Cutting‑plane algorithm for estimation of sparse Cox proportional hazards models

• Hiroki Saishu ^[1] ; Kota Kudo ^[1] ; Yuichi Takano ^[2]
  1. [1] University of Tsukuba

     University of Tsukuba

     Japón

     
  2. [2] Institute of Systems and Information Engineering, University of Tsukuba,Japan
• Localización: Top, ISSN-e 1863-8279, ISSN 1134-5764, Vol. 32, Nº. 1, 2024, págs. 57-82
• Idioma: inglés
• DOI: 10.1007/s11750-023-00658-4
• Enlaces
  • Texto completo
• Resumen

  • Survival analysis is a family of statistical methods for analyzing event occurrence times. We adopt a mixed-integer optimization approach to estimation of sparse Cox proportional hazards (PH) models for survival analysis. Specifcally, we propose a high-performance cutting-plane algorithm based on a reformulation of our sparse estimation problem into a bilevel optimization problem. This algorithm solves the upper-level problem using cutting planes that are generated from the dual lowerlevel problem to approximate an upper-level nonlinear objective function. To solve the dual lower-level problem efciently, we devise a quadratic approximation of the Fenchel conjugate of the loss function. We also develop a computationally efcient least-squares method for adjusting quadratic approximations to ft each dataset. Computational results demonstrate that our method outperforms regularized estimation methods in terms of accuracy for both prediction and subset selection especially for low-dimensional datasets. Moreover, our quadratic approximation of the Fenchel conjugate function accelerates the cutting-plane algorithm and maintains high generalization performance of sparse Cox PH models.

