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Extending Deep Cox Survival Models with Case-Cohort risk set Sampling

This paper introduces and evaluates the first integration of case-cohort risk-set sampling into deep Cox proportional hazards neural networks, demonstrating that while nested case-control sampling consistently approximates full-risk-set performance, mini-batch training enables case-cohort sampling to achieve comparable accuracy with substantial computational savings.

Original authors: Justine Nasejje, Michael P. Falk, Marvin N. Wright, Wende Clarence Safari, Pierre-Philippe Dechant, Raeesa Manjoo Docrat, Rukia Nuermaimaiti

Published 2026-09-17

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CC BY 4.0

Original authors: Justine Nasejje, Michael P. Falk, Marvin N. Wright, Wende Clarence Safari, Pierre-Philippe Dechant, Raeesa Manjoo Docrat, Rukia Nuermaimaiti

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of medical research and engineering, predicting when something will happen is often more valuable than knowing exactly when it will happen. Whether it is a machine part failing or a patient surviving an illness, scientists use a method called survival analysis to study the time until an event occurs. A major challenge in this field is that studies often end before every subject experiences the event; some people are still alive when the study closes, or a machine is still working. This is known as right-censored data, and it requires special statistical tools to handle the incomplete information without throwing away the valuable data that does exist. For decades, the standard tool for this has been a model that calculates risk based on a group of people who are still at risk at any given moment, known as the risk set.

As data has exploded in size, with modern studies tracking thousands of patients and millions of data points, these traditional calculations have become a bottleneck. To process the data, a computer must look at every single person still in the study every time an event happens, a process that demands immense computing power and memory. This creates a barrier for using modern, powerful artificial intelligence techniques, which thrive on large datasets but struggle when forced to perform these heavy, repetitive calculations. Researchers are now asking how to keep the accuracy of these advanced models while making them fast enough to run on the massive datasets of the twenty-first century.

A team of researchers from universities in South Africa, Germany, and the United Kingdom has tackled this problem by testing a new way to train deep neural networks, a type of artificial intelligence designed to find complex patterns in data. Their goal was to see if they could speed up the training process by not looking at every single person in the study at once, but instead by looking at a carefully chosen sample. They compared two different sampling strategies: one that picks a few new people to compare against each event every time, and another that picks a fixed group of people at the very beginning of the study and sticks with them. They tested these methods on both computer-generated data, where the true answer is known, and a real-world dataset of breast cancer patients, measuring how well the models predicted survival and how much computer memory and time they required.

The researchers found that the method which picks a fresh group of people for every event, known as nested case-control sampling, works almost perfectly. Whether they used the entire dataset or just a small sample, this approach produced predictions that were nearly identical to the slow, heavy method that looks at everyone. The model learned the correct patterns, and the predictions were just as accurate as the standard approach, but it did so without the massive computational cost. This suggests that for many large-scale problems, researchers can safely use this sampling trick to make their models run much faster without losing any predictive power.

The second method, called case-cohort sampling, behaved differently and revealed a more complex story. When the researchers used a very small fixed group of people to represent the whole study, the model struggled significantly. In tests with a tiny group, the model's ability to distinguish between high-risk and low-risk patients dropped sharply, and the predictions became unreliable. However, the researchers discovered that the way the computer processes the data changes everything. When they switched from processing the whole dataset at once to processing it in small, rapid batches, the performance of the small fixed group improved dramatically. With this faster processing style, even a small group of people could help the model learn effectively, recovering most of its accuracy while still saving a great deal of computing resources.

The study also looked at how these methods affect the computer itself. The traditional way of processing all data at once requires a massive amount of memory, often reaching six gigabytes or more, which can crash standard computers. By switching to the batched approach, the memory usage dropped to less than half a gigabyte, making these advanced models accessible on much smaller machines. The researchers observed that while the small fixed group method was unstable when the computer looked at everything at once, the batched approach smoothed out the errors, allowing the model to learn steadily. This trade-off between the size of the sample and the way the data is processed offers a new toolkit for scientists: they can choose to use a method that is always robust, or they can use a smaller, faster method if they are willing to adjust how the computer handles the data.

Ultimately, this work bridges the gap between classical statistics and modern artificial intelligence. It shows that the heavy lifting required to analyze survival data does not have to be done by looking at every single person in the study at every single moment. By using smart sampling techniques, particularly when combined with modern training methods, researchers can build powerful models that are both accurate and efficient. The findings suggest that for the future of survival analysis, the key is not just having more data, but knowing how to look at it in a way that respects the limits of our computers while preserving the truth hidden within the numbers.

Technical Summary: Extending Deep Cox Survival Models with Case-Cohort Risk Set Sampling

Problem Statement
Deep neural networks (DNNs) have demonstrated superior predictive performance in survival analysis when trained using the Cox partial likelihood. However, a significant bottleneck exists in scaling these models to large datasets: evaluating the full risk set at every observed event time incurs substantial computational and memory costs. This limitation restricts the scalability of deep Cox models. While risk-set sampling techniques like Nested Case-Control (NCC) have been adapted for neural networks to mitigate this, the integration of Case-Cohort (CC) sampling—a design where a static subcohort is sampled once at baseline—into deep Cox frameworks remains unexplored. This study addresses the gap in existing literature regarding the application and evaluation of CC sampling within deep survival learning, specifically investigating its trade-offs against NCC and full risk-set approaches in both full-batch and mini-batch training regimes.

