JDS Journal of Data Science 1683-86021680-743X1680-743X School of Statistics, Renmin University of China JDS1173 10.6339/25-JDS1173 Statistical Data Science Neural Network for Correlated Survival Outcomes Using Frailty Model ZhouRuiwen1 HeKevin2 WangDi2 LiuLili1 MaShujie3 QuAnnie4 MillerJ. Philip1 LiuLeilei.liu@wustl.edu1∗ Division of Biostatistics, Washington University in St. Louis, St. Louis, Missouri, USA Department of Biostatistics, University of Michigan, Ann Arbor, Michigan, USA Department of Statistics, University of California, Riverside, California, USA Department of Statistics, University of California, Irvine, California, USA Corresponding author. Email: lei.liu@wustl.edu. 20252632025234624637592024132025 2025 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.2025 Open access article under the CC BY license.

Extensive literature has been proposed for the analysis of correlated survival data. Subjects within a cluster share some common characteristics, e.g., genetic and environmental factors, so their time-to-event outcomes are correlated. The frailty model under proportional hazards assumption has been widely applied for the analysis of clustered survival outcomes. However, the prediction performance of this method can be less satisfactory when the risk factors have complicated effects, e.g., nonlinear and interactive. To deal with these issues, we propose a neural network frailty Cox model that replaces the linear risk function with the output of a feed-forward neural network. The estimation is based on quasi-likelihood using Laplace approximation. A simulation study suggests that the proposed method has the best performance compared with existing methods. The method is applied to the clustered time-to-failure prediction within the kidney transplantation facility using the national kidney transplant registry data from the U.S. Organ Procurement and Transplantation Network. All computer programs are available at https://github.com/rivenzhou/deep_learning_clustered.

 correlated survival outcomes deep learning prediction random effect This research is partly supported by NIH grants R21 EY031884, R21 EY033518, UL1 TR002345, R01 DK129539. References Aalen OO (1989). A linear regression model for the analysis of life times. Statistics in Medicine, 8(8): 907–925. https://doi.org/10.1002/sim.4780080803 Balan TA, Putter H (2020). A tutorial on frailty models. Statistical Methods in Medical Research, 29(11): 3424–3454. https://doi.org/10.1177/0962280220921889 Ching T, Zhu X, Garmire LX (2018). Cox-nnet: An artificial neural network method for prognosis prediction of high-throughput omics data. PLoS Computational Biology, 14(4): e1006076. https://doi.org/10.1371/journal.pcbi.1006076 Cook RJ, Lawless JF, et al. (2007). The Statistical Analysis of Recurrent Events. Springer. Fan J, Ma C, Zhong Y (2021). A selective overview of deep learning. Statistical Science: A Review Journal of the Institute of Mathematical Statistics, 36(2): 264. Faraggi D, Simon R (1995). A neural network model for survival data. Statistics in Medicine, 14(1): 73–82. https://doi.org/10.1002/sim.4780140108 Fine JP, Ying Z, Wei L (1998). On the linear transformation model for censored data. Biometrika, 85(4): 980–986. https://doi.org/10.1093/biomet/85.4.980 Gerds TA, Schumacher M (2006). Consistent estimation of the expected Brier score in general survival models with right-censored event times. Biometrical Journal, 48(6): 1029–1040. https://doi.org/10.1002/bimj.200610301 Glidden DV, Vittinghoff E (2004). Modelling clustered survival data from multicentre clinical trials. Statistics in Medicine, 23(3): 369–388. https://doi.org/10.1002/sim.1599 Hao J, Kim Y, Mallavarapu T, Oh JH, Kang M (2018). Cox-pasnet: Pathway-based sparse deep neural network for survival analysis. In: 2018 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) (H. Zheng, X. Hu, Z. Callejas, H. Schmidt, D. Griol, J. Baumbach, J. Dickerson, and L. Zhang, editors), 381–386. IEEE. Harrell Jr FE, Lee KL, Califf RM, Pryor DB, Rosati RA (1984). Regression modelling strategies for improved prognostic prediction. Statistics in Medicine, 3(2): 143–152. https://doi.org/10.1002/sim.4780030207 He K, Kalbfleisch JD, Li Y, Li Y (2013). Evaluating hospital readmission rates in dialysis facilities; adjusting for hospital effects. Lifetime Data Analysis, 19: 490–512. https://doi.org/10.1007/s10985-013-9264-6 Hens N, Wienke A, Aerts M, Molenberghs G (2009). The correlated and shared gamma frailty model for bivariate current status data: An illustration for cross-sectional serological data. Statistics in Medicine, 28(22): 2785–2800. https://doi.org/10.1002/sim.3660 Katzman JL, Shaham U, Cloninger A, Bates J, Jiang T, Kluger Y (2018). Deepsurv: Personalized treatment recommender