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Jang H, Koh H. A unified web cloud computing platform MiMedSurv for microbiome causal mediation analysis with survival responses. Sci Rep 2024; 14:20650. [PMID: 39232070 PMCID: PMC11374894 DOI: 10.1038/s41598-024-71852-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2024] [Accepted: 08/31/2024] [Indexed: 09/06/2024] Open
Abstract
In human microbiome studies, mediation analysis has recently been spotlighted as a practical and powerful analytic tool to survey the causal roles of the microbiome as a mediator to explain the observed relationships between a medical treatment/environmental exposure and a human disease. We also note that, in a clinical research, investigators often trace disease progression sequentially in time; as such, time-to-event (e.g., time-to-disease, time-to-cure) responses, known as survival responses, are prevalent as a surrogate variable for human health or disease. In this paper, we introduce a web cloud computing platform, named as microbiome mediation analysis with survival responses (MiMedSurv), for comprehensive microbiome mediation analysis with survival responses on user-friendly web environments. MiMedSurv is an extension of our prior web cloud computing platform, named as microbiome mediation analysis (MiMed), for survival responses. The two main features that are well-distinguished are as follows. First, MiMedSurv conducts some baseline exploratory non-mediational survival analysis, not involving microbiome, to survey the disparity in survival response between medical treatments/environmental exposures. Then, MiMedSurv identifies the mediating roles of the microbiome in various aspects: (i) as a microbial ecosystem using ecological indices (e.g., alpha and beta diversity indices) and (ii) as individual microbial taxa in various hierarchies (e.g., phyla, classes, orders, families, genera, species). To illustrate its use, we survey the mediating roles of the gut microbiome between antibiotic treatment and time-to-type 1 diabetes. MiMedSurv is freely available on our web server ( http://mimedsurv.micloud.kr ).
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Affiliation(s)
- Hyojung Jang
- Department of Applied Mathematics and Statistics, The State University of New York, Korea, Incheon, South Korea
| | - Hyunwook Koh
- Department of Applied Mathematics and Statistics, The State University of New York, Korea, Incheon, South Korea.
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Roy S, Daniels MJ, Roy J. A Bayesian nonparametric approach for multiple mediators with applications in mental health studies. Biostatistics 2024; 25:919-932. [PMID: 38332624 PMCID: PMC11247183 DOI: 10.1093/biostatistics/kxad038] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/15/2023] [Revised: 12/14/2023] [Accepted: 12/15/2023] [Indexed: 02/10/2024] Open
Abstract
Mediation analysis with contemporaneously observed multiple mediators is a significant area of causal inference. Recent approaches for multiple mediators are often based on parametric models and thus may suffer from model misspecification. Also, much of the existing literature either only allow estimation of the joint mediation effect or estimate the joint mediation effect just as the sum of individual mediator effects, ignoring the interaction among the mediators. In this article, we propose a novel Bayesian nonparametric method that overcomes the two aforementioned drawbacks. We model the joint distribution of the observed data (outcome, mediators, treatment, and confounders) flexibly using an enriched Dirichlet process mixture with three levels. We use standardization (g-computation) to compute all possible mediation effects, including pairwise and all other possible interaction among the mediators. We thoroughly explore our method via simulations and apply our method to a mental health data from Wisconsin Longitudinal Study, where we estimate how the effect of births from unintended pregnancies on later life mental depression (CES-D) among the mothers is mediated through lack of self-acceptance and autonomy, employment instability, lack of social participation, and increased family stress. Our method identified significant individual mediators, along with some significant pairwise effects.