• Referencias bibliográficas
  • Aalen O (1978) Nonparametric inference for a family of counting processes. Ann Stat 6(4):701–726
  • Arthanari TS, Dodge Y (1981) Mathematical programming in statistics. Wiley, Hoboken
  • Berk L, Bertsimas D (2019) Certifably optimal sparse principal component analysis. Math Program Comput 11(3):381–420
  • Bertsimas D, King A (2016) An algorithmic approach to linear regression. Oper Res 64(1):2–16
  • Bertsimas D, King A (2017) Logistic regression: from art to science. Stat Sci 32(3):367–384
  • Bertsimas D, Li ML (2020) Scalable holistic linear regression. Oper Res Lett 48(3):203–208
  • Bertsimas D, King A, Mazumder R (2016) Best subset selection via a modern optimization lens. Ann Stat 44(2):813–852 Bertsimas D, Pauphilet...
  • Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge
  • Bradburn MJ, Clark TG, Love SB, Altman DG (2003) Survival analysis part III: multivariate data analysis–choosing a model and assessing its...
  • Breslow N (1974) Covariance analysis of censored survival data. Biometrics 30(1):89–99 Buckley J, James I (1979) Linear regression with censored...
  • Clark TG, Bradburn MJ, Love SB, Altman DG (2003) Survival analysis part IV: further concepts and methods in survival analysis. Br J Cancer...
  • Cox DR (1972) Regression models and life-tables. J R Stat Soc Ser B (Methodol) 34(2):187–202
  • Cox DR (1975) Partial likelihood. Biometrika 62(2):269–276
  • Cozad A, Sahinidis NV, Miller DC (2014) Learning surrogate models for simulation-based optimization. AIChE J 60(6):2211–2227
  • Cutler SJ, Ederer F (1958) Maximum utilization of the life table method in analyzing survival. J Chronic Dis 8(6):699–712
  • Davidson-Pilon C (2019) Lifelines: survival analysis in Python. J Open Source Softw 4(40):1317
  • Demyanyk Y, Hasan I (2010) Financial crises and bank failures: a review of prediction methods. Omega 38(5):315–324
  • Deng L, Ding J, Liu Y, Wei C (2018) Regression analysis for the proportional hazards model with parameter constraints under case-cohort design....
  • Efron B (1977) The efciency of Cox’s likelihood function for censored data. J Am Stat Assoc 72(359):557–565
  • Fan J, Li R (2002) Variable selection for Cox’s proportional hazards model and frailty model. Ann Stat 30(1):74–99
  • Goeman JJ (2010) L1 penalized estimation in the Cox proportional hazards model. Biometr J 52(1):70–84
  • Gui J, Li H (2005) Penalized Cox regression analysis in the high-dimensional and low-sample size settings, with applications to microarray...
  • Harrell FE Jr, Lee KL, Mark DB (1996) Multivariable prognostic models: Issues in developing models, evaluating assumptions and adequacy, and...
  • Hastie T, Tibshirani R, Tibshirani RJ (2020) Best subset, forward stepwise or Lasso? Analysis and recommendations based on extensive comparisons....
  • Kamiya S, Miyashiro R, Takano Y (2019). Feature subset selection for the multinomial logit model via mixed-integer optimization. In: The 22nd...
  • Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53(282):457–481
  • Katzman JL, Shaham U, Cloninger A, Bates J, Jiang T, Kluger Y (2018) DeepSurv: personalized treatment recommender system using a Cox proportional...
  • Klein JP, Moeschberger ML (2003) Survival analysis: techniques for censored and truncated data. Springer, New York
  • Kobayashi K, Takano Y, Nakata K (2021) Bilevel cutting-plane algorithm for cardinality-constrained mean-CVaR portfolio optimization. J Glob...
  • Kobayashi K, Takano Y, Nakata K (2023) Cardinality-constrained distributionally robust portfolio optimization. Eur J Oper Res 309(3):1173–1182...
  • Konno H, Yamamoto R (2009) Choosing the best set of variables in regression analysis using integer programming. J Glob Optim 44(2):273–282
  • Kudo K, Takano Y, Nomura R (2020) Stochastic discrete frst-order algorithm for feature subset selection. IEICE Trans Inf Syst 103(7):1693–1702...
  • Lane WR, Looney SW, Wansley JW (1986) An application of the Cox proportional hazards model to bank failure. J Bank Financ 10(4):511–531
  • Lee S, Lim H (2019) Review of statistical methods for survival analysis using genomic data. Genom Inform 17(4):e41
  • Li R, Chang C, Justesen JM, Tanigawa Y, Qian J, Hastie T, Tibshirani R (2022) Fast Lasso method for large-scale and ultrahigh-dimensional...
  • Maldonado S, Pérez J, Weber R, Labbé M (2014) Feature selection for support vector machines via mixed integer linear programming. Inf Sci...
  • Mazumder R, Radchenko P, Dedieu A (2023) Subset selection with shrinkage: sparse linear modeling when the SNR is low. Oper Res 71(1):129–147...
  • Miyashiro R, Takano Y (2015a) Subset selection by Mallows’ Cp: a mixed integer programming approach. Expert Syst Appl 42(1):325–331
  • Miyashiro R, Takano Y (2015b) Mixed integer second-order cone programming formulations for variable selection in linear regression. Eur J...
  • Miyashiro R (2019) Feature subset selection for ordered logit model via tangent-plane-based approximation. IEICE Trans Inf Syst 102(5):1046–1053...
  • Nelson W (1972) Theory and applications of hazard plotting for censored failure data. Technometrics 14(4):945–966
  • Park MY, Hastie T (2007) L1-regularization path algorithm for generalized linear models. J R Stat Soc Ser B (Stat Methodol) 69(4):659–677
  • Park YW, Klabjan D (2020) Subset selection for multiple linear regression via optimization. J Glob Optim 77(3):543–574
  • Rosset S, Neumann E, Eick U, Vatnik N (2003) Customer lifetime value models for decision support. Data Min Knowl Discov 7(3):321–339
  • Saikia R, Barman MP (2017) A review on accelerated failure time models. Int J Stat Syst 12(2):311–322
  • Saishu H, Kudo K, Takano Y (2021) Sparse Poisson regression via mixed-integer optimization. PLoS One 16(4):e0249916
  • Sato T, Takano Y, Miyashiro R, Yoshise A (2016) Feature subset selection for logistic regression via mixed integer optimization. Comput Optim...
  • Sato T, Takano Y, Miyashiro R (2017) Piecewise-linear approximation for feature subset selection in a sequential logit model. J Oper Res Soc...
  • Simon N, Friedman J, Hastie T, Tibshirani R (2011) Regularization paths for Cox’s proportional hazards model via coordinate descent. J Stat...
  • Takano Y, Miyashiro R (2020) Best subset selection via cross-validation criterion. TOP 28(2):475–488
  • Tamura R, Kobayashi K, Takano Y, Miyashiro R, Nakata K, Matsui T (2017) Best subset selection for eliminating multicollinearity. J Oper Res...
  • Tamura R, Kobayashi K, Takano Y, Miyashiro R, Nakata K, Matsui T (2019) Mixed integer quadratic optimization formulations for eliminating...
  • Tamura R, Takano Y, Miyashiro R (2022) Feature subset selection for kernel SVM classifcation via mixed-integer optimization. arXiv preprint...
  • Tibshirani R (1997) The Lasso method for variable selection in the Cox model. Stat Med 16(4):385–395 Tobin J (1958) Estimation of relationships...
  • Uno H, Cai T, Pencina MJ, D’Agostino RB, Wei LJ (2011) On the C-statistics for evaluating overall adequacy of risk prediction procedures with...
  • Ustun B, Rudin C (2016) Supersparse linear integer models for optimized medical scoring systems. Mach Learn 102(3):349–391
  • Van De Vijver MJ, He YD, Van’t Veer LJ, Dai H, Hart AAM, Voskuil DW, Bernards R (2002) A geneexpression signature as a predictor of survival...
  • Van den Poel D, Larivière B (2004) Customer attrition analysis for fnancial services using proportional hazard models. Eur J Oper Res 157(1):196–217...
  • Van Wieringen WN, Kun D, Hampel R, Boulesteix AL (2009) Survival prediction using gene expression data: a review and comparison. Comput Stat...
  • Verweij PJ, Van Houwelingen HC (1994) Penalized likelihood in Cox regression. Stat Med 13(23–24):2427–2436
  • Vinzamuri B, Reddy CK (2013) Cox regression with correlation based regularization for electronic health records. In: 2013 IEEE 13th international...
  • Wächter A, Biegler LT (2006) On the implementation of an interior-point flter line-search algorithm for large-scale nonlinear programming....
  • Wang P, Li Y, Reddy CK (2019) Machine learning for survival analysis: a survey. ACM Comput Surv (CSUR) 51(6):1–36
  • Watanabe A, Tamura R, Takano Y, Miyashiro R (2023) Branch-and-bound algorithm for optimal sparse canonical correlation analysis. Expert Syst...
  • Wilson CM, Li K, Sun Q, Kuan PF, Wang X (2021) Fenchel duality of Cox partial likelihood with an application in survival kernel learning....
  • Zhang HH, Lu W (2007) Adaptive Lasso for Cox’s proportional hazards model. Biometrika 94(3):691–703