Simulation design

We generated exposure variables from the multivariate normal distribution with no correlations between exposure variables. Each exposure variable has mean 0.4 and standard deviation 0.5. Two covariates were generated as follows: . Mediator variables were then simulated from normal distribution with a proportion of and associated with . Specifically, 40% of the are associated with 10% of with an effect size of 0.8, as well as with Z₁ and Z₂ with effect sizes of 0.2 each, and a standard deviation of 0.5. Another 40% of the are associated with 10% of the , also with an effect size of 0.8, but with a standard deviation of 0.3. Additionally, 10% of the are associated only with Z₁ and Z₂, with effect sizes of 0.2 and 0.3, respectively, and a standard deviation of 0.5. The remaining 10% of the are not associated with any of the or , with a standard deviation of 0.3. Survival outcomes T were simulated using an AFT model, with T associated with randomly selected (effect size 0.8), (effect size 4.0), and (effect size 0.12). The error term in the AFT model followed a normal distribution, and the censoring times were drawn from an exponential distribution, calibrated to achieve a 25% censoring rate. As part of sensitivity analyses, we also explored censoring rates of 50% and 75%. In comparison with the SMAHP, we compared performance using two different approaches. The first approach begins with a univariate marginal mediation and outcome model, using SIS to identify exposures, followed by a second SIS approach, described as Step 2, which we will refer to as SIS + SIS. The second approach is a naïve modeling method, where marginal mediation and outcome models are fitted without any penalization or the application of the SIS procedure. In addition, we conducted sensitivity analyses incorporating correlated gene and protein structures, considering correlations among exposure genes and among mediator proteins, with correlation levels set to 0.4. We also examined robustness by sampling the exposures from a negative binomial distribution with dispersion parameter 3 instead of a multivariate normal distribution. We included additional simulation experiments to assess the robustness of the proposed method to outliers. Specifically, 2% of the exposure values were generated as outliers by sampling from a uniform distribution over the interval , where and denote the mean and standard deviation of the normal distribution used to generate the exposures. We further evaluated performance using AFT models with two alternative residual distributions (Gamma and logistic error). For each setting, we repeated the simulation 200 times.

Simulation results

In these simulation experiments, different combinations of n, p, and k were considered. Table 1 summarizes the simulation results when the censoring rate is 25%. Across all scenarios, the naïve method significantly inflated the FDR, even though it achieved high power. In contrast, SMAHP maintained high power while adequately controlling the FDR at the 5% level. The size difference in power and FDR was small between SMAHP and SIS + SIS in Scenarios I and II. However, this difference was more pronounced in Scenarios III and IV. For instance, in Scenario III, our proposed method achieved a power of 0.8296, whereas SIS + SIS had a power of 0.6423 with a smaller sample size (n = 200). When the censoring rate was increased to 50% (see Table 2), the SIS + SIS approach no longer controlled the FDR, unlike our proposed method. This issue worsened as the censoring rate increased to 75% (see Table 3), regardless of the sample size adjustment. In all scenarios with such high 75%, SIS + SIS exhibited lower power and inflated FDR compared to SMAHP. On the other hand, SMAHP also had an inflated FDR with a smaller sample size, but it quickly regained control over FDR when the sample size was increased to n = 400. For instance, when the censoring rate was 75%, SMAHP achieved a power of 0.7618 and an FDR of 0.1082, which improved to a power of 0.9843 and an FDR of 0.0580 as the sample size increased.

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Table 1. Simulation results using SMAHP (penalizations + SIS), SIS + SIS, and naïve approaches. The data were simulated with censoring rate of 25%.

https://doi.org/10.1371/journal.pcbi.1014217.t001

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Table 2. Simulation results using SMAHP (penalizations + SIS), SIS + SIS, and naïve approaches. The data were simulated with censoring rate of 50%.

https://doi.org/10.1371/journal.pcbi.1014217.t002

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Table 3. Simulation results using SMAHP (penalizations + SIS), SIS + SIS, and naïve approaches. The data were simulated with censoring rate of 75%.

