Simultaneous variable selection and estimation for a partially linear Cox model

Abstract

We consider simultaneous variable selection and estimation for a deep neural network-based partially linear Cox model and propose a novel penalized approach. In particular, a two-step iterative algorithm is developed with the use of the minimum information criterion to ensure sparse estimation. The proposed method circumvents the curse of dimensionality while facilitating the interpretability of linear covariate effects on survival, and the algorithm greatly reduces the computational burden by avoiding the need to select the optimal tuning parameters that is usually required by many other popular penalties. The convergence rate and asymptotic properties of the resulting estimator are established along with the consistency of variable selection. The performance of the procedure is demonstrated through extensive simulation studies and an application to a myeloma dataset.

Keywords

Variable selection partially linear Cox model minimum information criterion deep neural network right-censored

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References

Huang

. Efficient estimation of the partly linear additive Cox model. Ann Stat 1999; 27: 1536–1563.

Rong

Zhao

Zheng

, et al. Kernel Cox partially linear regression: Building predictive models for cancer patients’ survival. Stat Med 2024; 43: 1–15.

Zhong

Mueller

Wang

. Deep learning for the partially linear Cox model. Ann Stat 2022; 50: 1348–1375.

Schmidt-Hieber

. Nonparametric regression using deep neural networks with ReLU activation function. Ann Stat 2020; 48: 1875–1897.

Bauer

Kohler

. On deep learning as a remedy for the curse of dimensionality in nonparametric regression. Ann Stat 2019; 47: 2261–2285.

Breiman

. Heuristics of instability and stabilization in model selection. Ann Stat 1996; 24: 2350–2383.

Tibshirani

. Regression shrinkage and selection via the LASSO. J R Stat Soc Ser B 1996; 58: 267–288.

Fan

. Variable selection via nonconcave penalized likelihood and its oracle properties. J Am Stat Assoc 2001; 96: 1348–1360.

Fan

. A unified approach to model selection and sparse recovery using regularized least squares. Ann Stat 2009; 37: 3498–3528.

10.

Dicker

Huang

Lin

. Variable selection and estimation with the seamless-

L_{0}

penalty. Stat Sin 2013; 23: 929–962.

11.

Liu

. Efficient regularized regression with

L_{0}

penalty for variable selection and network construction. Comput Math Methods Med 2016; 2016: 1–11.

12.

Zhao

, et al. Simultaneous estimation and variable selection for interval-censored data with broken adaptive ridge regression. J Am Stat Assoc 2019; 115: 204–216.

13.

Sun

Kang

Haridas

, et al. Penalized deep partially linear Cox models with application to CT scans of lung cancer patients. Biometrics 2024; 80: 1–12.

14.

Wijayasinghe

Fan

, et al. Sparse estimation of Cox proportional hazards models via approximated information criteria. Biometrics 2016; 72: 751–759.

15.

Nair

Hinton

. Rectified linear units improve restricted boltzmann machines. In: Proceedings of the 27th international conference on machine learning (ICML-10). pp.807–814.

16.

Hastle

Tibshirani

Wainwright

. Statistical Learning with Sparsity: The LASSO and Generalizations. New York: CRC Press, 2015.

17.

Kingma

. Adam: A method for stochastic optimization, 2014. arXiv:1412.6980.

18.

Nocedal

Wright

. Numerical Optimization. New York: Springer, 2016.

19.

Kraft

. A Software Package for Sequential Quadratic Programming. Köln: Wiss. Berichtswesen d. DFVLR, 1988.

20.

Akaike

. A new look at the statistical model identification. IEEE Trans Automat Contr 1974; 19: 716–723.

21.

Harrell

Califf

Pryor

, et al. Evaluating the yield of medical tests. J Am Med Assoc 1982; 247: 2543–2546.

22.

Antolini

Boracchi

Biganzoli

. A time-dependent discrimination index for survival data. Stat Med 2005; 24: 3927–3944.

23.

Broyl

Hose

Lokhorst

, et al. Gene expression profiling for molecular classification of multiple myeloma in newly diagnosed patients. Blood 2010; 116: 2543–2553.

24.

Graf

Schmoor

Sauerbrei

, et al. Assessment and comparison of prognostic classification schemes for survival data. Stat Med 1999; 18: 2529–2545.

25.

Sun

. The Statistical Analysis of Interval-censored Failure Time Data. New York: Springer, 2006.

26.

Sun

Ding

. Neural network on interval-censored data with application to the prediction of Alzheimer’s disease. Biometrics 2023; 79: 2677–2690.

27.

Ahmed

Iqbal

Aldabbas

, et al. Images data practices for semantic segmentation of breast cancer using deep neural network. J Ambient Intell Humaniz Comput 2023; 14: 15227–15243.

28.

Liu

Wang

. Simultaneous variable selection and estimation for survival data via the Gaussian seamless-

L_{0}

penalty. Stat Med 2024; 43: 1509–1526.

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