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Affiliation(s)
- Samrat Roy
- Operations and Decision Sciences, Indian Institute of Management Ahmedabad, Gujarat, India
| | | | - Jason Roy
- Department of Biostatistics and Epidemiology, Rutgers University, New Brunswick, USA
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Abstract
The microbiome represents a hidden world of tiny organisms populating not only our surroundings but also our own bodies. By enabling comprehensive profiling of these invisible creatures, modern genomic sequencing tools have given us an unprecedented ability to characterize these populations and uncover their outsize impact on our environment and health. Statistical analysis of microbiome data is critical to infer patterns from the observed abundances. The application and development of analytical methods in this area require careful consideration of the unique aspects of microbiome profiles. We begin this review with a brief overview of microbiome data collection and processing and describe the resulting data structure. We then provide an overview of statistical methods for key tasks in microbiome data analysis, including data visualization, comparison of microbial abundance across groups, regression modeling, and network inference. We conclude with a discussion and highlight interesting future directions.
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Affiliation(s)
- Christine B Peterson
- Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, Texas, USA
| | - Satabdi Saha
- Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, Texas, USA
| | - Kim-Anh Do
- Department of Biostatistics, The University of Texas MD Anderson Cancer Center, Houston, Texas, USA
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Zhang H, Hong X, Zheng Y, Hou L, Zheng C, Wang X, Liu L. High-dimensional quantile mediation analysis with application to a birth cohort study of mother-newborn pairs. Bioinformatics 2024; 40:btae055. [PMID: 38290773 PMCID: PMC10873903 DOI: 10.1093/bioinformatics/btae055] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/16/2024] [Revised: 02/16/2024] [Accepted: 02/16/2024] [Indexed: 02/01/2024] Open
Abstract
MOTIVATION There has been substantial recent interest in developing methodology for high-dimensional mediation analysis. Yet, the majority of mediation statistical methods lean heavily on mean regression, which limits their ability to fully capture the complex mediating effects across the outcome distribution. To bridge this gap, we propose a novel approach for selecting and testing mediators throughout the full range of the outcome distribution spectrum. RESULTS The proposed high-dimensional quantile mediation model provides a comprehensive insight into how potential mediators impact outcomes via their mediation pathways. This method's efficacy is demonstrated through extensive simulations. The study presents a real-world data application examining the mediating effects of DNA methylation on the relationship between maternal smoking and offspring birthweight. AVAILABILITY AND IMPLEMENTATION Our method offers a publicly available and user-friendly function qHIMA(), which can be accessed through the R package HIMA at https://CRAN.R-project.org/package=HIMA.
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Affiliation(s)
- Haixiang Zhang
- Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
| | - Xiumei Hong
- Department of Population, Family and Reproductive Health, Center On the Early Life Origins of Disease, Johns Hopkins University Bloomberg School of Public Health, Baltimore, MD 21205, United States
| | - Yinan Zheng
- Department of Preventive Medicine, Northwestern University, Chicago, IL 60611, United States
| | - Lifang Hou
- Department of Preventive Medicine, Northwestern University, Chicago, IL 60611, United States
| | - Cheng Zheng
- Department of Biostatistics, University of Nebraska Medical Center, Omaha, NE 68198, United States
| | - Xiaobin Wang
- Department of Population, Family and Reproductive Health, Center On the Early Life Origins of Disease, Johns Hopkins University Bloomberg School of Public Health, Baltimore, MD 21205, United States
- Department of Pediatrics, Johns Hopkins University School of Medicine, Baltimore, MD 21205, United States
| | - Lei Liu
- Division of Biostatistics, Washington University in St. Louis, St. Louis, MO 63110, United States
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Jang H, Park S, Koh H. Comprehensive microbiome causal mediation analysis using MiMed on user-friendly web interfaces. Biol Methods Protoc 2023; 8:bpad023. [PMID: 37840574 PMCID: PMC10576642 DOI: 10.1093/biomethods/bpad023] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2023] [Revised: 09/19/2023] [Accepted: 10/02/2023] [Indexed: 10/17/2023] Open
Abstract
