Pith. sign in

REVIEW 3 major objections 5 minor 45 references

Bayesian Variable Selection for High-Dimensional Mediation Analysis: Application to Metabolomics Data in Epidemiological Studies

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Modeling mediator correlations in the selection prior increases the power of high-dimensional mediation analysis to find active exposure–mediator–outcome pathways.

desk verdict Solid method paper with a real tuning confound in its headline comparison; the power gain is not yet convincingly attributed to the MRF prior. read the letter →

arxiv 2411.17910 v1 pith:KMBGAI5C submitted 2024-11-26 stat.ME stat.AP

Youngho Bae , Chanmin Kim , Fenglei Wang , Qi Sun , Kyu Ha Lee This is my paper · ORCID
classification stat.MEstat.AP MSC 62F1562J0562P10
keywords Bayesianvariableselectionhigh-dimensionalmediationanalysisMarkovrandomfieldpriorsequentialsubsettingBernoullispike-and-slabmetabolomicsindirecteffectsphasetransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a Bayesian variable-selection method for mediation analysis with many correlated mediators, the setting of metabolomics studies where hundreds of small molecules sit between an exposure and a health outcome. Its central claim is that modeling the mediator correlation structure in the prior for which pathways are active—rather than treating each mediator's inclusion independently—increases the chance of finding genuine exposure–mediator–outcome pathways without raising false positives. In simulations the proposed MVN-MRF-SSB model detected 68% of active pathways versus 57% for the independent-Bernoulli version when mediators were correlated, and it kept the false positive rate for indirect effects at 0.7% versus 1.6%. In two cohort substudies it identified 15 metabolites as mediators of the Mediterranean diet–cardiometabolic risk association, where the comparison model found 8, and it estimated indirect effects with uncertainty intervals for each selected pathway.

What carries the argument

The load-bearing object is the MRF-SSB prior pair. The Markov random field prior for $\gamma_j$, the exposure–mediator inclusion indicator, is $\pi(\gamma_j \mid \gamma_{(-j)}, \Sigma) = \mathrm{logit}^{-1}(\theta_\gamma + \eta \sum_{r \neq j} |c_{r,j}| \gamma_r)$, so a mediator's inclusion probability rises when correlated mediators are included, with $c_{r,j}$ taken from the mediator covariance. The sequential subsetting Bernoulli prior for $\omega_j$, the mediator–outcome inclusion indicator, is $\pi(\omega_j) = \tilde{\gamma}_j \theta_\omega^{\omega_j}(1-\theta_\omega)^{1-\omega_j} + (1-\tilde{\gamma}_j) I_0$, where $\tilde{\gamma}_j$ is the realized exposure–mediator inclusion; this transmits mediator-model selections to the outcome model but not vice versa. These priors sit on spike-and-slab distributions for $\tau_j$ and $\delta_j$, with the mediator covariance estimated by a factor-analytic structure $\Sigma = \sigma_\Sigma^2(\lambda\lambda^\top + I_q)$, and the MRF coupling strength $\eta$ is placed below a phase-transition boundary found by posterior simulation. The MRF part is what converts mediator correlations into extra detection power; the SSB part is what keeps the eventual indirect-effect selection sparse.

What would settle it

Rerun the Scenario I comparison with the mediator indices randomly permuted in the MRF edge weights $c_{r,j}$—same mediator data, same correlations, but the prior's edges no longer point at the true correlation neighborhoods. If MVN-MRF-SSB still beats MVN-IB-SSB on the combined-pathway TPR, the correlation structure is not the source of the gain; if the advantage vanishes, the paper's mechanism is confirmed.

Watch

Extended reading notes

Core claim

The paper's central claim is that the standard practice of assigning independent Bernoulli priors to the binary selection indicators in high-dimensional mediation analysis discards useful information: mediators that are correlated with an active mediator are more likely to carry an exposure effect, and the proposed Markov random field prior exploits exactly that. Coupled with a sequential subsetting Bernoulli prior that feeds the selected exposure–mediator pathways forward into the mediator–outcome selection while cutting feedback in the reverse direction, the prior pair improves the true positive rate for detecting active pathways in correlated settings. In the paper's Scenario I the gain is 68.1% versus 56.9% TPR on the combined pathways, with a lower false positive rate for indirect effects (0.7% versus 1.6%). In the motivating metabolomics analysis the prior identifies seven additional diet-related metabolites not selected by the independent-prior model, among them fish-intake markers such as C22:6 lysophosphatidylcholine.

