Deep Learning for Survival Analysis: A Review

In plain words

Survival analysis is the statistics of "how long until something happens" (a death, a machine failure, a relapse). Its data are awkward: for some subjects the event has not happened by the time the study ends, or they leave the study early, so the exact time is unknown. Classic tools such as the Cox model have handled this for decades. In recent years, researchers have adapted deep neural networks to the same problem, and the number of such methods has grown quickly.

This paper collects those deep methods and sorts them by how they work: which classic statistical model each one builds on, what kind of network it uses, and which harder situations it can handle, such as several competing causes of an event, repeated events, or measurements that change over time. Many of the methods only handle the simplest case, one event type with a known cut-off time. The authors also say that the results different papers report cannot be compared directly, so the paper is a map of the field rather than a ranking. The sorted list is kept as an editable online table, and the authors invite others to add to it.

TL;DR

• Systematic review, not a new method. The authors searched Web of Science to 31 December 2022, screened the hits against four criteria, added papers they already knew, and ended with 61 methods §3.1.
• Most methods are classified by model class: 26 are Cox-based and 19 are discrete-time (the text's count; the discrete-time paragraph names 20). The rest are parametric, PEM-based, ODE-based, ranking-based or other §3.3.1 fig. 4.
• Single-risk, right-censored data is the common case. Competing risks appear in nine methods, interval-censoring is explicit in one, and left-censoring and right-truncation are not addressed by any method §3.4.1.
• The authors did no benchmarking and do not rank methods: reported results use different datasets, splits and metric variants §3.6.
• The companion repository holds the table's data and R scripts. It has no LICENSE file, so reuse terms for the table are unclear index.qmd.

Problem

Deep learning has been applied to survival tasks many times, but the authors found no general systematic review of these methods. Earlier reviews covered only genomics, only cancer survival prediction, only glioma, or only a few neural-network methods §1. Practitioners need to know which method fits their survival task, and researchers need to see where the gaps are §4.

Survival data are often censored (the event time is only partly known) or truncated (subjects are missing from the risk set at some times). Event types can be competing or multi-state, and features can change over time or be high-dimensional or multimodal §2.2. A method has to be checked against these aspects before it can be used for a given dataset §3.4.

Method

The review is a literature classification. It covers the survival concepts needed to read the methods, the screening process, and a classification of each method along several dimensions.

1. Background notation. Survival function S(t) = P(T > t), hazard h(t), and cumulative hazard H(t). For discrete time, the discrete hazard is the probability of an event in an interval given survival to its start §2.1 eq. 1 eq. 2.
2. Data aspects. Outcome types are interval-, right- and left-censoring, right- and left-truncation, competing risks, multi-state and recurrent events §2.2.1 tab. 2. Feature aspects are time-varying features (TVFs), time-varying effects (TVEs), high-dimensional features and multimodal inputs §2.2.2.
3. Classical estimation. Parametric models (such as the accelerated failure time model), the Cox model, piecewise exponential models (PEMs) and discrete-time models §2.3 eq. 3 eq. 4 eq. 6 eq. 7.
4. Search. One Web of Science query, with four groups of terms joined by AND: time-to-event terms, neural-network or deep-learning terms, model or method terms, and evaluation, comparison or benchmark terms. The cut-off is 31 December 2022. The query returned 211 articles §3.1.
5. Screening. A paper is kept only if it meets all four criteria: (a) it develops a new DL-based method rather than applying an existing one to new data; (b) it evaluates on at least one non-private benchmark dataset; (c) it reports time-to-event metrics such as the C-index or the integrated Brier score; (d) it estimates and predicts in a time-to-event setting with all parameters learned end-to-end in the network §3.1. The review excludes two-step approaches where DL only extracts features and a non-DL survival model does the rest §3.1.
6. Completion. Papers the authors already knew that met the same criteria were added. The total is 61 distinct methods §3.1 fig. 1.
7. Naming. A method is named by its published name when that is given and unique. Otherwise it gets a suffix: the first three letters of the first author's surname and the year §3.1.

The screening rule, in pseudocode. The paper gives no code for this step; the pseudocode restates the text.

