Survival analysis

Overview

Survival analysis predicts when an event will happen for each individual, from covariates. Some records are censored: observation stopped before the event happened, so the model only learns that the event came later than the stop time 1907.00825 §3. The output for one individual is a survival curve S(t | x) = exp(-H(t | x)), where H is the cumulative hazard 1907.00825 eq. 1.

The classic model is the Cox model. It writes the hazard as a shared baseline times a relative-risk term that depends only on covariates, h(t | x) = h0(t) exp[g(x)] 1907.00825 eq. 3. Fitting its exact partial likelihood sums over risk sets, which can hold almost the whole data set, so the fit does not batch well 1907.00825 §3.2. Proportional variants keep the relative-risk form. The time-as-input variant drops it 1907.00825 §3.4.

A 2023 review classifies 61 deep-learning methods for time-to-event data. Its stated counts put 26 of them in the Cox-based class and 19 in the discrete-time class 2305.14961 §3.3.1. Use it as a map of the field, not a ranking: the authors say reported results cannot be compared across papers 2305.14961 §3.6.

Variants

Three choices define most of the design space. Each has its own page. A fourth variant learns the input representation as well, for high-dimensional inputs.

1. Risk term. Keep a linear g(x), or replace it with a neural network. The paper's Cox-MLP (DeepSurv) keeps the proportional form and trains on a batched partial likelihood 1907.00825 §6.1.1. See Cox proportional hazards.
2. Training loss. Use the exact partial likelihood, or pair each event with one control sampled from its risk set. The sampled loss trains on mini-batches 1907.00825 §3.2. See Nested case-control sampling.
3. Time dependence. Keep proportional hazards, or feed time to the network as an input (Cox-Time), so the risk can change over time 1907.00825 §3.4. See Cox proportional hazards.
4. Learned representation. Compress high-dimensional inputs with a denoising autoencoder, and train the code jointly with a Cox partial-likelihood term through a weighted sum with the reconstruction loss 1810.03382 eq. 7. The input in that paper is cardiac motion: 11,514 displacement values per patient 1810.03382 eq. 2. Card: Deep learning cardiac motion analysis for human survival prediction. See also Denoising autoencoder.

The paper also compares against discrete-time networks (DeepHit) and random survival forests. Neither has a card here, so nothing from them is used on this page 1907.00825 §2 1907.00825 §6.1.1.

Evaluation

The papers score models with a mix of measures. All reported numbers are author-reported.

Measure	What it checks	Where it is defined
Time-dependent concordance, C^td	Whether higher predicted risk goes with earlier events, over time	1907.00825 §4.1 eq. 14
Harrell's C-index	Whether higher predicted risk goes with earlier events, for one fixed set of pairs (not time-dependent)	1810.03382 § Predictive performance
Integrated Brier score, IBS	Error of the predicted survival probabilities, averaged over time. Lower is better.	1907.00825 §4.2 eq. 15 eq. 16
Integrated binomial log-likelihood, IBLL	Fit of the predicted survival probabilities, averaged over time. Closer to zero is better.	1907.00825 §4.2 eq. 17

IBS and IBLL are weighted by a Kaplan-Meier estimate of the censoring distribution 1907.00825 tab. 3 1907.00825 tab. 4. Concordance does not weight by censoring, so it is not a check on the survival curve.

The cardiac-motion paper reports its C-index in two forms. The apparent value is 0.85 for its model and 0.61 for a Cox benchmark. After bootstrap optimism correction the values are 0.73 and 0.59 1810.03382 § Predictive performance. The corrected values are the ones to compare, and the correction repeats the full training and tuning on each bootstrap sample 1810.03382 § Internal Validation.

When to use which

The papers' results split by data set and by metric, so choose by constraint and then check on your own held-out data.

Situation	Use	Why
Hazards are plausibly proportional, and you want the smallest change from linear Cox	Proportional neural risk (Cox-MLP, DeepSurv-style)	Best IBLL on FLCHAIN among the compared methods 1907.00825 tab. 4
Large data, mini-batch training, a per-event loss	Sampled-control loss	One control per event, no full risk-set sum 1907.00825 §3.2
Covariate effects may change over time, or you need curves at many times	Time as input (Cox-Time)	Drops proportional hazards 1907.00825 §3.4; best IBS and IBLL on KKBox 1907.00825 tab. 7
Ranking of individuals is the main goal	Compare with DeepHit or RSF	DeepHit has the best KKBox C^td but the worst IBS and IBLL 1907.00825 tab. 7
Small data, proportionality unknown	Fit a proportional and a time-as-input model, then compare IBS and IBLL	Cox-Time is best on three of four small sets, but trails the proportional models on FLCHAIN 1907.00825 tab. 3 1907.00825 tab. 4
Tight prediction latency, many individuals	Proportional model	Its cumulative hazard factorises into baseline times exp[g(x)] 1907.00825 §3.5
Raw, high-dimensional inputs (images, motion tracks) with a modest number of events	Learned representation with a Cox term	Improved over a Cox model on manual volumetric indices in one centre's 302 patients, internal validation only 1810.03382 § Predictive performance; the tuned values are not reported 1810.03382 § Hyperparameter Tuning
Choosing among many deep methods before any experiment	Read the review's classification first	It sorts methods by model class, architecture and outcome types 2305.14961 §3.3 2305.14961 §3.4

The paper reports no wall-clock or memory figures for the first six rows. The latency row rests on the cost argument alone: Cox-Time needs one network call per distinct event time up to the horizon 1907.00825 §3.5. The learned-representation row reports a cost: the full pipeline, including the search and the bootstrap, took 76 hours on one GPU 1810.03382 § Network Design and Training.

