS}^\ast)\hat{\alpha}_x^{\text{ERS}}\hat{\beta}_m^{\text{ERS}},$$0, which behaved well with respect to censoring, increased sensibly with stronger mediation structure, satisfied nesting, and had workable uncertainty quantification via delta method or bootstrap (Shi et al., 2022). This is especially pertinent when an environmental or lifestyle risk score is linked to time-to-event outcomes through a correlated mediator block.

Taken together, these developments place ERS-MA as a supervised, mixture-level, scalar-summary approach that is most appropriate when exposures are correlated, a compact summary is desirable, and the scientific target is a global mediated effect of combined environmental burden. It is less appropriate when the goal is to identify specific causal agents, to recover complex exposure-specific nonlinearity or interactions not encoded in the score, or to represent mediator structure that is inherently multivariate rather than effectively scalar.