Load-bearing premise

The method's causal interpretation relies on the Imai–Yamamoto assumption that every mediator operates through an independent pathway, with no causal effect of one mediator on another and no unmeasured common cause of mediators and outcome; if mediators affect each other, the estimated indirect effects are not identified as causal.

Editorial extensions

If this is right

  • In correlated-mediator settings, the proposed MVN-MRF-SSB model detects active exposure–mediator–outcome pathways at a higher true positive rate than the independent-Bernoulli counterpart (68.1% versus 56.9% in Scenario I).
  • The power gain disappears when mediators are uncorrelated (Scenario II), showing the improvement is attributable to the correlation structure rather than to generally more aggressive selection.
  • Under a null model with no active pathways (Scenario III), the proposed method selects essentially no pathways (mean NVS 0.4), so the correlation-aware prior does not inflate false discoveries in the absence of true effects.
  • The sequential subsetting prior keeps the final indirect-effect false positive rate low even when the MRF prior raises the candidate pool at the exposure–mediator stage (0.7% versus 1.6% in Scenario I).
  • In the metabolomics application the method reports 15 active mediators of the Mediterranean diet–cardiometabolic risk association, including additional fish-intake markers, with posterior-median indirect effects and 95% intervals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension the authors mention but do not develop is to replace the continuous mediator model with generalized linear models for binary or count mediators; the same MRF-SSB prior should carry over and the power gain can be checked in simulations.
  • Because the SSB prior cuts feedback from the outcome model back to exposure–mediator selection, the method is asymmetric by design: a mediator with a strong outcome association but no detected exposure association can never enter the final pathway set; analysts should decide whether that asymmetry fits their question.
  • The phase-transition calibration for $\eta$ is a heuristic posterior-simulation step; an automatic data-driven rule would remove the most delicate tuning choice and would clarify how much of the reported gain depends on tuning rather than on the prior's structure.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a Bayesian variable selection framework for high-dimensional mediation analysis with correlated mediators. The method combines a Markov random field (MRF) prior on the exposure-mediator selection indicators, which leverages mediator correlations, with sequential subsetting Bernoulli (SSB) priors on the mediator-outcome selection indicators, which encourage joint selection of active pathways. The authors present simulation studies comparing the proposed MVN-MRF-SSB model with MVN-IB-SSB and Normal-IB-SSB baselines across four scenarios, and apply the method to metabolomics data from the HPFS and NHSII cohorts to identify metabolites mediating the relationship between Mediterranean diet adherence and cardiometabolic risk. The central empirical claim is that the MRF prior improves power to detect active exposure-mediator-outcome pathways while maintaining specificity.

Significance. If the central claim holds, the proposed method would be a useful addition to the toolkit for high-dimensional mediation analysis in correlated omics data. The paper includes publicly available code, a realistic application, and a comparison against simpler variants of its own model. The methodological contribution is the integration of MRF and SSB priors in a joint mediation model, which is a sensible extension of existing Bayesian variable selection ideas. However, the strength of the evidence for the main claim is limited by the confounded comparison protocol and the omission of the most relevant existing competitor, so the practical significance is currently not fully established.

