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A review of survival stacking: a method to cast survival regression analysis as a classification problem

  • Erin Craig ORCID logo EMAIL logo , Chenyang Zhong und Robert Tibshirani
Veröffentlicht/Copyright: 28. März 2025
The International Journal of Biostatistics
Aus der Zeitschrift The International Journal of Biostatistics Band 21 Heft 1

Abstract

While there are many well-developed data science methods for classification and regression, there are relatively few methods for working with right-censored data. Here, we review survival stacking, a method for casting a survival regression analysis problem as a classification problem, thereby allowing the use of general classification methods and software in a survival setting. Inspired by the Cox partial likelihood, survival stacking collects features and outcomes of survival data in a large data frame with a binary outcome. We show that survival stacking with logistic regression is approximately equivalent to the Cox proportional hazards model. We further illustrate survival stacking on real and simulated data. By reframing survival regression problems as classification problems, survival stacking removes the reliance on specialized tools for survival regression, and makes it straightforward for data scientists to use well-known learning algorithms and software for classification in the survival setting. This in turn lowers the barrier for flexible survival modeling.

Keywords: survival regression analysis; Cox regression; censored data
Corresponding author: Erin Craig, Department of Biomedical Data Science, Stanford University, Stanford, USA, E-mail: 

Funding source: National Science Foundation

Award Identifier / Grant number: 19 DMS1208164

Funding source: National Institutes of Health

Award Identifier / Grant number: 5R01 EB001988-16

Acknowledgments

The authors thank Terry Therneau for the argument in Section 2.3, and Terry Therneau, Thomas Gerds, Lu Tian, Trevor Hastie and Stephen Pfohl for helpful discussions. We also thank the peer reviewers for their valuable suggestions.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: National Science Foundation, award number 19 DMS1208164; National Institutes of Health, award number 5R01 EB001988-16.

  7. Data availability: Not applicable.

Appendix A: The Cox partial and profile likelihoods

Recall, the Cox partial likelihood is:

L partial ( β ) = ∏ i : d i = 1 P i  has the event at  t i ∣ risk set  R ( t i ) , one subject has the event at  t i = ∏ i : d i = 1 exp x i T β ∑ j ∈ R ( t i ) exp x j T β

(10) ℓ partial ( β ) = log ( L partial ( β ) ) = ∑ i : d i = 1 x i T β − log ∑ j ∈ R ( t i ) exp x j T β .

For simplicity, we have assumed (and will continue to assume) that there are no tied times: no two subjects have the event at the exact same time. Once fitted, the coefficients β are often used to describe the relative risk between subjects for different values of x. Optimizing the partial likelihood, however, does not allow us to say anything about the absolute risk for any individual subject: the baseline hazard λ 0(t) does not appear anywhere in the partial likelihood.

To jointly model the baseline hazard, we can look at the full log-likelihood for the Cox model. We will assume that the baseline hazard is discrete: the function λ 0(t) takes values λ t 1 , λ t 2 , … , λ t k at observed event times t 1, …, t k , and λ 0(t) = 0 at all other times. The full log-likelihood for the Cox model is then:

(11) ℓ full { λ t i } i = 1 k , β = ∑ i : d i = 1 log ( λ t i ) + x i T β − λ t i ∑ j ∈ R ( t i ) exp x j T β ,

where t i is the final observation time for subject i.

We can use the full likelihood to estimate the baseline hazard as a function of β. We optimize the full likelihood (Equation (11)) for λ t i to obtain:

(12) λ t i ( β ) = 1 ∑ j ∈ R ( t i ) exp x j T β .

Equation (12) is known as Breslow’s estimate of the baseline hazard [38], and it is the most common method of estimating the baseline hazard. We first estimate β ̂ by maximizing the partial likelihood, and then estimate the baseline hazard as { λ t i ( β ̂ ) } i = 1 k .

Lastly, if we plug Equation (12) back in to the full likelihood, we obtain the profile likelihood:

(13) ℓ profile ( β ) = ∑ i : d i = 1 x i T β − log ∑ j ∈ R ( t i ) exp x j T β − 1 ,

which coincides with the partial likelihood (Equation (2)), up to a constant.

Appendix B: Software: learning methods for survival data

Table 2:

Common R software packages for fitting survival models, and their support for attributes of survival data.

Package Function Time dep. covs. Truncation Sample weights Time-varying effects Non-linear Missing data
Linear models
glmnet [39] glmnet ✓ ✓ ✓
survival [40] coxph ✓ ✓ ✓
Parametric models
flexsurv [41] flexsurvreg ✓ ✓ ✓ ✓
survival survreg ✓ ✓ ✓
Trees & random forests
aorsf [42] orsf ✓ ✓
grf [43] survival_forest ✓ ✓ ✓
LTRCforests [44] ltrccif ✓ ✓ ✓ ✓ ✓
partykit [45] ctree ✓ ✓ ✓
randomForestSRC [34] rfsrc ✓ ✓ ✓
ranger [46] ranger ✓ ✓ ✓
rpart [47] rpart ✓ ✓ ✓
Boosting
CoxBoost [48] CoxBoost ✓
gbm [49] gbm ✓ ✓ ✓
mboost [50] glmboost ✓
gamboost ✓ ✓
xgboost [51] xgboost ✓ ✓ ✓
SVMs
survivalsvm [52] survivalsvm ✓
Neural nets
survivalmodels [53] coxtime ✓ ✓
deephit ✓ ✓
deepsurv ✓ ✓
dnnsurv ✓ ✓
loghaz (Nnet-survival) ✓ ✓
pchazard ✓ ✓
Ecosystems
mlr3proba [54] ✓ ✓
Table 3:

Common Python software packages, and their support for attributes of survival data.

