Discrete-Time Survival Model

• Discrete-Time Survival Models are statistical methods for analyzing time-to-event data on discrete intervals, defining hazards, survival functions, and event probabilities.
• They use generalized linear approaches with link functions like logit and cloglog to flexibly incorporate covariates, competing risks, and interval censoring.
• Modern extensions employ neural networks, federated learning, and regularization to improve accuracy and privacy in complex, real-world settings.

A discrete-time survival model characterizes the distribution of time-to-event data observed on a finite, ordered grid, typically integer-valued time points. Such models express the hazard, survival, and event probabilities conditional on remaining at risk up to each time interval, and offer flexibility for modeling covariate effects, competing risks, recurrent events, interval censoring, and nonlinear relationships. Discrete-time approaches are theoretically rigorous alternatives to continuous-time methods, enabling robust likelihood-based inference, efficient federated and privacy-preserving computation, principled neural extensions, and precise handling of grouped or imprecise event times.

1. Core Definitions and Survival Quantities

Let $T_i\in\{1,2,...,T\}$ denote the discrete event time (possibly censored) for subject $i$ and ${\bf x}_i$ its covariate vector. At each interval $t=1,...,T$:

• Discrete-Time Hazard Rate:

$$h_i(t \mid {\bf x}_i) = P(T_i = t \mid T_i \ge t,\; {\bf x}_i)$$

• Survival Function:

$$S_i(t \mid {\bf x}_i) = P(T_i > t \mid {\bf x}_i) = \prod_{j=1}^{t} [1 - h_i(j \mid {\bf x}_i)]$$

• Event Probability Mass Function:

$$p_i(t \mid {\bf x}_i) = h_i(t \mid {\bf x}_i) \cdot S_i(t-1 \mid {\bf x}_i)$$

Discrete-time models elegantly accommodate left truncation (delayed entry), right censoring, and event indicators. The basic likelihood for $n$ independent subjects, possibly censored, is: $$L = \prod_{i=1}^{n} \left( \prod_{j=1}^{t^*_i - 1} [1-h_i(j \mid {\bf x}_i)] \cdot [ h_i(t^*_i \mid {\bf x}_i) ]^{\delta_i} \right)$$ with $t^*_i = \min(T_i, C_i)$ and event indicator $\delta_i$.

2. Model Classes and Link Functions

Discrete-time survival models use a generalized linear approach for $h_i(t \mid {\bf x}_i)$, with common link functions:

• Logit/Proportional Odds:

$$\text{logit}(h_i(t \mid {\bf x}_i)) = \alpha_t + {\bf x}_i^\top \beta$$

This is the discrete analog of Cox's proportional hazards, yielding

$$h_i(t \mid {\bf x}_i) = \frac{\exp(\alpha_t + {\bf x}_i^\top \beta)}{1 + \exp(\alpha_t + {\bf x}_i^\top \beta)}$$

• Complementary Log-Log (cloglog):

$$\text{cloglog}(h_i(t \mid {\bf x}_i)) = \alpha_t + {\bf x}_i^\top \beta$$

with

$$h_i(t \mid {\bf x}_i) = 1 - \exp\left[-\exp(\alpha_t + {\bf x}_i^\top \beta)\right]$$

• Multinomial Logit (competing risks):

For $M$ event types,

$$\lambda_{t,r}({\bf x}_i) = \frac{ \exp( \alpha_{t,r} + {\bf x}_i^\top \beta_r ) }{ 1 + \sum_{k=1}^M \exp( \alpha_{t,k} + {\bf x}_i^\top \beta_k ) }$$

where $\lambda_{t,r}$ is the cause-specific hazard.

The choice of link function depends on the modeling objective (probability ratios, odds, proportionality) and theoretical properties (bounds, interpretability).

3. Estimation, Regularization and Inference

Numerical estimation is performed via maximum likelihood, conditional likelihood, or efficient estimating functions.

• Unconditional MLE / Pooled Logistic Regression:

Joint estimation of all intervals' intercepts and regression parameters.

• Conditional Likelihood (partial likelihood):

Conditioning out baseline hazards yields estimators akin to Cox's partial-likelihood. Computationally challenging with many tied events.

• Breslow–Peto and Weighted Mantel–Haenszel Estimators:

Consistent and robust alternatives for hazard-probability and hazard-odds models (Tan, 2020).

• Regularization:

Lasso ($\ell_1$), ridge ($\ell_2$), and elastic-net penalties are seamlessly incorporated (notably in PyDTS (Meir et al., 2022, Meir et al., 2023)), enabling feature selection and shrinkage, with tuning via grid search or cross-validation optimizing integrated AUC/Brier score.

• Variance Estimation:

Robust ("sandwich") and model-based (Fisher information) SEs provide inference under model correctness or mild misspecification (Tan, 2020).

• Privacy-Preserving Computation:

Differentially private output and objective perturbation, and posterior sampling via pSGLD allow formal privacy guarantees for discrete-time regression (Nguyên et al., 2017).

4. Discrete-Time Survival in Complex Settings

4.1 Competing Risks

Competing event types require joint modeling of cause-specific hazards. The full likelihood involves all event types and intervals (Meir et al., 2023, Meir et al., 2022, Boom et al., 2023).

4.2 Interval-Censored Covariates

When key covariates (e.g., HIV serostatus) are only intermittently measured, joint modeling of the censoring mechanism, covariate, and outcome hazard is essential. Markov-type assumptions and likelihood marginalization over possible covariate paths yield valid inference (Kenny et al., 14 Aug 2024).