major comments (3)
  1. [Sections 3.2 and 4.3, Table 1] The comparison between MVN-MRF-SSB and MVN-IB-SSB is confounded by asymmetric prior calibration. The MRF prior's η is selected per simulated dataset by the phase-transition algorithm in Section 3.2, which by construction chooses the largest grid value below the first η at which the posterior mean inclusion count exceeds the η=0 count by 0.05. In contrast, the IB prior uses a fixed θγ = logit^{-1}(0.1) with no analogous calibration. In Scenario I this yields NVS(γ)=24.5 versus 18.3 and TPR(γ)=68.7% versus 58.4%, but also FPR(γ)=1.4% versus 0.3%. The observed power gain may therefore reflect the sensitivity boost from data-dependent tuning rather than the MRF's exploitation of mediator correlations. The authors should re-run the comparison with matched calibration—for example, choosing θγ for the IB prior to achieve the same expected model size, or applying the same phase-transition algorithm to a model with η=0—and report whether the TPR gap persists.
  2. [Sections 4.1 and 4.2] The simulation study omits the most relevant existing competitor. The Introduction cites Song et al. (2021a) as a joint mediation framework that handles correlated mediators through a latent neighboring structure, but the simulation comparisons include only Normal-IB-SSB and MVN-IB-SSB, which are simplifications of the proposed model rather than independent published methods. Without a comparison against Song et al. (2021a) or another existing joint mediation method, the abstract's claim of 'superior power' over existing approaches is unsupported. The authors should add such a comparison or restrict the claim to the comparison with the independent-Bernoulli baseline.
  3. [Section 4.3, Table 1] The claim that specificity is maintained is not supported at the exposure-mediator stage. In Scenario I, MVN-MRF-SSB increases FPR(γ) from 0.3% to 1.4%, more than a fourfold increase, and in Scenario IV from 0.3% to 0.7% with a standard deviation of 1.6. The lower FPR for the final γ×ω pathway (0.7% versus 1.6% in Scenario I) is produced by the SSB prior's pruning in the outcome model, not by the MRF prior. The text should distinguish the two stages and avoid describing the FPR increase as 'slight'; the tradeoff should be reported transparently.
minor comments (5)
  1. [Section 1] The statement that 'no existing statistical methods for joint mediation analysis leverage mediator correlations through specialized priors or penalty terms' is inconsistent with the later citation of Song et al. (2021a), which is described as employing a neighboring structure for correlated mediators; please reword to acknowledge prior work.
  2. [Section 3.2] The definition of γ(ηg) as the '50-th percentile of the posterior mean estimates of γ' is ambiguous; please clarify whether it is the median over mediators of the posterior inclusion probabilities or the median over MCMC iterations.
  3. [Section 5.3, Table 3] Several reported 95% HPDIs for indirect effects include zero (e.g., C51:3 TG: 0.15 (0.00, 0.25); C22:6 LPC: -0.08 (-0.13, 0.00)), while these pathways are listed as selected; since selection is based on PPI thresholds, please add a note explaining how to interpret such intervals.
  4. [Section 1, last paragraph] There is a typo in 'provides an detailed analysis'; it should be 'a detailed analysis'.
  5. [Section 4.2] The paper would benefit from a table or figure showing the distribution of the selected η values across the 120 simulated datasets, to help readers assess the variability in the data-dependent calibration.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the simulation and real-data results are new posterior quantities from fitted models, and no equation reduces to a fitted parameter by construction.

full rationale

The paper's central claim is an empirical comparison (MVN-MRF-SSB vs MVN-IB-SSB) on independently generated data (Section 4) and a real-data illustration (Section 5). The reported TPR/FPR, PPIs, indirect-effect estimates, and selected pathways are computed after fitting the Bayesian model; none of these outputs is defined in terms of the headline result. The MRF prior p_j(γ,Σ) uses the estimated mediator correlation matrix Σ, but this is an empirical-Bayes prior construction rather than a target quantity being predicted, and the SSB prior's dependence of ω_j on γ_j is a transparent modelling choice that the paper does not disguise as a finding. The only notable concern is that η for the proposed method is selected per dataset by the Section 3.2 phase-transition algorithm, which is designed to increase posterior inclusion counts, while the comparator uses a fixed θγ=0.1; this asymmetry is a legitimate threat to the fairness of the power comparison, but it is not circular because the reported TPR gain is an empirical outcome, not an analytical consequence of the tuning rule, and the paper discloses the tuning procedure. Self-citations (Lee et al. 2017) for the calibration and factor-analytic robustness are not load-bearing in a circular sense: the algorithm is restated in the text and Scenario IV independently probes misspecification. The Discussion openly lists limitations (no sequential mediator ordering, restrictive FA covariance, illustrative real-data analysis), none of which hides a definitional equivalence. No circular step meeting the quotation-and-reduction standard was found.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard causal inference assumptions (SUTVA, no mediator-mediator causation), a linear additive model, and a factor-analytic covariance structure. The MRF tuning parameter eta is selected data-dependently, and prior inclusion probabilities are fixed by hand. No new physical or conceptual entities are introduced.

