Survival stacking is a method that transforms a right-censored survival dataset into a classification dataset. This procedure is detailed in (4); here, we provide a brief overview.

Given a survival dataset D=(Zi,ti,δi), the goal is to construct a classification dataset D′. The resulting dataset consists of a matrix of independent variables X and a target vector y containing boolean outcomes. The matrix X includes all covariates from D, along with additional columns representing risk set indicators corresponding to each event time point ti where δi=1. D′ is built iteratively, by “stacking” the risk sets at the different failure times. More specifically, for each event time point ti where δi=1, the procedure is as follows: 1.

Identify the risk set R(ti), which includes all subjects still under observation at time ti.

Federated learning of logistic regression enables multiple parties to collaboratively train a logistic regression model without sharing their local data. This decentralized approach preserves privacy while allowing the computation of a global model across distributed datasets. In the following, we outline the basic procedure for federated logistic regression.

Consider K parties, each holding a local dataset Dk=(Xk,yk), where Xk represents the feature matrix and yk is the binary target variable. The goal is to jointly estimate the logistic regression parameters β by minimizing the following loss function:L(β)=∑k=1K∑i=1nk[ykilog⁡(yki′)−(1−yki)log⁡(1−yki′)],where 1.

nk is the number of subjects in Dk

We can combine survival stacking with federated learning to collaboratively fit a Cox proportional hazards model without sharing raw data. Each participating party independently applies survival stacking to its dataset. Subsequently, the parties engage in a federated logistic regression procedure to fit a logistic regression model on their stacked datasets. The resulting logistic regression coefficients are approximations of the Cox model coefficients. To perform survival stacking, each party must know the time points of the failures. In some cases, these might be considered sensitive data. A solution to this problem is to collaboratively compute the highest time point [by sharing the maximum time point, or using multi -party computation (5)]. Next, we split the time frame into a predetermined number of time points. For each time point, we take the risk set at that point and set the values to 1 for subjects who experience a failure close to that time frame (6). This also reduces the size of the stacked dataset, which can result in more accurate models. In Section 3, we also demonstrate that this method effectively estimates the Cox model coefficients in a federated setting. The code includes the computation of several statistics, such as the Wald statistic and p-values.

We describe our novel approach to securely compute the Schoenfeld residuals in a federated setting. We assume that every party has access to the trained model coefficients β, as well as their own data. The protocol consists of four steps: preprocessing, sharing of failure times, precomputation, and the MPC calculation. 1.

Preprocessing: each dataset contains one event per row. An event can be either a failure or a censoring. Every row also contains the time of the event and the individual’s covariates. In this step, each party perturbs all of their event times by a small random value. This is only done to ensure that there are no duplicate times in the complete dataset. Then the party’s data is sorted by the new perturbed event times.

With this approach, the parties can securely validate the federated Cox model. This is very useful, as this validation is almost always done in practice, but doing it in the clear would nullify the privacy enhancement of the federated training. Hence, being able to do both parts securely makes the whole model more valuable.

As mentioned, the standard approach in evaluation Schoenfeld residuals is to plot them and inspect them visually. As an example, we plot the residuals for the age covariate of the Rotterdam Tumor Bank dataset in Figure 1, as computed by the approach described above. We see that the residuals do not show a clear time dependence in this case, also indicated by the fact that the fitted line is close to the constant line through zero. This indicates that indeed the residuals are independently and identically distributed around zero and so the Proportional Hazards Assumption holds. However, we note that this is often a quite subjective measurement and is difficult to quantify exactly.

We begin by comparing the effectiveness of different optimization methods for fitting a logistic regression model on stacked data. Survival stacking significantly increases the size of the dataset, resulting in a high proportion of zero values in the risk set indicator columns. Furthermore, the target vector contains very few positive labels, leading to a sparse dataset with a highly imbalanced class distribution. As a consequence, a model that sets all coefficients to zero already achieves a high accuracy score. To assess the performance of our approach, we therefore compare it against the coxph implementation in the lifelines software package (10). We compare with a reference implementation, instead of using scores like a C-index or calibration plots, as evaluating the accuracy of a Cox model purely through numerical metrics is challenging and often requires contextual and domain knowledge. By comparing our approach to established reference implementations, we can better quantify any inaccuracies introduced by survival stacking relative to directly applying a Cox model.

The results of our evaluation on two datasets are presented in Tables 1 and 2. Each experiment was repeated 10 times until convergence and the results were subsequently averaged.

The findings indicate that both the Adam optimizer and the Newton–Raphson method converge to values close to the reference Cox parameters. Adam requires a greater number of communication rounds, whereas Newton–Raphson involves transmitting the Hessian matrix, which can be large. The choice between these methods thus represents a trade-off between the frequency and the size of communication. This trade-off largely depends on the size of the Hessian matrix, which is primarily influenced by the number of time bins used in the analysis.