https://doi.org/10.1371/journal.pcbi.1014217.t003

As a whole, SMAHP required the most computation time compared with the SIS + SIS and naïve methods, as we recorded the average computational time. This is expected, as SMAHP leverages a penalization technique. See Tables 1 to 3. In addition, we explored different penalties for the mediation model in SMAHP, with the MCP-penalization used as the default. We investigated whether we would obtain similar or substantially different results with other penalties, such as the elastic-net and Lasso penalties. S1 Table summarizes this experiment, showing similar power and FDR, as well as comparable computational time. Additional sensitivity analyses, suggested by the reviewers, were conducted to further evaluate the performance of SMAHP. Specifically, we examined settings with correlated exposure genes and correlated mediator proteins (S2 Table). Across these settings, the results were broadly consistent with the primary simulation results, with SMAHP showing stable power and false discovery rate (FDR) control under low to moderate correlation levels. We also evaluated performance when exposures were sampled from a negative binomial distribution (S3 Table). In this setting, SMAHP showed similar behavior to the primary simulations, with satisfactory power and FDR control around the nominal 5% level. Sensitivity analyses incorporating outliers (S4 Table) indicated that SMAHP continued to perform well as the sample size increased. Finally, under two alternative residual distributions, Gamma (S5 Table) and logistic (S6 Table), results were consistent with the primary simulations, showing that the SMAHP maintained high power and controlled the FDR. The entire set of simulations was run on the high-performance supercomputer Minerva at the Icahn School of Medicine at Mount Sinai, utilizing 10 CPU cores and 8 GB of RAM per node.

Application study: Analysis of clinical proteomic tumor analysis consortium data

We are motivated by the problem of human papillomavirus-negative (HPV-Neg) HNSCC, which remains insufficiently studied, as highlighted by recent clinical research [17,53], despite HNSCC being ranked as the sixth most prevalent epithelial cancer globally [54]. Specifically, a recent CPTAC HNSCC study [17] has emphasized that a comprehensive understanding of how transcriptomic and molecular changes contribute to tumor phenotypes is still lacking, highlighting the critical need for further investigation in this area.

In this study, pre-processed RNA-Seq, proteomics, and clinical data (metadata) from the National Cancer Institute-initiated CPTAC were downloaded from the Proteomic Data Commons (https://pdc.cancer.gov/pdc/cptac-pancancer), which is one of the largest public repositories of proteogenomic data. The CPTAC has been utilized to increase understanding of the molecular mechanisms of cancer through its large-scale, mass spectrometry-based proteomic profiling data of tumor samples, which were previously analyzed by the Cancer Genome Atlas (TCGA) [52]. In comparison to TCGA, CPTAC provides a more extensive proteome coverage.

The overall survival (OS) is defined as the time from cancer diagnosis to death. Patients were censored if the event had not occurred by the last time of follow-up. The median OS is 46.27 months, and the Kaplan-Meier curve is presented in S1 Fig. The original sample size consisted of 109 patients, with 60,669 genes and 9,469 proteins available in the RNA-Seq and proteomics data, respectively. Prior to the application of the SMAHP, a univariate AFT model was fitted to each gene and protein separately for pre-screening purposes in this ultra high-dimensional data. The top 100 genes and top 200 proteins with the smallest p-values were then selected for analysis using SMAHP. Seven patients did not have OS data available and were excluded, resulting in a final sample size of 102 patients, with a censoring rate of 67.6%. After applying SMAHP, there was indirect effect (p = 0.001) of late cornified envelope 3E protein (LCE3E; Ensembl ID: ENSG00000185966.4) on the association between high-mobility group box 1 pseudogene 23 (HMGB1P23; Ensembl ID: ENSG00000253770.1) and OS, with age included as an additional covariate. Fig 1 provides a summary of the analysis. While the direct effect on the survival time is positive, it appears that the indirect effect through the mediator is negative. Such a pattern may arise when the exposure influences survival through multiple pathways, with the mediator capturing an effect in the opposite direction to other pathways such as the direct pathway. This observation highlights the complexity of the underlying biological processes and is provided to aid interpretation of the estimated effects.

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Fig 1. A directed acyclic graph summarizing the analysis of CPTAC HNSCC data, highlighting direct and indirect effects with 95% confidence intervals in parentheses using SMAHP.

https://doi.org/10.1371/journal.pcbi.1014217.g001

Although members of the high-mobility group box family have been implicated in carcinogenesis (e.g., high expression of HMGB1 in human nasopharyngeal carcinoma) and autoinflammatory diseases [55], no published literature exists on diseases or disorders specifically associated with HMGB1P23. A search of the MalaCards human disease database (https://www.malacards.org/) [56] also did not yield relevant findings. Interestingly, the MalaCards revealed that LCE3E is associated with plantar warts, which are caused by HPV. However, given that the study samples tested HPV-negative, this association should be interpreted with caution. Reported LCE3E-related pathways include keratinization and nervous system development. The total runtime to complete the analysis using SMAHP was 2.28 minutes. The analysis was performed on a 2023 MacBook Pro equipped with an M3 processor and 16GB of RAM.