free parameters (4)
  • eta (MRF correlation strength) = set to eta_{g-1} below phase transition boundary, chosen via posterior simulations over a grid of eta values on the data
    Data-dependent tuning of the edge strength in the Markov random field prior; the value directly controls how much mediator correlations influence selection.
  • theta_gamma (prior inclusion probability for exposure-mediator selection) = logit^{-1}(theta_gamma) = 0.1
    Chosen by hand as a prior guess for the proportion of exposure-mediator associations; it controls model size along with v_j^2.
  • theta_omega (prior inclusion probability for mediator-outcome selection) = 0.1
    Chosen by hand as the prior proportion of active mediator-outcome associations, used in the sequential subsetting Bernoulli prior.
  • v_j^2 and psi_j^2 (spike-and-slab slab variances) = 9
    Chosen by hand for the slab variance in the spike-and-slab priors on tau and delta; larger values allow larger effect sizes.
assumptions (4)
  • domain assumption Stable Unit Treatment Value Assumption (SUTVA)
    Links potential mediators and outcomes to observed data; invoked in Section 2 without discussion of interference or hidden treatment variants.
  • domain assumption Causally independent mediators (Imai-Yamamoto assumption)
    Assumes no causal effects among mediators and that each mediator is independent of the potential outcome given confounders; needed for nonparametric identification of each indirect effect. The authors explicitly assume independent pathways in Section 2.
  • domain assumption Linear additive mediator and outcome models
    Equations (1) and (2) specify linear models for mediators and outcome, with no mediator-mediator interactions and no exposure-mediator interaction in the outcome model. This structure is required for the formula IE_j = (a-a') delta_j tau_j.
  • domain assumption Factor analytic covariance structure for mediators
    Section 3.3 assumes Sigma = sigma_Sigma^2 (lambda lambda^T + I_q), a one-factor model. The authors note this is restrictive but computationally feasible, and cite robustness under misspecification from prior work.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bayesian Variable Selection for High-Dimensional Mediation Analysis: Application to Metabolomics Data in Epidemiological Studies." pith.science (2026). https://pith.science/paper/KMBGAI5C

@misc{pith2026241117910,
  author       = {Pith},
  title        = {Pith review of: Bayesian Variable Selection for High-Dimensional Mediation Analysis: Application to Metabolomics Data in Epidemiological Studies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMBGAI5C}},
  note         = {Machine review of arXiv:2411.17910}
}
read the original abstract

In epidemiological research, causal models incorporating potential mediators along a pathway are crucial for understanding how exposures influence health outcomes. This work is motivated by integrated epidemiological and blood biomarker studies, investigating the relationship between long-term adherence to a Mediterranean diet and cardiometabolic health, with plasma metabolomes as potential mediators. Analyzing causal mediation in such high-dimensional omics data presents substantial challenges, including complex dependencies among mediators and the need for advanced regularization or Bayesian techniques to ensure stable and interpretable estimation and selection of indirect effects. To this end, we propose a novel Bayesian framework for identifying active pathways and estimating indirect effects in the presence of high-dimensional multivariate mediators. Our approach adopts a multivariate stochastic search variable selection method, tailored for such complex mediation scenarios. Central to our method is the introduction of a set of priors for the selection: a Markov random field prior and sequential subsetting Bernoulli priors. The first prior's Markov property leverages the inherent correlations among mediators, thereby increasing power to detect mediated effects. The sequential subsetting aspect of the second prior encourages the simultaneous selection of relevant mediators and their corresponding indirect effects from the two model parts, providing a more coherent and efficient variable selection framework, specific to mediation analysis. Comprehensive simulation studies demonstrate that the proposed method provides superior power in detecting active mediating pathways. We further illustrate the practical utility of the method through its application to metabolome data from two cohort studies, highlighting its effectiveness in real data setting.

Figures

Figures reproduced from arXiv: 2411.17910 by the authors.