We also evaluate the accuracy of our survival stacking approach, first in a centralized setting and then in a federated one. Rather than evaluating our implementation using independent measures, we benchmark our results against reference implementations in lifelines and R.

A key factor influencing the accuracy of survival stacking is the number of time bins used. In the original stacking approach, a new stack was added for each failure event. However, this strategy can lead to excessively large models, which is impractical, particularly in the context of federated learning. To address this issue, we employ time binning, as described in Section 2.2.3. We therefore evaluate the performance of survival stacking across different numbers of time bins. The results for various datasets are presented in Tables 3 and 4.

We see that more time bins lead to better results, up to a certain value. Introducing more time bins beyond that point leads to a more sparse dataset, without adding much more information. For both the Rotterdam and Colon dataset, the optimal number of time bins was 100. The optimal number of bins also depends on the size of the dataset. Interestingly, time-binning can result in models closer to the reference Cox model than the original stacking method.

In this experiment, we evaluate the advantages of federated learning over training a Cox model locally. We use the colon dataset (11) and split the dataset uniformly at random, so that every party has nearly the same number of data points. To ensure robustness, we repeat the experiment 10 times with different splits. The results are in Table 5. Examples are also shown in Figure 2, which presents partial results visually.

In Table 5, we assess the benefits of federated learning in terms of model accuracy, by performing a comprehensive comparison. This table presents the accuracy loss of the federated models alongside that of individually trained models. To ensure a fair comparison, each pair of federated and individual models (corresponding to the same row and number of parties) is trained on identical data partitions.

We measure the accuracy of a model by comparing it to the ideal central Cox model (trained on all of the combined data) and take the accuracy loss to be the Manhattan distance between the coefficients of both models. That is, let β∗=(β1∗,…,βm∗) be the optimal coefficients of the central Cox model, and consider some other model with coefficients β=(β1,…,βm), then we define that model’s accuracy loss as∑i=1m|βi−βi∗|.A lower value corresponds to a better approximation of the optimal model. We opt for this approach since we wish to assess how well we can fit a Cox model in a federated setting. Therefore, we compare with a reference implementation, instead of using scores like a C-index or calibration plots. Additionally, evaluating the accuracy of a Cox model purely through numerical metrics is challenging and often requires contextual and domain knowledge.

Our findings show that federated models consistently yield results significantly closer to those of the central Cox model than their individually trained counterparts. Although some variance is observed, primarily due to randomness in data partitioning, the accuracy loss in the individual models is approximately 5 to 8 times greater than that of the corresponding federated models. Moreover, for both federated and individual models, accuracy loss increases with the number of parties. This is likely due to the fixed size of the total dataset: as n increases, the data available to each individual party decreases, making it more challenging to fit accurate models. However, in a real-world scenario, adding more parties would typically contribute additional data rather than merely redistributing a fixed dataset. As a result, we expect that increasing the number of parties in practice would enhance model quality by incorporating more diverse information.

Furthermore, Figure 2 shows a particular example to illustrate the difference between the models. We see that in this case the individual parties’ best models are quite inaccurate and in most cases not comparable to the ideal centralized model. For instance, the model trained by Party 2 overestimates the coefficients for the covariates sex, obstruct, and perfor, while underestimating the coefficient for surg. Consequently, this misestimate leads to over- and underestimation of the effects of these covariates on survival probabilities, resulting in less reliable predictions. This highlights that models trained on a limited subset of the data generally perform significantly worse than those trained on the complete dataset. This demonstrates the clear benefits of collaborative approaches such as federated learning, which enable improved model performance without requiring data centralization.

We now evaluate the accuracy of the federated Cox regression model. Survival stacking and data federation can both lead to inaccuracies in the outcome model. Here we assess this loss in accuracy, by testing it on the colon dataset with 25 bins.

Figure 3 presents the results of the central Cox model and the central logistic regression model, both of which assume a fully centralized setting where all data is aggregated in a single location. Alongside these, it shows the outcomes of the federated Cox regression for varying numbers of participating parties. Note that the central Cox model is the same as in Figure 2. However, while Figure 2 examines individual Cox models trained by a single party on a fraction of the complete dataset (with a fixed number of parties, n=4), here we evaluate federated models trained across all n parties, where 2≤n≤5.

This figure indicates that the federated models closely approximate the centrally trained Cox model. Furthermore, the sources of error introduced by survival stacking and federated learning do not necessarily compound. For example, for the covariate sex, the central logistic regression model overestimates the coefficient relative to the central Cox regression model, whereas the federated regression model underestimates it relative to the central logistic regression model. This interaction results in a slight improvement in the overall estimation. The p-values give an indication of the error.

Our results demonstrate that federated models produce significantly more accurate predictions than models trained independently by individual parties. These findings underscore the benefits of collaborative model training, particularly in scenarios where direct data sharing is not feasible. Federated learning thus presents a viable approach for improving Cox model accuracy while preserving data privacy.