Pseudo-Value Neural Regression

• Pseudo-value neural regression is a neural network-based approach that reformulates censored survival analysis into a standard regression problem using jackknife-derived surrogate targets.
• It leverages conventional feed-forward architectures with mean-squared-error loss to estimate survival probabilities for both single-state and multi-state analysis without specialized censoring functions.
• Empirical evaluations show that methods like msPseudo and DNNSurv outperform traditional Cox-based models in accuracy and robustness, especially under high censoring conditions.

Pseudo-value neural regression refers to a family of neural network-based methods that leverage pseudo-values—jackknife-derived surrogate targets—to perform regression in contexts where the true outcomes are partially missing or censored, most notably in survival analysis and multi-state time-to-event modeling. Rather than relying on specialized loss functions for censoring or proportional hazards assumptions, these methods recast survival prediction as standard regression on pseudo-values, enabling the use of generic neural network architectures and optimizers. This framework has been explicitly formulated for multi-state survival analysis (Rahman et al., 2022) and single-state survival analysis (Zhao et al., 2019).

1. Foundations of Pseudo-Values in Regression

Pseudo-values are jackknife-based estimates constructed by evaluating a consistent estimator on the full data and then recomputing it with each observation omitted in turn. For a survival quantity $Y$, the pseudo-value for individual $i$ at time $t^*$ is

$$Y_i^{\mathrm{pseudo}}(t^*) = n\,\widehat Y(t^*) - (n-1)\,\widehat Y^{-i}(t^*)$$

where $\widehat Y(t^*)$ is a consistent estimator (e.g., Aalen–Johansen or Kaplan–Meier), and $\widehat Y^{-i}(t^*)$ is the estimator with subject $i$ left out (Rahman et al., 2022, Zhao et al., 2019). For multi-state survival analysis, pseudo-values can be defined for state occupation probabilities ($\pi_k(t)$) or transition probabilities ($\mathcal{P}_{j\rightarrow k}(s,t)$).

In single-state survival contexts, pseudo-values target conditional survival probabilities over prespecified time intervals, enabling the reduction of a censored regression to a standard regression problem (Zhao et al., 2019).

2. Transformation of Censored Regression to Supervised Learning

The use of pseudo-values allows censored outcomes to be replaced by surrogate numeric targets, yielding a dataset $\{(X_i, Y_i^{\mathrm{pseudo}}(t^*))\}_{i=1}^n$ for regression. Asymptotically, regression on pseudo-values recovers the conditional expectation function for the true uncensored outcome (Rahman et al., 2022). This transformation simplifies network architectures and training protocols, as censoring is handled in the pseudo-value construction and not through the loss function.

In DNNSurv (Zhao et al., 2019), each subject is expanded into multiple pseudo-observations over discrete time intervals, further increasing the effective sample size and enabling time-specific conditional probability estimation.

3. Neural Architectures for Pseudo-Value Regression

Pseudo-value neural regression employs conventional feed-forward networks:

• Input: Baseline covariates $i$0.
• Hidden layers: Typically 2–3 fully connected layers with ReLU activations (width 32–128 units) and optional dropout (0.1–0.4).
• Output: For multi-state settings, a $i$1 matrix, where $i$2 is the number of targeted quantities (e.g., states or transitions) and $i$3 is the number of time points. The output is constrained to $i$4 by sigmoidal activation (Rahman et al., 2022). Single-state settings such as DNNSurv concatenate covariates and time indicators for outputting survival probability at each interval (Zhao et al., 2019).

Formally, the predicted pseudo-value for sample $i$5 is

$i$6

where $i$7 is the nonlinear activation.

4. Training Procedures and Loss Functions

Training reduces to mean-squared-error (MSE) regression:

$i$8

No inverse-probability-of-censoring weighting is required; pseudo-values inherently adjust for censoring. Hyperparameters are optimized using validation set metrics such as integrated Brier score (iBS) or time-dependent C-index, with early-stopping to avoid overfitting. Standard optimizers (e.g., Adam) are used with typical learning rates ($i$9–$t^*$0). Robustness to censoring and flexibility with non-Markov/non-proportional hazards data are achieved by the pseudo-value approach rather than by modifying deep learning workflows (Rahman et al., 2022, Zhao et al., 2019).

5. Empirical Evaluation and Comparative Performance

Pseudo-value neural regression, exemplified by msPseudo (Rahman et al., 2022) and DNNSurv (Zhao et al., 2019), consistently achieves or exceeds state-of-the-art predictive accuracy under a variety of settings:

• Non-Markov, nonlinear multi-state data: msPseudo achieves integrated AUC ≈ 0.83, outperforming CoxPH (≈0.67) and SurvNODE (≈0.71).
• Real-world medical datasets: On METABRIC, msPseudo yields iBS = 0.11 for state occupation prediction, better than Cox-based methods.
• High censoring (75%): msPseudo maintains average iAUC ≈ 0.77 vs. msCox ≈ 0.53.
• Single-state survival: DNNSurv matches or surpasses Cox-based deep nets especially under model mis-specification or covariate-dependent censoring, aided by IPCW pseudo-values (Zhao et al., 2019).

Metrics for evaluation include integrated AUC, Brier score, dynamic concordance index, and time-resolved versions of both.

Method	Primary Task Context	Censoring Handling	Key Result
msPseudo	Multi-state survival analysis	Jackknife pseudo	iBS=0.11 (METABRIC SOP, real data)
DNNSurv	Single-state survival analysis	Jackknife pseudo	Outperforms DeepSurv (non-PH data)

6. Choice of Estimator and Practical Guidelines

The performance of pseudo-value neural regression depends on the choice of consistent estimator for the survival quantity. The Aalen–Johansen (AJ) estimator is used when the Markov property holds (validated via score tests), while the landmark-AJ (LMAJ) estimator is applied in non-Markov settings (Rahman et al., 2022). For single-state contexts with potential covariate-dependent censoring, inverse-probability-of-censoring-weighted estimators can be substituted (Zhao et al., 2019). For transitions with low numbers at risk, reverting to AJ helps control variance.

Guidelines for application include:

• Testing model assumptions (Markov, PH) and selecting appropriate estimators based on hypothesis tests.
• Grid-search and cross-validation for depth, width, regularization, and time-grid selection.
• Output calibration as probabilities via sigmoid mapping.
• Early stopping based on validation iBS or iAUC.
• No special weighting or ad-hoc loss adjustment is needed to address censoring.

7. Significance and Extensions

Pseudo-value neural regression provides a principled method for regression in partially observed and censored regimes by converting the problem to standard supervised learning. Unlike proportional hazards neural nets or models requiring ad-hoc loss modifications, this framework is robust to non-linear, non-Markov dynamics and covariate-dependent censoring, provided a consistent estimator exists for the target quantity (Rahman et al., 2022, Zhao et al., 2019). It generalizes across multi-state and single-state survival contexts and is amenable to standard deep learning hardware and software. A plausible implication is that similar pseudo-value transformations could be extended to other structured prediction tasks in which the outcome is only indirectly observable due to censoring, missingness, or competing risks.

In summary, pseudo-value neural regression recharacterizes censored regression as classical regression with statistically sound surrogate targets, enabling deep neural models to be applied directly and efficiently to complex survival and time-to-event data without the limitations of existing hazard-based methods or restrictive model assumptions.