Fine-Gray Model in Competing Risks

• Fine-Gray formulation is a semiparametric model for competing risks that parameterizes the subdistribution hazard for direct estimation of the cumulative incidence function.
• It uses a modified risk set construction and IPCW-weighted partial likelihood to yield interpretable regression coefficients for individual-specific risk prediction.
• Modern extensions incorporate deep learning and dynamic feature selection to address high-dimensional covariates and improve model transparency in biomedical applications.

The Fine-Gray formulation is a semiparametric model for analyzing survival data with competing risks, directly parameterizing the covariate effect on the cumulative incidence function (CIF) for each event type by modeling the subdistribution hazard. This approach modifies risk-set construction and estimation compared to cause-specific hazards, yielding interpretable regression coefficients for risk prediction and inference. The model is widely applied in biomedical studies and has recently been extended to incorporate neural network architectures, high-dimensional inference, and multiple imputation for complex data regimes.

1. Subdistribution Hazard, Cumulative Incidence Function, and Risk Set Construction

Given a time-to-event pair $(T,E)$, where $E\in\{1,\dots,K\}$ indexes the event type or is zero for censoring, the cumulative incidence function (CIF) for cause $k$, conditional on covariates $X$, is defined as: $F_k(t\mid X) = \Pr(T \le t, E = k \mid X)\,.$ The Fine-Gray subdistribution hazard for cause $k$ is the instantaneous rate of experiencing event $k$ at time $t$, with the risk set retaining subjects who have failed from other causes but not from $k$: $$\lambda_k^{\mathrm{sub}}(t\mid X) = \lim_{\Delta t\to 0} \frac{ \Pr(t \le T < t+\Delta t, E = k \mid T\ge t \lor (T < t, E \ne k), X) }{\Delta t}\,.$$ This construction accommodates the competing risks framework by altering the risk set to include subjects censored and those experiencing other events but not the event of interest. The relationship between CIF and subdistribution hazard is: $$\lambda_k^{\mathrm{sub}}(t\mid X) = \frac{\partial_t F_k(t\mid X)}{1-F_k(t\mid X)},$$ and therefore,

$$F_k(t \mid X) = 1 - \exp\left(-\int_0^t \lambda_k^{\mathrm{sub}}(s\mid X)\,ds\right)\,.$$

Risk set for estimation is defined as: $$\mathcal R_k^{\mathrm{sub}}(t) = \{j: T_j \ge t\} \cup \{j: T_j < t, E_j \neq k, E_j \neq 0\}\,.$$ This risk set and hazard definition are foundational for all Fine-Gray model extensions (Ramachandram et al., 16 Nov 2025, Jeanselme et al., 2023, Shen et al., 8 Aug 2025).

2. Proportional Subdistribution Hazards and Estimation Procedures

A proportional subdistribution hazards assumption is used: $$\lambda_k^{\mathrm{sub}}(t\mid X) = \lambda_{0k}^{\mathrm{sub}}(t)\,\exp(\eta_k(X)),$$ where $\lambda_{0k}^{\mathrm{sub}}(t)$ is an unspecified baseline subdistribution hazard and $\eta_k(X)$ is typically linear ($\beta^\top X$) or more generally a risk score possibly modeled by a neural network.

Estimation is performed via IPCW-weighted partial likelihood: $$\mathcal L_k^{\mathrm{FG}} = -\sum_{i:E_i = k} \left[ \eta_k(X_i) - \log \Big( \sum_{j \in \mathcal{R}_k^{\mathrm{sub}}(T_i)} w_j(T_i) \exp(\eta_k(X_j)) \Big) \right],$$ where $w_j(T_i)$ are inverse-probability-censoring weights: $$w_j(T_i) = \mathbf{1}\{C_j \ge T_i\} / \hat{G}(T_i),$$ with $\hat{G}(t)$ the Kaplan-Meier estimate for censoring. Baseline subdistribution hazards or CIFs are estimated via Breslow-type estimators: $$\hat{F}_k(t\mid X) = 1 - [1 - \hat{F}_{0k}(t)]^{\exp(\eta_k(X))},$$

$$\hat{F}_{0k}(t) = \sum_{i: T_i \le t, E_i = k} \frac{d_i}{ \sum_{j \in \mathcal{R}_k^{\mathrm{sub}}(T_i)} \exp(\eta_k(X_j)) }\,.$$

This framework yields interpretable regression for individual-specific risk prediction (Ramachandram et al., 16 Nov 2025, Bonneville et al., 26 May 2024, Dietrich et al., 2023).

3. Modern Extensions: Deep Learning and Dynamic Feature Selection

Recent advances utilize neural architectures to increase expressiveness and interpretability:

• Neural Additive Models (CRISPNAM-FG): Each scalar feature is encoded by a small multilayer perceptron ("FeatureNet"), and risk-specific projections aggregate feature contributions via additive shape functions:

$\eta_k(X) = \sum_{i=1}^p s_{i,k}(x_i)$

These shape functions enable transparent auditing and feature importance quantification:

$$\mathcal{I}_{i,k} = \frac{1}{N} \sum_{j=1}^N |s_{i,k}(x_{ij})|$$

(Ramachandram et al., 16 Nov 2025).

