Annals of Applied Statistics

Robust elastic net estimators for variable selection and identification of proteomic biomarkers

Gabriela V. Cohen Freue, David Kepplinger, Matías Salibián-Barrera, and Ezequiel Smucler

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Abstract

In large-scale quantitative proteomic studies, scientists measure the abundance of thousands of proteins from the human proteome in search of novel biomarkers for a given disease. Penalized regression estimators can be used to identify potential biomarkers among a large set of molecular features measured. Yet, the performance and statistical properties of these estimators depend on the loss and penalty functions used to define them. Motivated by a real plasma proteomic biomarkers study, we propose a new class of penalized robust estimators based on the elastic net penalty, which can be tuned to keep groups of correlated variables together in the selected model and maintain robustness against possible outliers. We also propose an efficient algorithm to compute our robust penalized estimators and derive a data-driven method to select the penalty term. Our robust penalized estimators have very good robustness properties and are also consistent under certain regularity conditions. Numerical results show that our robust estimators compare favorably to other robust penalized estimators. Using our proposed methodology for the analysis of the proteomics data, we identify new potentially relevant biomarkers of cardiac allograft vasculopathy that are not found with nonrobust alternatives. The selected model is validated in a new set of 52 test samples and achieves an area under the receiver operating characteristic (AUC) of 0.85.

Article information

Source
Ann. Appl. Stat., Volume 13, Number 4 (2019), 2065-2090.

Dates
Received: March 2018
Revised: February 2019
First available in Project Euclid: 28 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1574910036

Digital Object Identifier
doi:10.1214/19-AOAS1269

Mathematical Reviews number (MathSciNet)
MR4037422

Zentralblatt MATH identifier
07160931

Keywords
Robust estimation regularized estimation penalized estimation elastic net penalty proteomics biomarkers

Citation

Cohen Freue, Gabriela V.; Kepplinger, David; Salibián-Barrera, Matías; Smucler, Ezequiel. Robust elastic net estimators for variable selection and identification of proteomic biomarkers. Ann. Appl. Stat. 13 (2019), no. 4, 2065--2090. doi:10.1214/19-AOAS1269. https://projecteuclid.org/euclid.aoas/1574910036