4.3 Recurrent Events and Frailty

For subjects with multiple events, Gamma frailty models capture unobserved heterogeneity. Feed-Forward Panel Estimation delivers closed-form panel likelihood and sequential variational Bayesian updates for frailty (Bateni et al., 25 Oct 2024).

4.4 Time-Varying Covariates

Discrete-time models easily accommodate time-updated predictors, either via GLM/forest expansion (Moradian et al., 2021) or recurrent deep architectures (Fuhlert et al., 2022).

4.5 Recall Error and Reporting Uncertainty

Multistage discrete survival models incorporating certainty ratings and gaps between event and reporting time enhance bias and MSE under retrospective ascertainment (Salehabadi et al., 2018).

5. Neural and Machine-Learning Extensions

Modern machine-learning techniques extend discrete-time survival modeling beyond parametric forms.

• Neural Network Parameterizations:

Logistic-Hazard, PMF, and dual mixture-of-experts architectures (feature and hazard MoE) flexibly model nonlinear hazards and subgroup heterogeneity (Gensheimer et al., 2018, Kvamme et al., 2019, Lee et al., 29 Oct 2025).

• Discrete Calibrated Survival (DCS):

A deep net integrating calibration-oriented loss (rank-probability score), discrimination kernel loss, and time-node quantile allocation achieves state-of-the-art discrimination and calibration (Fuhlert et al., 2022).

• Monotonicity-Preserving Survival (SurvSurf):

Architecture and loss enforce monotonicity over time and event order, robust to intermittently observed data and supporting consistent integrated Brier score evaluation (Chen et al., 7 Apr 2025).

• Federated Learning:

The person–period reformulation renders the loss perfectly separable, enabling federated SGD, communication-efficient privacy-preserving survival modeling (Andreux et al., 2020).

• Random Forests for Dynamic Estimation:

As person–period classifiers, forests using Hellinger splitting (superpp, separate, pooled) yield robust, data-driven hazard estimation that dynamically updates as covariate information arrives (Moradian et al., 2021).

6. Evaluation Metrics and Performance Measures

Standard and advanced evaluation metrics for discrete-time survival include:

• Time-dependent Concordance Index (C-index):

Measures the accuracy of event ordering, often calculated dynamically over time.

• Integrated and Cause-Specific AUC:

Assesses discrimination for specific event types, integrated over time intervals.

• Brier Score:

Quantifies calibration by comparing predicted vs observed survival/event probabilities, with IPCW adjustment for censoring.

• Integrated Brier Score (IBS):

Summarizes calibration performance over all time points (Chen et al., 7 Apr 2025).

• Distributional Divergence for Calibration (DDC):

KL divergence between predicted and uniform distributions of survival at event times (Fuhlert et al., 2022).

7. Applications and Extensions

Discrete-time survival models find application in numerous substantive domains:

• Clinical risk prediction:

Length-of-stay analysis, mortality prediction with competing outcomes (e.g., ICU studies) (Meir et al., 2023, Meir et al., 2022).

• Policy evaluation and epidemiology:

Modeling impacts of medical interventions, drug rollouts, and disease progression, especially with interval-censored covariates (Kenny et al., 14 Aug 2024).

• Financial asset pricing:

Estimation of time-to-event distributions for contingent cash flow valuation under left-truncation/right-censoring, with asymptotically normal hazard estimator vectors (Lautier et al., 2022).

• Fecundity and reproductive studies:

Joint modeling of cyclic longitudinal features with discrete survival times on nested timescales, supporting individualized prediction via empirical Bayes and quadrature (Saha et al., 2021).

References to Key Papers

Area	Key Reference	arXiv ID
General Theory	Tan	(Tan, 2020)
GLM, Competing Risks	Meir et al	(Meir et al., 2022, Meir et al., 2023)
Federated Learning	Andreux et al	(Andreux et al., 2020)
Deep Neural Methods	Fotso (Nnet-survival), Kvamme et al, DCS, Dual MoE	(Gensheimer et al., 2018, Kvamme et al., 2019, Fuhlert et al., 2022, Lee et al., 29 Oct 2025)
Survival Forests	Moradian et al	(Moradian et al., 2021)
Interval Censoring	Kenny et al	(Kenny et al., 14 Aug 2024)
Frailty, Recurrent Events	FFPSurv	(Bateni et al., 25 Oct 2024)
Change-Point Detection	Multivariate Bernoulli	(Boom et al., 2023)
Privacy Preserving	Dwork et al	(Nguyên et al., 2017)
Financial Applications	Huang et al	(Lautier et al., 2022)
Recall Error	Mirzaei Salehabadi et al	(Salehabadi et al., 2018)
Joint Longitudinal–Survival	Saha et al	(Saha et al., 2021)
Monotonic Neural Survival	SurvSurf	(Chen et al., 7 Apr 2025)

Concluding Perspective

Discrete-time survival analysis provides a flexible, rigorous modeling foundation for a spectrum of modern time-to-event problems. Beyond its theoretical tractability in hazard specification and likelihood factorization, it is ideally suited for integration with high-dimensional, federated, and privacy-sensitive datasets, supports nuanced clinical and financial applications, and underpins advanced neural architectures for discrimination, calibration, monotonicity, and interpretability. Its continued development, including joint modeling, competing risks, recurrent phenomena, and robust computational approaches, positions it as an essential tool in contemporary survival statistics and applied research.