system using a Cox proportional hazards deep neural network. BMC Medical Research Methodology, 18(1): 1–12. https://doi.org/10.1186/s12874-017-0458-6 Kvamme H, Borgan Ø, Scheel I (2019). Time-to-event prediction with neural networks and Cox regression. Journal of Machine Learning Research, 20(129): 1–30. Lee H, Ha I, Lee Y (2023). Deep neural networks for semiparametric frailty models via h-likelihood. arXiv preprint: https://arxiv.org/2307.06581/. Liao L, Ahn Hi (2016). Combining deep learning and survival analysis for asset health management. International Journal of Prognostics and Health Management, 7(4), 1–10. Lin J, Luo S (2022). Deep learning for the dynamic prediction of multivariate longitudinal and survival data. Statistics in Medicine, 41(15): 2894–2907. https://doi.org/10.1002/sim.9392 Lin X, Taylor JM, Ye W (2008). A penalized likelihood approach to joint modeling of longitudinal measurements and time-to-event data. Statistics and its Interface, 1(1): 33–45. https://doi.org/10.4310/SII.2008.v1.n1.a4 Liu L, He K, Wang D, Ma S, Qu A, Lin L, et al. (2023). Healthcare center clustering for Cox’s proportional hazards model by fusion penalty. Statistics in Medicine, 42(20), 3685–3698. Maas AL, Hannun AY, Ng AY, et al. (2013). Rectifier nonlinearities improve neural network acoustic models. In: Proc. Icml, Volume 30, Atlanta, GA (S. Dasgupta, D. McAllester, editors), 3. Mandel F, Ghosh RP, Barnett I (2023). Neural networks for clustered and longitudinal data using mixed effects models. Biometrics, 79(2): 711–721. https://doi.org/10.1111/biom.13615 Martinsson E (2017). Wtte-rnn: Weibull time to event recurrent neural network a model for sequential prediction of time-to-event in the case of discrete or continuous censored data, recurrent events or time-varying covariates. Doctoral dissertation, Chalmers University of Technology and University of Gothenburg. Normand SLT, Shahian DM (2007). Statistical and clinical aspects of hospital outcomes profiling. Statistical Science, 22(2), 206–226. Paik MC, Tsai WY, Ottman R (1994). Multivariate survival analysis using piecewise gamma frailty. Biometrics, 50(4), 975–988. https://doi.org/10.2307/2533437 Ranganath R, Perotte A, Elhadad N, Blei D (2016). Deep survival analysis. In: Machine Learning for Healthcare Conference (F. Doshi-Velez, J. Fackler, D. Kale, B. Wallace, and J. Wiens, editors), 101–114. PMLR. Ripatti S, Palmgren J (2000). Estimation of multivariate frailty models using penalized partial likelihood. Biometrics, 56(4): 1016–1022. https://doi.org/10.1111/j.0006-341X.2000.01016.x Rizopoulos D, Molenberghs G, Lesaffre EM (2017). Dynamic predictions with time-dependent covariates in survival analysis using joint modeling and landmarking. Biometrical Journal, 59(6): 1261–1276. https://doi.org/10.1002/bimj.201600238 Shih JH, Louis TA (1995). Assessing gamma frailty models for clustered failure time data. Lifetime Data Analysis, 1: 205–220. https://doi.org/10.1007/BF00985771 Sun T, Wei Y, Chen W, Ding Y (2020). Genome-wide association study-based deep learning for survival prediction. Statistics in Medicine, 39(30): 4605–4620. https://doi.org/10.1002/sim.8743 Tanner KT, Sharples LD, Daniel RM, Keogh RH (2021). Dynamic survival prediction combining landmarking with a machine learning ensemble: Methodology and empirical comparison. Journal of the Royal Statistical Society. Series A. Statistics in Society, 184(1): 3–30. https://doi.org/10.1111/rssa.12611 Wiegrebe S, Kopper P, Sonabend R, Bender A (2023). Deep learning for survival analysis: A review. arXiv preprint: https://arxiv.org/2305.14961. Wu R, Qiao J, Wu M, Yu W, Zheng M, Liu T, et al. (2024). Neural frailty machine: Beyond proportional hazard assumption in neural survival regressions. In: Advances in Neural Information Processing Systems (A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, editors), 36. Yu Z, Liu L (2011). A joint model of recurrent events and a terminal event with a nonparametric covariate function. Statistics in Medicine, 30(22): 2683–2695. https://doi.org/10.1002/sim.4297 Yu Z, Liu L, Bravata DM, Williams LS (2014). Joint model of recurrent events and a terminal event with time-varying coefficients. Biometrical Journal, 56(2): 183–197. https://doi.org/10.1002/bimj.201200160 Yu Z, Liu L, Bravata DM, Williams LS, Tepper RS (2013). A semiparametric recurrent events model with time-varying coefficients. Statistics in Medicine, 32(6): 1016–1026. https://doi.org/10.1002/sim.5575 Zhong Q, Mueller JW, Wang JL (2021). Deep extended hazard models for survival analysis. Advances in Neural Information Processing Systems, 34: 15111–15124. Zhong Q, Wang JL (2023). Neural networks for partially linear quantile regression. Journal of Business & Economic Statistics, 42(2), 603–614.