It is a central goal of human microbiome studies to see the roles of the microbiome as a mediator that transmits environmental, behavioral, or medical exposures to health or disease outcomes. Yet, mediation analysis is not used as much as it should be. One reason is because of the lack of carefully planned routines, compilers, and automated computing systems for microbiome mediation analysis (MiMed) to perform a series of data processing, diversity calculation, data normalization, downstream data analysis, and visualizations. Many researchers in various disciplines (e.g. clinicians, public health practitioners, and biologists) are not also familiar with related statistical methods and programming languages on command-line interfaces. Thus, in this article, we introduce a web cloud computing platform, named as MiMed, that enables comprehensive MiMed on user-friendly web interfaces. The main features of MiMed are as follows. First, MiMed can survey the microbiome in various spheres (i) as a whole microbial ecosystem using different ecological measures (e.g. alpha- and beta-diversity indices) or (ii) as individual microbial taxa (e.g. phyla, classes, orders, families, genera, and species) using different data normalization methods. Second, MiMed enables covariate-adjusted analysis to control for potential confounding factors (e.g. age and gender), which is essential to enhance the causality of the results, especially for observational studies. Third, MiMed enables a breadth of statistical inferences in both mediation effect estimation and significance testing. Fourth, MiMed provides flexible and easy-to-use data processing and analytic modules and creates nice graphical representations. Finally, MiMed employs ChatGPT to search for what has been known about the microbial taxa that are found significantly as mediators using artificial intelligence technologies. For demonstration purposes, we applied MiMed to the study on the mediating roles of oral microbiome in subgingival niches between e-cigarette smoking and gingival inflammation. MiMed is freely available on our web server (http://mimed.micloud.kr).
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Affiliation(s)
- Hyojung Jang
- Department of Applied Mathematics and Statistics, The State University of New York, Korea, Incheon, South Korea
| | - Solha Park
- Department of Applied Mathematics and Statistics, The State University of New York, Korea, Incheon, South Korea
| | - Hyunwook Koh
- Department of Applied Mathematics and Statistics, The State University of New York, Korea, Incheon, South Korea
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Wang C, Ahn J, Tarpey T, Yi SS, Hayes RB, Li H. A microbial causal mediation analytic tool for health disparity and applications in body mass index. MICROBIOME 2023; 11:164. [PMID: 37496080 PMCID: PMC10373330 DOI: 10.1186/s40168-023-01608-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2023] [Accepted: 06/22/2023] [Indexed: 07/28/2023]
Abstract
BACKGROUND Emerging evidence suggests the potential mediating role of microbiome in health disparities. However, no analytic framework can be directly used to analyze microbiome as a mediator between health disparity and clinical outcome, due to the non-manipulable nature of the exposure and the unique structure of microbiome data, including high dimensionality, sparsity, and compositionality. METHODS Considering the modifiable and quantitative features of the microbiome, we propose a microbial causal mediation model framework, SparseMCMM_HD, to uncover the mediating role of microbiome in health disparities, by depicting a plausible path from a non-manipulable exposure (e.g., ethnicity or region) to the outcome through the microbiome. The proposed SparseMCMM_HD rigorously defines and quantifies the manipulable disparity measure that would be eliminated by equalizing microbiome profiles between comparison and reference groups and innovatively and successfully extends the existing microbial mediation methods, which are originally proposed under potential outcome or counterfactual outcome study design, to address health disparities. RESULTS Through three body mass index (BMI) studies selected from the curatedMetagenomicData 3.4.2 package and the American gut project: China vs. USA, China vs. UK, and Asian or Pacific Islander (API) vs. Caucasian, we exhibit the utility of the proposed SparseMCMM_HD framework for investigating the microbiome's contributions in health disparities. Specifically, BMI exhibits disparities and microbial community diversities are significantly distinctive between reference and comparison groups in all three applications. By employing SparseMCMM_HD, we illustrate that microbiome plays a crucial role in explaining the disparities in BMI between ethnicities or regions. 20.63%, 33.09%, and 25.71% of the overall disparity in BMI in China-USA, China-UK, and API-Caucasian comparisons, respectively, would be eliminated if the between-group microbiome profiles were equalized; and 15, 18, and 16 species are identified to play the mediating role respectively. CONCLUSIONS The proposed SparseMCMM_HD is an effective and validated tool to elucidate the mediating role of microbiome in health disparity. Three BMI applications shed light on the utility of microbiome in reducing BMI disparity by manipulating microbial profiles. Video Abstract.