Notations and assumptions

Herein, we consider a proteogenomic data with a survival outcome T, a vector of proteomes as mediators, a vector of genes as exposures, and additional covariates to be adjusted for such as age, sex, smoking history, alcohol consumption categories, immune score, and histologic grades from n i.i.d. observations. For better readability, the subject-level subscripts are suppressed unless otherwise stated.

In this paper, our causal mediation model is implemented within the counterfactual framework [22,23]. In counterfactual notations, we will let denote the potential outcome of the kth proteome when each is set to x, and let be the potential outcome of T with an observed expression levels for all genes when .

The causal effects are assessed by taking the mean expected difference in counterfactual outcomes that would have been observed [24,25]. In the log-scale, we define the interventional indirect (IIE) and interventional direct effect [61,62] with any two levels of continuous exposure by the decomposition of a total effect [26].

where for all genes in and or is a random draw from the distribution of or . The exposure values are rescaled in order to assess the changes in causal quantities after the unit increase in original exposure value. The estimates of IDE and IIE are obtained from the mediation and outcome models, respectively, which are discussed in Model Specifications subsection.

We will in addition assume that the IDE and IIE are estimated under the assumptions of no-unmeasured confounders [27–29] for the exposure-outcome (Eq 1), mediator-outcome (Eq 2), and exposure-mediator (Eq 3). That is, for each exposure,

(1)

(2)

(3)

Model specifications

We consider the following models to describe the causal relationships illustrated in Fig 2. For the outcome model (Eq 4), the high-dimensional accelerated failure time (AFT) model [30–32] is used to estimate and test the effect of mediator (proteomes) in the causal pathway between continuous exposures (gene expressions) on the survival outcome T, while accounting for clinically meaningful covariates . The data we use to fit this outcome model also includes , the censoring indicator for the log censoring time . Furthermore, without loss of generality, the features are centered to eliminate an intercept. For the mediator model (Eq 5), the linear regression is fitted to model the association between and each mediator for .

(4)

(5)

where is the regression coefficients vector for the effect of on T; is a vector of regression coefficients for the covariates; is the regression coefficients vector of the effect of on T with the presence of ; and is a random error variable following the log-Weibull distribtution (flexible such as log-normal and other distributions) with the scale parameter b [33]. is a vector of regression coefficients for the effect of on each ; is the regression coefficients vector for covariates; and is normal random error. It is important to note that estimates of lead to the IDE of on T (i.e., ). The IIE of on the causal pathway between and T (i.e., ) is defined by the product rule of estimates [32,34,45].

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Fig 2. A causal diagram to describe the framework of the mediation analysis of high-dimensional mediators with multivariable exposures and survival endpoint.

Here, the two types of effects are assessed: (1) the global indirect effect between and T, and specific mediation effect that is mediated by a mediator variable for each proteome that is selected from the penalized variable selection (nodes colored in blue) and (2) the direct effect between multivariable exposures RNA-Seq gene expressions (nodes colored in orange) and survival outcome T (a node colored in red).

https://doi.org/10.1371/journal.pcbi.1014217.g002

Step 1: Penalization for selecting mediator and exposure candidates

Penalization for outcome model.

It is likely that proteomes are highly correlated with one another, as are genes. Moreover, it is essential to assess the associations between these biological entities and the outcome. Therefore, we consider penalized regression technique to identify proteomic mediators and genomic exposures, following the recent work [35] on the penalized AFT model using a variation of the alternating direction method of multipliers algorithm. For the ith subject, we can derive the following equations from the outcome model (Eq 4)

where the outcome model penalization is applied separately for and , respectively. The estimated active sets, denoted as and , will be determined from each of these models. refers to the active set of pre-screened significant genes for , and refers to the active set of pre-screened significant proteins for . For more details on the penalized AFT regression, refer to the S1 Appendix.

Penalization for mediation model.