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References

  • Alfons, A. (2016). robustHD: Robust Methods for High-Dimensional Data. R package version 0.5.1.
  • Alfons, A., Croux, C. and Gelper, S. (2013). Sparse least trimmed squares regression for analyzing high-dimensional large data sets. Ann. Appl. Stat. 7 226–248.
    Zentralblatt MATH: 06171270
    Digital Object Identifier: doi:10.1214/12-AOAS575
    Project Euclid: euclid.aoas/1365527197
  • Clarke, F. H. (1990). Optimization and Nonsmooth Analysis, 2nd ed. Classics in Applied Mathematics 5. SIAM, Philadelphia, PA.
    Zentralblatt MATH: 0696.49002
  • Cohen Freue, G. V. and Borchers, C. H. (2012). Multiple Reaction Monitoring (MRM). Circ. Cardiovasc. Genet. 5 378.
  • Cohen Freue, G. V, Kepplinger, D., Salibián-Barrera, M. and Smucler, E. (2019). Supplement to “Robust elastic net estimators for variable selection and identification of proteomic biomarkers.” DOI:10.1214/19-AOAS1269SUPP.
  • Domanski, D., Percy, A. J., Yang, J., Chambers, A. G., Hill, J. S., Freue, G. V. C. and Borchers, C. H. (2012). MRM-based multiplexed quantitation of 67 putative cardiovascular disease biomarkers in human plasma. Proteomics 12 1222–1243.
  • Donoho, D. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann. Wadsworth Statist./Probab. Ser. 157–184. Wadsworth, Belmont, CA.
    Zentralblatt MATH: 0523.62032
  • Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R. (2004). Least angle regression. Ann. Statist. 32 407–499.
    Zentralblatt MATH: 1091.62054
    Digital Object Identifier: doi:10.1214/009053604000000067
    Project Euclid: euclid.aos/1083178935
  • Fan, J., Li, Q. and Wang, Y. (2017). Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 247–265.
    Zentralblatt MATH: 1414.62178
    Digital Object Identifier: doi:10.1111/rssb.12166
  • Fan, J. and Peng, H. (2004). Nonconcave penalized likelihood with a diverging number of parameters. Ann. Statist. 32 928–961.
    Zentralblatt MATH: 1092.62031
    Digital Object Identifier: doi:10.1214/009053604000000256
    Project Euclid: euclid.aos/1085408491
  • Friedman, J., Hastie, T. and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33 1–22.
  • Khan, J. A., Van Aelst, S. and Zamar, R. H. (2007). Robust linear model selection based on least angle regression. J. Amer. Statist. Assoc. 102 1289–1299.
    Zentralblatt MATH: 1332.62240
    Digital Object Identifier: doi:10.1198/016214507000000950
  • Koller, M. and Stahel, W. A. (2017). Nonsingular subsampling for regression S estimators with categorical predictors. Comput. Statist. 32 631–646.
    Zentralblatt MATH: 1417.65043
    Digital Object Identifier: doi:10.1007/s00180-016-0679-x
  • Lin, D., Freue, G. C., Hollander, Z., Mancini, G. B. J., Sasaki, M., Mui, A., Wilson-McManus, J., Ignaszewski, A., Imai, C. et al. (2013). Plasma protein biosignatures for detection of cardiac allograft vasculopathy. J. Heart Lung Transplant. 32 723–733.
  • Maronna, R. A. (2011). Robust ridge regression for high-dimensional data. Technometrics 53 44–53.
    Mathematical Reviews (MathSciNet): MR2791951
    Digital Object Identifier: doi:10.1198/TECH.2010.09114
  • Maronna, R. A., Martin, R. D. and Yohai, V. J. (2006). Robust Statistics. Wiley Series in Probability and Statistics: Theory and Methods. Wiley, Chichester.
    Zentralblatt MATH: 1094.62040
  • Maronna, R. A. and Yohai, V. J. (2010). Correcting MM estimates for “fat” data sets. Comput. Statist. Data Anal. 54 3168–3173.
    Zentralblatt MATH: 1284.62425
    Digital Object Identifier: doi:10.1016/j.csda.2009.09.015
  • Osborne, M. R., Presnell, B. and Turlach, B. A. (2000). On the LASSO and its dual. J. Comput. Graph. Statist. 9 319–337.
  • Peña, D. and Yohai, V. (1999). A fast procedure for outlier diagnostics in large regression problems. J. Amer. Statist. Assoc. 94 434–445.
    Zentralblatt MATH: 1072.62618
  • Rousseeuw, P. J. (1984). Least median of squares regression. J. Amer. Statist. Assoc. 79 871–880.
    Zentralblatt MATH: 0547.62046
    Digital Object Identifier: doi:10.1080/01621459.1984.10477105
  • Rousseeuw, P. and Yohai, V. (1984). Robust regression by means of S-estimators. In Robust and Nonlinear Time Series Analysis (Heidelberg, 1983). Lect. Notes Stat. 26 256–272. Springer, New York.
    Zentralblatt MATH: 0567.62027
  • Salibian-Barrera, M. and Yohai, V. J. (2006). A fast algorithm for S-regression estimates. J. Comput. Graph. Statist. 15 414–427.
  • Schmauss, D. and Weis, M. (2008). Cardiac allograft vasculopathy. Circ. 117 2131–2141.
  • Smucler, E. and Yohai, V. J. (2017). Robust and sparse estimators for linear regression models. Comput. Statist. Data Anal. 111 116–130.
    Zentralblatt MATH: 06914538
    Digital Object Identifier: doi:10.1016/j.csda.2017.02.002
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
    Zentralblatt MATH: 0850.62538
    Digital Object Identifier: doi:10.1111/j.2517-6161.1996.tb02080.x
  • Tibshirani, R. J. (2013). The lasso problem and uniqueness. Electron. J. Stat. 7 1456–1490.
    Zentralblatt MATH: 1337.62173
    Digital Object Identifier: doi:10.1214/13-EJS815
  • Tomioka, R., Suzuki, T. and Sugiyama, M. (2011). Super-linear convergence of dual augmented Lagrangian algorithm for sparsity regularized estimation. J. Mach. Learn. Res. 12 1537–1586.
    Zentralblatt MATH: 1280.68206
  • Yohai, V. J. (1987). High breakdown-point and high efficiency robust estimates for regression. Ann. Statist. 15 642–656.
    Zentralblatt MATH: 0624.62037
    Digital Object Identifier: doi:10.1214/aos/1176350366
    Project Euclid: euclid.aos/1176350366
  • Yohai, V. J. and Zamar, R. H. (1988). High breakdown-point estimates of regression by means of the minimization of an efficient scale. J. Amer. Statist. Assoc. 83 406–413.
    Zentralblatt MATH: 0648.62036
    Digital Object Identifier: doi:10.1080/01621459.1988.10478611
  • Zou, H. and Hastie, T. (2005). Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B. Stat. Methodol. 67 301–320.
    Zentralblatt MATH: 1069.62054
    Digital Object Identifier: doi:10.1111/j.1467-9868.2005.00503.x
  • Zou, H. and Zhang, H. H. (2009). On the adaptive elastic-net with a diverging number of parameters. Ann. Statist. 37 1733–1751.
    Zentralblatt MATH: 1168.62064
    Digital Object Identifier: doi:10.1214/08-AOS625
    Project Euclid: euclid.aos/1245332831

Supplemental materials

  • Supplementary material for “Robust elastic net estimators for variable selection and identification of proteomic biomarkers”. We provide additional details on PENSE algorithm, properties and mathematical proofs.
    Digital Object Identifier: doi:10.1214/19-AOAS1269SUPP
    Supplemental PDF (147 KB)

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