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Affiliation(s)
- Chan Wang
- Department of Population Health, Division of Biostatistics, New York University Grossman School of Medicine, New York, NY, 10016, USA
| | - Jiyoung Ahn
- Department of Population Health, Division of Epidemiology, New York University Grossman School of Medicine, New York, NY, 10016, USA
| | - Thaddeus Tarpey
- Department of Population Health, Division of Biostatistics, New York University Grossman School of Medicine, New York, NY, 10016, USA
| | - Stella S Yi
- Department of Population Health Section for Health Equity, New York University Grossman School of Medicine, New York, 10016, USA
| | - Richard B Hayes
- Department of Population Health, Division of Epidemiology, New York University Grossman School of Medicine, New York, NY, 10016, USA
| | - Huilin Li
- Department of Population Health, Division of Biostatistics, New York University Grossman School of Medicine, New York, NY, 10016, USA.
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Hong Q, Chen G, Tang ZZ. PhyloMed: a phylogeny-based test of mediation effect in microbiome. Genome Biol 2023; 24:72. [PMID: 37041566 PMCID: PMC10088256 DOI: 10.1186/s13059-023-02902-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2022] [Accepted: 03/15/2023] [Indexed: 04/13/2023] Open
Abstract
Microbiome data from sequencing experiments contain the relative abundance of a large number of microbial taxa with their evolutionary relationships represented by a phylogenetic tree. The compositional and high-dimensional nature of the microbiome mediator challenges the validity of standard mediation analyses. We propose a phylogeny-based mediation analysis method called PhyloMed to address this challenge. Unlike existing methods that directly identify individual mediating taxa, PhyloMed discovers mediation signals by analyzing subcompositions defined on the phylogenic tree. PhyloMed produces well-calibrated mediation test p-values and yields substantially higher discovery power than existing methods.
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Affiliation(s)
- Qilin Hong
- Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI 53715 USA
| | - Guanhua Chen
- Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI 53715 USA
| | - Zheng-Zheng Tang
- Department of Biostatistics and Medical Informatics, University of Wisconsin-Madison, Madison, WI 53715 USA
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Wang C, Ahn J, Tarpey T, Yi SS, Hayes RB, Li H. A microbial causal mediation analytic tool for health disparity and applications in body mass index. RESEARCH SQUARE 2023:rs.3.rs-2463503. [PMID: 36712075 PMCID: PMC9882678 DOI: 10.21203/rs.3.rs-2463503/v1] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 01/15/2023]
Abstract
Background: Emerging evidence suggests the potential mediating role of microbiome in health disparities. However, no analytic framework is available to analyze microbiome as a mediator between health disparity and clinical outcome, due to the unique structure of microbiome data, including high dimensionality, sparsity, and compositionality. Methods: Considering the modifiable and quantitative features of microbiome, we propose a microbial causal mediation model framework, SparseMCMM_HD, to uncover the mediating role of microbiome in health disparities, by depicting a plausible path from a non-manipulable exposure (e.g. race or region) to a continuous outcome through microbiome. The proposed SparseMCMM_HD rigorously defines and quantifies the manipulable disparity measure that would be eliminated by equalizing microbiome profiles between comparison and reference groups. Moreover, two tests checking the impact of microbiome on health disparity are proposed. Results: Through three body mass index (BMI) studies selected from the curatedMetagenomicData 3.4.2 package and the American gut project: China vs. USA, China vs. UK, and Asian or Pacific Islander (API) vs. Caucasian, we exhibit the utility of the proposed SparseMCMM_HD framework for investigating microbiome’s contributions in health disparities. Specifically, BMI exhibits disparities and microbial community diversities are significantly distinctive between the reference and comparison groups in all three applications. By employing SparseMCMM_HD, we illustrate that microbiome plays a crucial role in explaining the disparities in BMI between races or regions. 11.99%, 12.90%, and 7.4% of the overall disparity in BMI in China-USA, China-UK, and API-Caucasian comparisons, respectively, would be eliminated if the between-group microbiome profiles were equalized; and 15, 21, and 12 species are identified to play the mediating role respectively. Conclusions: The proposed SparseMCMM_HD is an effective and validated tool to elucidate the mediating role of microbiome in health disparity. Three BMI applications shed light on the utility of microbiome in reducing BMI disparity by manipulating microbial profiles.