# Screening as the review describes it  [§3.1]
# Input:  hits from the Web of Science query (cut-off 31 Dec 2022)
# Output: the reviewed set (61 methods in the paper)

def screen(hits, known_papers):
    # Keep a paper only if it meets all four criteria  [§3.1]:
    # (a) new DL method, (b) non-private benchmark dataset,
    # (c) time-to-event metric, (d) end-to-end network estimation
    selected = [p for p in hits if meets_all_four(p)]
    # Add known papers that meet the same criteria  [§3.1];
    # the number of papers added is not stated
    selected += [p for p in known_papers if meets_all_four(p)]
    return selected

8. Classification of estimation. Each method is described by three concepts: the model class (the classical survival model it builds on), the loss function, and the parametrization (which part of the model the network outputs). The loss is usually the negative log-likelihood of that model class §3.3.1.
9. Model classes. Cox-based methods replace the linear predictor of the Cox model with a network and minimize the Cox partial likelihood. Discrete-time methods turn event times into binary outcomes per interval and usually minimize the negative log-likelihood. Parametric methods output the parameters of a distribution, such as Weibull or log-normal. PEM-based methods use piecewise-constant hazards and a Poisson likelihood. ODE-based methods use neural ODEs for the hazard or transition rates. Ranking-based methods use a ranking loss as the central part. "Other" covers the remaining methods, such as GAN-based or quantile-regression methods §3.3.1.
10. Network architecture. Architectures named include feed-forward (FFNN), convolutional (CNN), recurrent (RNN, often LSTM), autoencoder (AE), GAN, transformer, neural ODE, normalizing flow, fuzzy and diffusion networks §3.2. Many methods use a pre-trained CNN, such as ResNet18, and fine-tune it on the target data §3.2. The generalized predictor adds nonlinear time-varying effects of tabular features and embeddings of unstructured data to the linear part eq. 9 §3.2. For multimodal input, each modality gets its own subnetwork and the outputs are fused in the head §3.2 fig. 3. For competing risks, shared and cause-specific subnetworks are used §3.2 fig. 2.
11. Supported tasks. For each method, the review records which outcome types and feature aspects the paper explicitly supports §3.4. The results are summarized as Venn diagrams fig. 6 fig. 7.
12. Interpretability. Some methods are interpretable by construction, for example by assigning biological meaning to network nodes or by keeping a Cox part for the tabular features. Others rely on post-hoc methods such as SHAP, LIME or LRP, which the review says may be applicable but are not always adjusted for censoring §3.5.
13. Evaluation and reproducibility. The C-index (Harrell's or Uno's) is the most common metric. Brier or Graf scores are used for distributional predictions. The right-censored log-loss and D-calibration are seldom reported §3.6. Code availability is recorded per method §3.8.

The sketch below shows the batch-wise Cox loss that the review describes for DeepSurv and Cox-Time. The restriction to the current batch is the review's description of DeepSurv. The paper gives no code for this step.

# Batch-wise negative Cox partial likelihood  [§3.3.1] [eq. 7]
# Input:  log-risk per individual in the batch, event flags, times
# Output: scalar loss to minimise

def batch_cox_loss(log_risk, times, events):
    # Risk set restricted to individuals in this batch  [§3.3.1]
    loss = 0
    for i in batch:
        if events[i] == 1:
            # Individuals still at risk at time times[i]  [eq. 7]
            at_risk = [j for j in batch if times[j] >= times[i]]
            loss -= log_risk[i] - log(sum(exp(log_risk[at_risk])))
    return loss

Key equations, as numbered in the paper (see the Pitfalls note on numbering):

h_T(t) = P(T = t \mid T \geq t) eq. 2

Pl(\boldsymbol{\theta}) = \sum_{m=1}^M \left( g(\mathbf{x}_{(m)}, \boldsymbol{\beta}) - \log \sum_{j \in \mathcal{R}(t_{(m)})} \exp\left(g(\mathbf{x}_j, \boldsymbol{\beta})\right) \right) eq. 7

\eta = g(\mathbf{x}, \mathbf{z}, t) = f(\mathbf{x}, t) + \gamma_1 d_1(\mathbf{z}_1) + \dots + \gamma_G d_G(\mathbf{z}_G) eq. 9

Hyperparameters

The review trains no model. The protocol values that affect the set of methods are listed here.