Methodology
The authors propose a training framework for Cox-based DNNs that incorporates risk-set sampling. The core statistical foundation is the Cox proportional hazards model, where the network output $\eta_i = f_\theta(X_i)$ replaces the linear predictor $\beta^\top X_i$ . The model minimizes the negative log-partial likelihood.

The study evaluates three sampling strategies for constructing the risk set $R(t_j)$ at each event time $t_j$ :

Full Risk Set: The baseline approach using all individuals at risk ( $R(t_j) = \{i : T_i \ge t_j\}$ ).
Nested Case-Control (NCC): For each event, a fixed number of controls ( $m$ ) are sampled randomly from the current risk set.
Case-Cohort (CC): A static subcohort of size $m$ is sampled once at baseline. At each event time, the risk set consists of the event case plus the members of the subcohort who remain at risk.

The framework is implemented using a feed-forward neural network (three hidden layers with ReLU activations, dropout regularization) trained via the Adam optimizer. The evaluation covers two distinct training regimes:

Full-Batch: The entire dataset is used for each gradient update.
Mini-Batch: Data is partitioned into batches; risk sets are restricted to individuals present within the current batch. For CC in mini-batch, the risk set is the intersection of the fixed subcohort and the current batch, plus the event case.

Experiments were conducted on:

Synthetic Data: Generated with known ground-truth latent risk structures ( $N=6,000$ ) to evaluate the recovery of the true linear predictor and latent risk distribution.
Real-World Data: The METABRIC breast cancer dataset ( $N \approx 1,800$ ) to assess performance on complex clinical data.

Performance was measured using the Concordance Index (C-index), Integrated Brier Score (IBS), Pearson and Spearman correlations (for synthetic data), and computational resource utilization (CPU and memory).

Key Contributions
To the best of the authors' knowledge, this study provides the first integration and systematic evaluation of Case-Cohort risk-set sampling within deep Cox proportional hazards neural networks. The specific contributions include:

Novel Integration: Adapting the classical Case-Cohort design into deep neural network implementations of the Cox model, addressing a gap in deep survival learning frameworks.
Training Formulation: Developing a training formulation that incorporates Case-Cohort sampling within both full-batch and mini-batch stochastic optimization, enabling scalable learning.
Systematic Comparison: Providing the first empirical comparison of full risk set, NCC, and CC strategies within deep Cox models, analyzing the trade-offs between computational cost and predictive accuracy.

Results

Full-Batch Training:
- NCC: Consistently approximated the full risk-set baseline across all configurations. In synthetic data, it achieved C-index values (~0.78) nearly identical to the baseline, with log-hazard correlations approaching 0.99.
- CC: Showed high sensitivity to subcohort size ( $k$ ). Severe down-sampling ( $k=5$ ) resulted in a significant performance drop (C-index reduction of ~23% relative to baseline) and poor recovery of the latent risk structure. However, increasing the subcohort size (e.g., to 1,000 in synthetic data or 500 in METABRIC) mitigated overfitting and recovered baseline performance, though at the cost of increased training time.
Mini-Batch Training:
- NCC: Remained robust, tracking the full-risk baseline with high stability.
- CC: Mini-batch execution substantially reduced the degradation observed in full-batch CC. A small subcohort ( $k=5$ ) that performed poorly in full-batch achieved a C-index of ~0.738 (only a ~5.3% drop from baseline) in mini-batch. Increasing the subcohort size to 15 or higher allowed CC to recover near-baseline C-index and IBS scores.
- Resource Efficiency: Mini-batching drastically reduced peak memory usage (from ~6 GB in full-batch to ~0.5 GB in mini-batch). While increasing the subcohort size in mini-batch did not significantly increase memory usage, it did increase wall-clock training time.
Risk Structure Recovery: In synthetic experiments, NCC consistently preserved the latent risk structure. CC with small subcohorts struggled to recover the true risk ranking in full-batch but showed marked improvement in mini-batch settings.

Significance and Claims
The paper claims that risk-set sampling can provide substantial computational savings in deep Cox models without sacrificing predictive performance, provided the sampling strategy is chosen appropriately.

NCC is presented as a robust method offering consistent performance across sampling configurations.
Mini-batch CC is identified as an effective alternative when the subcohort size is appropriately chosen, offering a balance between computational efficiency and statistical accuracy.
The study highlights that CC sampling is particularly valuable when the underlying data naturally arises from a case-cohort study design, offering a natural analytical strategy for such data.
The findings suggest that while full-batch CC requires large subcohorts to avoid performance degradation, the stochastic nature of mini-batch training allows CC to synthesize global risk dynamics from far fewer samples, achieving full predictive capacity at a fraction of the computational cost.

The authors conclude that their framework extends the classical Cox partial likelihood to modern large-scale learning, recovering exact formulations in full-batch settings while integrating efficient sampling designs for scalable training.

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