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Affiliation(s)
- Chan Wang
- Division of Biostatistics, Department of Population Health, New York University Grossman School of Medicine, New York, 10016, NY, USA
| | - Jiyoung Ahn
- Division of Epidemiology, Department of Population Health, New York University Grossman School of Medicine, New York, 10016, NY, USA
| | - Thaddeus Tarpey
- Division of Biostatistics, Department of Population Health, New York University Grossman School of Medicine, New York, 10016, NY, USA
| | - Stella S. Yi
- Department of Population Health Section for Health Equity, New York University Grossman School of Medicine, New York, 10016, USA
| | - Richard B. Hayes
- Division of Epidemiology, Department of Population Health, New York University Grossman School of Medicine, New York, 10016, NY, USA
| | - Huilin Li
- Division of Biostatistics, Department of Population Health, New York University Grossman School of Medicine, New York, 10016, NY, USA,Correspondence:
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Perera C, Zhang H, Zheng Y, Hou L, Qu A, Zheng C, Xie K, Liu L. HIMA2: high-dimensional mediation analysis and its application in epigenome-wide DNA methylation data. BMC Bioinformatics 2022; 23:296. [PMID: 35879655 PMCID: PMC9310002 DOI: 10.1186/s12859-022-04748-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2021] [Accepted: 05/23/2022] [Indexed: 11/28/2022] Open
Abstract
Mediation analysis plays a major role in identifying significant mediators in the pathway between environmental exposures and health outcomes. With advanced data collection technology for large-scale studies, there has been growing research interest in developing methodology for high-dimensional mediation analysis. In this paper we present HIMA2, an extension of the HIMA method (Zhang in Bioinformatics 32:3150-3154, 2016). First, the proposed HIMA2 reduces the dimension of mediators to a manageable level based on the sure independence screening (SIS) method (Fan in J R Stat Soc Ser B 70:849-911, 2008). Second, a de-biased Lasso procedure is implemented for estimating regression parameters. Third, we use a multiple-testing procedure to accurately control the false discovery rate (FDR) when testing high-dimensional mediation hypotheses. We demonstrate its practical performance using Monte Carlo simulation studies and apply our method to identify DNA methylation markers which mediate the pathway from smoking to reduced lung function in the Coronary Artery Risk Development in Young Adults (CARDIA) Study.
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Affiliation(s)
- Chamila Perera
- Division of Biostatistics, Washington University in St. Louis, St. Louis, MO, 63110, USA
| | - Haixiang Zhang
- Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China
| | - Yinan Zheng
- Department of Preventive Medicine, Northwestern University, Chicago, IL, 60611, USA
| | - Lifang Hou
- Department of Preventive Medicine, Northwestern University, Chicago, IL, 60611, USA
| | - Annie Qu
- Department of Statistics, University of California, Irvine, CA, 92697, USA
| | - Cheng Zheng
- Department of Biostatistics, University of Nebraska Medical Center, Omaha, NE, 68198, USA
| | - Ke Xie
- Division of Biostatistics, Washington University in St. Louis, St. Louis, MO, 63110, USA
| | - Lei Liu
- Division of Biostatistics, Washington University in St. Louis, St. Louis, MO, 63110, USA.