Results

Author-reported. This is a literature classification of 61 methods, not an experiment. Setup: Web of Science query with a cut-off of 31 December 2022, then the screening in the Method section §3.1.

Paper vs code

Code. No method code is covered here. The paper links the companion repository survival-org/DL4Survival §4. It holds the review's data, R scripts and the rendered site, not the methods' code. The card cites commit feed79ae (code_commit). Per-method code links are in the code column of the data file raw.csv.

Discrepancies between the paper and its repository.

• The paper says all data and scripts are in the repository §4. The R scripts read the table from a Google Sheet bar_and_venn.R. The read from data/raw.csv is commented out bar_and_venn.R. The CSV in the repository may therefore not be the version behind the paper's figures.
• The text gives 61 methods §3.1. The CSV has 102 lines with a PASS field raw.csv, and the PRISMA script filters on PASS == "PASS" prisma_diagram.R. I did not resolve how these map to the 61 methods.
• The repository has no LICENSE file. The site text says the table is CC BY index.qmd. The license on this card covers the paper only, not the table's rows.

Pitfalls

• Counts do not add up. The stated class counts (26 Cox-based, 19 discrete-time, 3 PEM-based, 2 ODE-based, 2 ranking-based, 5 other) plus the five parametric methods named in the text total 62, not 61. The discrete-time paragraph names 20 methods, not 19 §3.3.1.
• The selection cannot be reproduced from the text. The number of the 211 hits that pass all four criteria is not stated. The number of papers added outside the query is not stated either §3.1.
• One database. Only Web of Science is named. Methods published elsewhere, or after the cut-off, are not in the review §3.1.
• Some classifications are ambiguous. SSCNN is listed among discrete-time methods that use a ranking loss as an auxiliary term, and also as one of two continuous-time ranking-based methods §3.3.1. The classification is the authors' judgement.
• TVE count. The text states seven methods handle time-varying effects but names five: Nnet-survival, MultiSurv, RNN-Surv, Cox-Time and DeepPAMM. It also says DSM and SSMTL give no detail on TVEs §3.4.2.
• No ranking. The authors say reported benchmark results cannot be compared, because datasets, preprocessing, missing-value handling, resampling, and metric variants differ. Selective reporting is also a concern §3.6. Do not use this review to pick a method by performance.
• Metrics. Harrell's and Uno's C-index differ, and the integration window of the integrated Brier score varies. Some common survival evaluation measures are not proper scoring rules §3.6.
• Parametric losses are hard to optimise. A Weibull with extreme-value errors is poorly conditioned in gradient descent at low precision. The authors cite a recommendation to use the log-normal, because other distributions have densities with numerically unstable forms §4.
• Batching is not free for Cox. The Cox loss uses the full risk set, so batch-wise training is only approximate unless the batch is large enough §3.3.1. Cox-Time approximates the risk set with a large subset of individuals at risk §3.3.1.
• Missing values are rarely discussed. Only MultiSurv and SurvNet handle them explicitly §4.
• Sample size requirements for DL survival models are not addressed by the reviewed papers §3.7.
• Equation numbers were counted from the LaTeX source, since the numbering is not in the text. Check them against the compiled PDF before citing them elsewhere.
• Version. This card covers arXiv v4 (last updated 22 February 2024). The authors say the online table will be extended, so counts may have changed since. The repository README links a journal version README.md that was not checked.
• Figures are described in words only, because the site does not yet support images. fig. 1 to fig. 7 are cited by number only.

Lineage

• Builds on: the Kaplan–Meier estimator and the Cox proportional hazards model, both cited as classical baselines §1 §2.3. Early neural survival models from the 1990s: Faraggi and Simon's Cox-with-network model, Liestbl et al.'s PEM as a network, and PLANN (Biganzoli et al.) as a discrete-time network §3. None has a card in this wiki.
• Method-level lineage named in the review: DeepSurv extends Faraggi's model; Cox-Time extends DeepSurv §3.3.1. Dynamic-DeepHit extends DeepHit, and RDSM extends DSM §3.3.2.
• Compared against: earlier surveys on genomics data (Lee and Lim, 2019), cancer survival prediction (Deepa and Gunavathi, 2022), glioma (Wijethilake et al., 2021), and the benchmarking study by Zhang et al. (2022) §1.
• Superseded by: not known from this wiki's content.