Implementation sketch

The pseudocode for each variant is on its sub-page: the Cox-Time training loop and Breslow survival curve are on Cox proportional hazards, and the sampled-control loss is on Nested case-control sampling. Both follow the paper's equations 1907.00825 §3.2 1907.00825 §3.4. The hybrid loss of the learned-representation variant is on its card, which gives the forward pass and loss in pseudocode.

Pitfalls

• Censoring. IBS and IBLL assume censoring is independent of the survival time 1907.00825 §4.2. The paper does not test this assumption.
• Duration definition. On KKBox, time starts at the first subscription or earliest transaction record. A customer who churned and came back counts as a new customer. This differs from the Kaggle task, which uses calendar dates 1907.00825 app. B.1 1907.00825 §6.2.
• Fixed covariates. All KKBox covariates are fixed at subscription time. The authors call this a simplification 1907.00825 app. B.1.
• Data preparation. The paper replaces the WHAS set with FLCHAIN and drops rows with missing values, so its FLCHAIN numbers are not comparable with results that keep those rows 1907.00825 §6.1.
• Run-to-run variance. Proportional neural models were unstable across fits on KKBox. The paper reports medians of five fits for that reason 1907.00825 §6.2.2.
• Sign convention. The paper reports IBLL as negative [arxiv:1907.00825 tab. 7]. The official code returns the negative of it, so its values are positive. See the Paper vs code section of the card before comparing numbers across the two.
• Batching the Cox term is approximate. The full risk set is what the partial likelihood needs, so a batch-wise version only approximates it unless the batch is large 2305.14961 §3.3.1. The cardiac-motion paper's formula uses the whole training sample as the risk set 1810.03382 eq. 5. Its public code computes risk sets within each mini-batch; the card lists the difference.
• Apparent versus corrected C. Optimism inflates the apparent C-index. The gap in the cardiac-motion paper is 0.85 against 0.73 for its own model 1810.03382 § Predictive performance.
• Single centre. The cardiac-motion paper uses one centre's cohort and bootstrap validation only, and its authors list external validation as future work 1810.03382 § Discussion.
• Review counts. The review's stated class counts add to 62, not 61, and its discrete-time paragraph names 20 methods, not the stated 19 2305.14961 §3.3.1. Cite them as "stated in the review", not as verified totals.
• No ranking. The review says benchmark results cannot be compared, because datasets, preprocessing, missing-value handling, resampling and metric variants differ 2305.14961 §3.6. Do not use it to pick a method by performance.

Contradictions

Each side is cited. None of these are resolved here.

• Concordance and calibration disagree on KKBox. DeepHit has the top C^td (0.888) and the worst IBS and IBLL 1907.00825 tab. 7.
• No single concordance winner on the small sets. DeepHit is best on SUPPORT and METABRIC, Cox-Time on Rot. & GBSG, and Cox-MLP (CC) on FLCHAIN 1907.00825 tab. 2.
• Control set. The sampled set is said to include the case 1907.00825 §3.2, but the control loss draws from the risk set without it 1907.00825 eq. 9. Details are on the nested case-control sampling page.
• Risk sets in training. The cardiac-motion paper defines the risk set over the training sample 1810.03382 eq. 5. The loss of 1907.00825 §3.2 samples one control per event instead. The two papers' C-index values are on different cohorts, so they are not a direct comparison 2305.14961 §3.6.

Papers

• Time-to-Event Prediction with Neural Networks and Cox Regression: replaces the Cox risk term with a neural network, trains it on a sampled-control loss, and adds Cox-Time, which takes time as an input.
• Deep learning cardiac motion analysis for human survival prediction: learns a latent code from right-ventricle motion with a denoising autoencoder and a Cox term, and reports internal bootstrap C-index results on one centre's cohort.
• Deep Learning for Survival Analysis: A Review: a systematic review that classifies 61 deep-learning methods by model class, architecture and outcome types. It ranks nothing.

Related

ConceptCox proportional hazardsNeural versions of the Cox survival model: a neural risk term trained on a sampled-control loss, a Cox partial-likelihood term added to an autoencoder loss, or time as an input that drops proportional hazards. Author-reported gains on KKBox churn data and small survival sets; the autoencoder paper has one centre and internal validation only.ConceptNested case-control samplingReplaces the full risk-set sum in a survival likelihood with one control drawn per event, so the loss can be trained on mini-batches. The paper reports that one control stays close to the exact likelihood on simulations, and that more controls lower training error at higher compute cost.ConceptDenoising autoencoderTrain an encoder and decoder to rebuild a clean input from a corrupted copy, so the code keeps the structure needed to rebuild it. Variants differ in the corruption, what the decoder may read, and what the code feeds: a survival loss, a sentence vector, a shape penalty, sparse-input pre-training, or time-series image repair for imputation. Choose by input type and label count.

Source and license

Compiled from the papers listed on this page, all published under CC BY, CC BY-SA or CC0, with citations. See each paper's page for its authors and license.