Figure 1
Figure 1. Empirical correlations among the measures of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. A directed acyclic graph illustrating high-dimensional mediators. Each mediator [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Estimated marginal posterior probabilities of inclusion (PPI) associated with [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 39 canonical work pages

  1. [1]

    T., Song, Y., Ferguson, K

    Aung, M. T., Song, Y., Ferguson, K. K., Cantonwine, D. E., Zeng, L., McElrath, T. F., Pennathur, S., Meeker, J. D., and Mukherjee, B. (2020). Application of an analytical framework for multivariate mediation analysis of environmental data. Nature C ommunications , 11(1):5624

    work page 2020

  2. [2]

    L., Lenart, E

    Bao, Y., Bertoia, M. L., Lenart, E. B., Stampfer, M. J., Willett, W. C., Speizer, F. E., and Chavarro, J. E. (2016). Origin, methods, and evolution of the three nurses' health studies. American Journal of Public Health , 106(9):1573--1581

    work page 2016

  3. [3]

    and Dunson, D

    Bhattacharya, A. and Dunson, D. B. (2011). Sparse B ayesian infinite factor models. Biometrika , 98(2):291--306

    2011

  4. [4]

    and Friedman, J

    Breiman, L. and Friedman, J. (1997). Predicting multivariate responses in multiple linear regression. Journal of the Royal Statistical Society: Series B , 59(1):3--54

    work page 1997

  5. [5]

    Castelletti, F. (2020). Bayesian model selection of G aussian directed acyclic graph structures. International Statistical Review , 88(3):752--775

    work page 2020

  6. [6]

    Y., Crainiceanu, C., Ogburn, E

    Ch \'e n, O. Y., Crainiceanu, C., Ogburn, E. L., Caffo, B. S., Wager, T. D., and Lindquist, M. A. (2018). High-dimensional multivariate mediation with application to neuroimaging data. Biostatistics , 19(2):121--136

    2018

  7. [7]

    G., Khondker, Z

    Chu, H., Ibrahim, J. G., Khondker, Z. S., Lin, W., and Zhu, H. (2013). The B ayesian covariance lasso. Statistics and its Interface , 6(2):243--259

    work page 2013

  8. [8]

    M., De Stavola, B

    Daniel, R. M., De Stavola, B. L., Cousens, S. N., and Vansteelandt, S. (2015). Causal mediation analysis with multiple mediators. Biometrics , 71(1):1--14

    work page 2015

Show all 45 references
  1. [9]

    M., Chen, T.-H., and Sampson, J

    Derkach, A., Pfeiffer, R. M., Chen, T.-H., and Sampson, J. N. (2019). High dimensional mediation analysis with latent variables. Biometrics , 75(3):745--756

    2019

  2. [10]

    Forastiere, L., Mealli, F., and VanderWeele, T. J. (2016). Identification and estimation of causal mechanisms in clustered encouragement designs: disentangling bed nets using B ayesian principal stratification. Journal of the American Statistical Association , 111(514):510--525

    2016

  3. [11]

    L., and Cui, Y

    Gao, Y., Yang, H., Fang, R., Zhang, Y., Goode, E. L., and Cui, Y. (2019). Testing mediation effects in high-dimensional epigenetic studies. Frontiers in genetics , 10:1195

    2019

  4. [12]

    B., Stern, H

    Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2013). Bayesian Data Analysis . Chapman and Hall/CRC Boca Raton

    2013

  5. [13]

    and McCulloch, R

    George, E. and McCulloch, R. (1997). Approaches for B ayesian variable selection. Statistica Sinica , 7:339--374

    1997

  6. [14]

    George, E. I. and McCulloch, R. E. (1993). Variable selection via Gibbs sampling . Journal of the American Statistical Association , 88(423):881--889

    1993

  7. [15]

    Hogan, J. W. and Tchernis, R. (2004). Bayesian factor analysis for spatially correlated data, with application to summarizing area-level material deprivation from census data. Journal of the American Statistical Association , 99(466):314--324

    2004

  8. [16]

    M., Sawyer, S., Kubzansky, L

    Huang, T., Trudel-Fitzgerald, C., Poole, E. M., Sawyer, S., Kubzansky, L. D., Hankinson, S. E., Okereke, O. I., and Tworoger, S. S. (2019). The mind--body study: Study design and reproducibility and interrelationships of psychosocial factors in the nurses' health study II . Ca...