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Sohn MB, Lu J, Li H. A compositional mediation model for a binary outcome: Application to microbiome studies. Bioinformatics 2021; 38:16-21. [PMID: 34415327 DOI: 10.1093/bioinformatics/btab605] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2021] [Revised: 07/06/2021] [Accepted: 08/18/2021] [Indexed: 02/03/2023] Open
Abstract
MOTIVATION The delicate balance of the microbiome is implicated in our health and is shaped by external factors, such as diet and xenobiotics. Therefore, understanding the role of the microbiome in linking external factors and our health conditions is crucial to translate microbiome research into therapeutic and preventative applications. RESULTS We introduced a sparse compositional mediation model for binary outcomes to estimate and test the mediation effects of the microbiome utilizing the compositional algebra defined in the simplex space and a linear zero-sum constraint on probit regression coefficients. For this model with the standard causal assumptions, we showed that both the causal direct and indirect effects are identifiable. We further developed a method for sensitivity analysis for the assumption of the no unmeasured confounding effects between the mediator and the outcome. We conducted extensive simulation studies to assess the performance of the proposed method and applied it to real microbiome data to study mediation effects of the microbiome on linking fat intake to overweight/obesity. AVAILABILITY AND IMPLEMENTATION An R package can be downloaded from https://github.com/mbsohn/cmmb. SUPPLEMENTARY INFORMATION Supplementary files are available at Bioinformatics online.
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Affiliation(s)
- Michael B Sohn
- Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY 14642, USA
| | - Jiarui Lu
- Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, PA 19104, USA
| | - Hongzhe Li
- Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, PA 19104, USA
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Zhang H, Zheng Y, Hou L, Zheng C, Liu L. Mediation analysis for survival data with high-dimensional mediators. Bioinformatics 2021; 37:3815-3821. [PMID: 34343267 PMCID: PMC8570823 DOI: 10.1093/bioinformatics/btab564] [Citation(s) in RCA: 15] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2021] [Revised: 07/18/2021] [Accepted: 07/29/2021] [Indexed: 11/12/2022] Open
Abstract
MOTIVATION Mediation analysis has become a prevalent method to identify causal pathway(s) between an independent variable and a dependent variable through intermediate variable(s). However, little work has been done when the intermediate variables (mediators) are high-dimensional and the outcome is a survival endpoint. In this paper, we introduce a novel method to identify potential mediators in a causal framework of high-dimensional Cox regression. RESULTS We first reduce the data dimension through a mediation-based sure independence screening method. A de-biased Lasso inference procedure is used for Cox's regression parameters. We adopt a multiple-testing procedure to accurately control the false discovery rate when testing high-dimensional mediation hypotheses. Simulation studies are conducted to demonstrate the performance of our method. We apply this approach to explore the mediation mechanisms of 379 330 DNA methylation markers between smoking and overall survival among lung cancer patients in The Cancer Genome Atlas lung cancer cohort. Two methylation sites (cg08108679 and cg26478297) are identified as potential mediating epigenetic markers. AVAILABILITY AND IMPLEMENTATION Our proposed method is available with the R package HIMA at https://cran.r-project.org/web/packages/HIMA/. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Haixiang Zhang
- Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
| | - Yinan Zheng
- Department of Preventive Medicine, Northwestern University, Chicago, IL 60611, USA
| | - Lifang Hou
- Department of Preventive Medicine, Northwestern University, Chicago, IL 60611, USA
| | - Cheng Zheng
- Department of Biostatistics, University of Nebraska Medical Center, Omaha, NE 68198, USA
| | - Lei Liu
- Division of Biostatistics, Washington University in St. Louis, St. Louis, MO 63110, USA
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