Package Function Time dep. covs. Truncation Sample weights Time-varying effects Non-linear Missing data
Linear models
lifelines [55] CoxPHFitter
CoxTimeVaryingFitter ✓ ✓
PySurvival [56] CoxPHModel
LinearMultiTaskModel ✓
scikit-survival [57] CoxPHSurvivalAnalysis
CoxnetSurvivalAnalysis
Trees & random forests
PySurvival RandomSurvivalForestModel ✓ ✓ ✓
ExtraSurvivalTreesModel ✓ ✓ ✓
ConditionalSurvivalForestModel ✓ ✓ ✓
scikit-survival RandomSurvivalForest ✓ ✓
Boosting
scikit-survival GradientBoostingSurvivalAnalysis ✓ ✓
Neural nets
pycox [58] CoxPH (DeepSurv) ✓ ✓
LogisticHazard (Nnet-survival) ✓ ✓
DeepHit ✓ ✓
N-MTLR ✓ ✓
PySurvival NeuralMultiTaskModel ✓ ✓
Table 4:

Common R software packages to reshape survival data, and their support for attributes of survival data. Though there exist packages to fit survival models in Python, we were unable to identify packages with support for reshaping data.

Language Package Function Time dep. covs. Truncation
R discSurv [59] dataLong
dataLongTimeDep ✓
survival [40] survSplit ✓ ✓
pammtools [60] as_ped ✓ ✓

Survival stacking allows the modeling of survival data – with time-varying covariates and truncation – using linear and non-linear models, and it naturally enables the modeling of time-varying effects. Moreover, this is now possible using familiar, well-developed software for classification and regression. This is important, as there are very few survival software packages that are equally flexible.

Here, we examine the support for various features of survival data in common survival software packages in R and Python, with a focus on methods discussed in this work (Tables 2–4). We note that, though individual methods may support a particular feature of survival data (e.g. Nnet-survival supports time-varying covariates), it is not always the case that the corresponding software follows suit.

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Received: 2022-05-06
Accepted: 2025-01-07
Published Online: 2025-03-28

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Prognostic adjustment with efficient estimators to unbiasedly leverage historical data in randomized trials
  4. Homogeneity test and sample size of response rates for AC 1 in a stratified evaluation design
  5. A review of survival stacking: a method to cast survival regression analysis as a classification problem
  6. DsubCox: a fast subsampling algorithm for Cox model with distributed and massive survival data
  7. A hybrid hazard-based model using two-piece distributions
  8. Regression analysis of clustered current status data with informative cluster size under a transformed survival model
  9. Bayesian covariance regression in functional data analysis with applications to functional brain imaging
  10. Risk estimation and boundary detection in Bayesian disease mapping
  11. An improved estimator of the logarithmic odds ratio for small sample sizes using a Bayesian approach
  12. Short Communication
  13. A multivariate Bayesian learning approach for improved detection of doping in athletes using urinary steroid profiles
  14. Research Articles
  15. Guidance on individualized treatment rule estimation in high dimensions
  16. Weighted Euclidean balancing for a matrix exposure in estimating causal effect
  17. Penalized regression splines in Mixture Density Networks
https://doi.org/10.1515/ijb-2022-0055
Schlagwörter für diesen Artikel
survival regression analysis; Cox regression; censored data

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Prognostic adjustment with efficient estimators to unbiasedly leverage historical data in randomized trials
  4. Homogeneity test and sample size of response rates for AC 1 in a stratified evaluation design
  5. A review of survival stacking: a method to cast survival regression analysis as a classification problem
  6. DsubCox: a fast subsampling algorithm for Cox model with distributed and massive survival data
  7. A hybrid hazard-based model using two-piece distributions
  8. Regression analysis of clustered current status data with informative cluster size under a transformed survival model
  9. Bayesian covariance regression in functional data analysis with applications to functional brain imaging
  10. Risk estimation and boundary detection in Bayesian disease mapping
  11. An improved estimator of the logarithmic odds ratio for small sample sizes using a Bayesian approach
  12. Short Communication
  13. A multivariate Bayesian learning approach for improved detection of doping in athletes using urinary steroid profiles
  14. Research Articles
  15. Guidance on individualized treatment rule estimation in high dimensions
  16. Weighted Euclidean balancing for a matrix exposure in estimating causal effect
  17. Penalized regression splines in Mixture Density Networks
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