We identify important mediators and exposures in the outcome model through the penalization approach detailed in Penalization for Outcome Model subsection earlier. Concurrently, we consider penalized mediation models for each mediator, minimizing the penalties based on the minimax concave penalty (MCP) approach [37] for the marginal mediation model (Eq 5). For ith subject, we express the marginal model for mediators as follows

See the S1 Appendix for more details on MCP estimates. From this penalized mediation model, the estimated active set will be identified, consisting of pre-screened significant genes for . In our simulation study, we compared its performance with that of elastic-net [38] and L₂ regularization (Lasso) [36] when modeling the mediator marginally.

Furthermore, we can rewrite the outcome and mediation models derived from these penalization steps as follows:

(6)(7)(8)

where the focus is shifted to (i) being the estimated active set (i.e., non-zero coefficients) of K mediators selected as candidate proteomes to be extensively studied based on the penalized outcome model in Eq 6; (ii) is the estimated active set of P exposures selected as candidate genes from the penalized outcome model in Eq 7; and lastly, (iii) is the estimated active set of P exposures selected as candidate genes from the penalized mediation model in Eq 8.

Step 2: Screening important mediators with control for exposures

In Step 1, candidate proteomes and genes are selected from the penalized AFT and MCP regression models. However, these candidates are chosen based on either the disjoint indirect effect (i.e., and , separately) or the direct effect (i.e., ). Hence, in Step 2, we reformulate the models defined in Eqs 6–8 into those specified for the causal mediation framework in Eqs 4 and 5. The objective of this section is to identify “important” proteomic mediators that account for exposures and to further screen potential mediators in order to reduce dimensionality in high-dimensional data, thereby boosting computational efficiency.

For each , we consider the outcome and mediation models as redefined

where can be estimated by the maximum likelihood estimators of the AFT outcome model, and can be estimated by the ordinary least squares estimators of the linear regression mediation model, respectively.

Leveraging the sure independence screening (SIS) approach [39], the pairs ( among the top largest values of are screened. If a pair exhibits meaningful mediation effects, the mediators and genes are selected and denoted as and , where is a subset of , and is the subset of for any , respectively. Otherwise, pairs without replacing “with minimal” mediation effects are dropped. SIS is a computationally efficient method that quickly reduces the dimensionality while retaining the variables in the model with higher correlations [40]. The threshold for selecting pairs are typically and , as used without formal theoretical justificaiton in the original SIS methodology paper [39]. However, the threshold can be increased to a multiple of , such as or , to increase the probability of identifying important mediators, as demonstrated in other studies [41–44].

Step 3: Hypothesis Testing

In the earlier subsections, penalized regression approaches are used for variable selection (Step 1) and SIS for identifying “important” mediators (Step 2). By incorporating these mediators, exposures, and clinically meaningful covariates, we can express the outcome and mediation models as

(9)

(10)

where and are the active sets of “important” mediators and exposures identified in Step 2, respectively. The coefficient corresponds to estimates for the mediators within . We have identified exposures that impose greater causal effect when paired with derived from the SIS. These exposures are subsequently regressed in the mediation model as described in Eq 10.

The joint significance (JS) test [47] is adopted to test whether a particular proteomic mediator lies in the causal pathway from an genomic exposure to a survival outcome (i.e., vs. for , where denotes the cardinality of a set ). The JS test (also referred to as the JS-uniform test) has been utilized in several studies [42,46] and is recognized for its ability to control the type I error rate while maintaining statistical power [48]. The primary distinction between our study and the aforementioned studies [42,46] lies in the number of exposures being tested for each mediator. Accordingly, for , where genes, we can articulate the null and research hypotheses as follows

(11)

to determine the gene-specific (or elementwise) indirect effect for each proteome mediator. The null hypothesis in Eq 11 can be further decomposed to three disjoint null sub-hypotheses

Let denote a matrix of the mediation-exposure associations from r mediation models, be a r-vector of mediator-outcome associations from the penalized outcome model, and be a matrix containing elementwise p-values. The elementwise p-value for testing hypotheses in Eq 11 can be obtained by (i) comparing the p-values from the sth row of with that from and (ii) taking the maximum of the two p-values.

(12)

where and are defined as

where and are regression coefficient estimates from fitted models using Eqs 9 and 10, respectively. and are estimates of standard error for and . is a standard normal cumulative distribution.

It is important to appropriately control the false discovery rate (FDR) for multiple hypothesis testing. Therefore, the BH-adjusted p-value [50] is applied on the matrix using stats R package.