• Monotonic Neural Networks (Neural Fine-Gray): CIFs are modeled directly by constrained monotonic networks guaranteeing exact likelihood maximization without numerical integration. Each event type is represented by a monotonic sub-CDF, and the sum of CIFs respects the unit constraint (Jeanselme et al., 2023).
• Stepwise Fine and Gray: Dynamic feature selection is addressed by fitting patient-specific, phasewise subdistribution hazard models, allowing the model to adaptively utilize richer, time-varying features only when they add prognostic value. Neural networks parameterize the risk score, and subject/horizon-specific thresholds govern the inclusion of dynamic features (Shen et al., 8 Aug 2025).

These adaptations maintain the interpretability of risk contributions, enable nonparametric baseline estimation, and yield competitive performance on clinical benchmarks.

4. Handling High-Dimensional Covariates and Incomplete Data

High-dimensional modeling and missingness are addressed using specialized estimation/inference frameworks:

• Regularized Estimation and One-Step Bias Correction: In settings with $p \gg n$, an $\ell_1$-penalized Fine-Gray estimator is used:

$$\hat{\beta}^{\mathrm{ini}}(\lambda) = \arg\min_{\beta} \{-m(\beta) + \lambda \|\beta\|_1\}$$

One-step bias-correction with estimated negative Hessian (nodewise Lasso) yields valid confidence intervals and asymptotic normality (Hou et al., 2017).

• Multiple Imputation and Bootstrapping: For incomplete covariates or censoring times, the substantive-model-compatible multiple imputation (FG-SMC) procedure first imputes missing censoring times, then covariates, to ensure compatibility with the Fine-Gray model. Inference leverages wild bootstrap adapted for the Fine-Gray estimating equations and multiple imputed datasets. Efficiency gains in estimation and narrow confidence intervals are empirically demonstrated (Bonneville et al., 26 May 2024, Dietrich et al., 2023).
• Risk Set and Weighting Schemes: All methods hinge on the correct definition of the subdistribution risk set and use of IPCW to address random censoring, with advanced concentration inequalities underpinning the validity of inference in high dimensions (Hou et al., 2017).

5. Interpretability, Transparency, and Clinical Implications

Fine-Gray models yield directly interpretable regression coefficients for each event-specific CIF, permitting transparent risk prediction and clinical decision support. Modern architectures such as CRISPNAM-FG further advance interpretability by:

• Providing feature-wise shape function plots that visualize the effect of covariates on the log-subdistribution hazard for each event type.
• Computing feature importance scores for each risk, enhancing explainability relevant to AI safety and clinician trust.
• Allowing patient-specific, time-dynamic feature selection so that complex time-varying features are applied only when demonstrably informative for prognosis.

Contrasts with cause-specific hazard models highlight that Fine-Gray provides direct estimation of absolute risk in the presence of competing events, an essential property for individual-level prediction and intervention planning (Ramachandram et al., 16 Nov 2025, Shen et al., 8 Aug 2025).

6. Limitations, Assumptions, and Open Challenges

The Fine-Gray formulation assumes proportional subdistribution hazards for each event type. Critical model aspects include:

• The modified risk set construction ("unnatural" set including failures from competing events) is required for identifiability but does not yield a true hazard function in the classical sense (Shen et al., 8 Aug 2025).
• Proportionality assumption is central; deviations may lead to bias, especially in imputation or prediction tasks. Empirical studies suggest that correct specification at the analysis phase is more critical than the imputation procedure used (Bonneville et al., 26 May 2024).
• Dynamic modeling with subject-specific time snapshots, especially in stepwise extensions, heuristically treats snapshots as i.i.d., which may introduce subtle dependence structures (Shen et al., 8 Aug 2025).
• Estimation of absolute risk (CIF) requires robust handling of censoring and competing risks, particularly when outcome-specific follow-up is limited or high-dimensional feature space is sparse.
• Thresholding rules for dynamic feature inclusion and regularization for neural architectures require careful validation to avoid overfitting and preserve model stability.

A plausible implication is that ongoing methodological research will continue to address these limitations, seeking robust, interpretable, and scalable competing risks models for complex survival data.

7. Summary Table: Fine-Gray Formulation—Model and Extensions

Component	Classical Formulation	Modern Extension (Neural/Stepwise)
Subdistribution hazard	$$\lambda_k^{\mathrm{sub}}(t|X) = \lambda_{0k}^{\mathrm{sub}}(t)\exp(\beta^\top X)$$	Neural risk scores $\eta_k(X)$, monotonic NN (Ramachandram et al., 16 Nov 2025, Jeanselme et al., 2023)
CIF estimation	$$F_k(t|X) = 1 - \exp(-\exp(\beta^\top X)\Lambda_{0k}(t))$$	CIF modeled directly by MLP, stepwise selection (Shen et al., 8 Aug 2025)
Risk set	$T_j \ge t \cup (T_j < t, E_j \ne k, E_j \ne 0)$	Dynamic, phase-specific subdistribution (Shen et al., 8 Aug 2025)
Regularization/Inference	IPCW, bootstrap, multiple imputation	L1/L2, nodewise Lasso, wild bootstrap, SMC MI (Hou et al., 2017, Bonneville et al., 26 May 2024)
Interpretability	Regression coefficients	FeatureNet shape functions, incremental phasewise scores (Ramachandram et al., 16 Nov 2025, Shen et al., 8 Aug 2025)

The Fine-Gray formulation remains foundational in survival analysis with competing risks, with active research extending its empirical utility, statistical rigor, and transparency for clinical and biomedical applications.