    2019

  9. [17]

    and Pan, W.-C

    Huang, Y.-T. and Pan, W.-C. (2016). Hypothesis test of mediation effect in causal mediation model with high-dimensional continuous mediators. Biometrics , 72(2):402--413

    2016

  10. [18]

    and Yamamoto, T

    Imai, K. and Yamamoto, T. (2013). Identification and sensitivity analysis for multiple causal mechanisms: Revisiting evidence from framing experiments. Political Analysis , 21(2):141--171

    2013

  11. [19]

    Kim, H.-Y. (2013). Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis. Restorative D entistry & E ndodontics , 38(1):52--54

    2013

  12. [20]

    H., Coull, B

    Lee, K. H., Coull, B. A., Moscicki, A.-B., Paster, B. J., and Starr, J. R. (2020). Bayesian variable selection for multivariate zero-inflated models: Application to microbiome count data. Biostatistics , 21(3):499--517

    2020

  13. [21]

    H., Tadesse, M

    Lee, K. H., Tadesse, M. G., Baccarelli, A. A., Schwartz, J., and Coull, B. A. (2017). Multivariate Bayesian variable selection exploiting dependence structure among outcomes: Application to air pollution effects on DNA methylation . Biometrics , 73(1):232--241

    2017

  14. [22]

    and Zhang, N

    Li, F. and Zhang, N. (2010). Bayesian variable selection in structured high-dimensional covariate spaces with applications in genomics. Journal of the American Statistical Association , 105(491):1202--1214

    2010

  15. [23]

    Lunn, D., Best, N., Spiegelhalter, D., Graham, G., and Neuenschwander, B. (2009). Combining mcmc with ‘sequential’pkpd modelling. Journal of P harmacokinetics and P harmacodynamics , 36:19--38

    2009

  16. [24]

    Perera, C., Zhang, H., Zheng, Y., Hou, L., Qu, A., Zheng, C., Xie, K., and Liu, L. (2022). Hima2: high-dimensional mediation analysis and its application in epigenome-wide dna methylation data. BMC B ioinformatics , 23(1):296

    2022

  17. [25]

    B., Giovannucci, E

    Rimm, E. B., Giovannucci, E. L., Willett, W. C., Colditz, G. A., Ascherio, A., Rosner, B., and Stampfer, M. J. (1991). Prospective study of alcohol consumption and risk of coronary disease in men. The Lancet , 338(8765):464--468

    1991

  18. [26]

    Rubin, D. B. (1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology , 66(5):688

    1974

  19. [27]

    Rubin, D. B. (1990). Formal mode of statistical inference for causal effects. Journal of Statistical Planning and Inference , 25(3):279--292

    1990

  20. [28]

    C., Smilde, A

    Saccenti, E., Hoefsloot, H. C., Smilde, A. K., Westerhuis, J. A., and Hendriks, M. M. (2014). Reflections on univariate and multivariate analysis of metabolomics data. Metabolomics , 10:361--374

    2014

  21. [29]

    Y., Bhupathiraju, S

    Shan, Z., Wang, F., Li, Y., Baden, M. Y., Bhupathiraju, S. N., Wang, D. D., Sun, Q., Rexrode, K. M., Rimm, E. B., Qi, L., et al. (2023). Healthy eating patterns and risk of total and cause-specific mortality. JAMA I nternal M edicine , 183(2):142--153

    2023

  22. [30]

    T., Zhang, M., Zhao, W., Needham, B

    Song, Y., Zhou, X., Kang, J., Aung, M. T., Zhang, M., Zhao, W., Needham, B. L., Kardia, S. L., Liu, Y., Meeker, J. D., et al. (2021a). Bayesian hierarchical models for high-dimensional mediation analysis with coordinated selection of correlated mediators. Statistics in M edici...

    2021

  23. [31]

    T., Zhang, M., Zhao, W., Needham, B

    Song, Y., Zhou, X., Kang, J., Aung, M. T., Zhang, M., Zhao, W., Needham, B. L., Kardia, S. L., Liu, Y., Meeker, J. D., et al. (2021b). Bayesian sparse mediation analysis with targeted penalization of natural indirect effects. Journal of the Royal Statistical Society Series C: ...

    2021

  24. [32]

    L., Roux, A

    Song, Y., Zhou, X., Zhang, M., Zhao, W., Liu, Y., Kardia, S. L., Roux, A. V. D., Needham, B. L., Smith, J. A., and Mukherjee, B. (2020). Bayesian shrinkage estimation of high dimensional causal mediation effects in omics studies. Biometrics , 76(3):700--710

    2020

  25. [33]

    Stingo, F., Chen, Y., Tadesse, M., and Vannucci, M. (2011). Incorporating biological information into linear models: a Bayesian approach to the selection of pathways and genes . The Annals of Applied Statistics , 5(3):1978--2002

    2011

  26. [34]

    and Vansteelandt, S

    VanderWeele, T. and Vansteelandt, S. (2014). Mediation analysis with multiple mediators. Epidemiologic M ethods , 2(1):95--115

    2014

  27. [35]

    D., Argiento, R., Guindani, M., Galloway-Pena, J., Shelburne, S

    Wadsworth, W. D., Argiento, R., Guindani, M., Galloway-Pena, J., Shelburne, S. A., and Vannucci, M. (2017). An integrative B ayesian D irichlet-multinomial regression model for the analysis of taxonomic abundances in microbiome data. BMC B ioinformatics , 18:1--12

    2017

  28. [36]

    D., Nguyen, L

    Wang, D. D., Nguyen, L. H., Li, Y., Yan, Y., Ma, W., Rinott, E., Ivey, K. L., Shai, I., Willett, W. C., Hu, F. B., et al. (2021). The gut microbiome modulates the protective association between a mediterranean diet and cardiometabolic disease risk. Nature M edicine , 27(2):333--343

    2021

  29. [37]

    Y., Guasch-Ferr \'e , M., Wittenbecher, C., Li, J., Li, Y., Wan, Y., Bhupathiraju, S

    Wang, F., Baden, M. Y., Guasch-Ferr \'e , M., Wittenbecher, C., Li, J., Li, Y., Wan, Y., Bhupathiraju, S. N., Tobias, D. K., Clish, C. B., et al. (2022). Plasma metabolite profiles related to plant-based diets and the risk of type 2 diabetes. Diabetologia , 65(7):1119--1132

    2022

  30. [38]

    J., Tessier, A.-J., Mei, Z., Haslam, D

    Wang, F., Glenn, A. J., Tessier, A.-J., Mei, Z., Haslam, D. E., Guasch-Ferr \'e , M., Tobias, D. K., Eliassen, A. H., Manson, J. E., Clish, C., et al. (2024). Integration of epidemiological and blood biomarker analysis links haem iron intake to increased type 2 diabetes risk. ...

    2024

  31. [39]

    Wei, R., Wang, J., Su, M., Jia, E., Chen, S., Chen, T., and Ni, Y. (2018). Missing value imputation approach for mass spectrometry-based metabolomics data. Scientific R eports , 8(1):663

    2018

  32. [40]

    Zhang, H., Zheng, Y., Zhang, Z., Gao, T., Joyce, B., Yoon, G., Zhang, W., Schwartz, J., Just, A., Colicino, E., et al. (2016). Estimating and testing high-dimensional mediation effects in epigenetic studies. Bioinformatics , 32(20):3150--3154

    2016

  33. [41]

    Zhang, Q. (2022). High-dimensional mediation analysis with applications to causal gene identification. Statistics in B iosciences , 14(3):432--451

    2022

  34. [42]

    A., and Caffo, B

    Zhao, Y., Lindquist, M. A., and Caffo, B. S. (2020). Sparse principal component based high-dimensional mediation analysis. Computational S tatistics & D ata A nalysis , 142:106835

    2020

  35. [43]

    and Luo, X

    Zhao, Y. and Luo, X. (2022). Pathway lasso: Pathway estimation and selection with high-dimensional mediators. Statistics and its Interface , 15(1):39

    2022

  36. [44]

    Zhao, Z., Banterle, M., Lewin, A., and Zucknick, M. (2024). Multivariate Bayesian structured variable selection for pharmacogenomic studies . Journal of the Royal Statistical Society Series C: Applied Statistics , 73(2):420--443

    2024

  37. [45]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence '...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.