Accurate training of the Cox proportional hazards model on vertically-partitioned data while preserving privacy

Background Analysing distributed medical data is challenging because of data sensitivity and various regulations to access and combine data. Some privacy-preserving methods are known for analyzing horizontally-partitioned data, where different organisations have similar data on disjoint sets of people. Technically more challenging is the case of vertically-partitioned data, dealing with data on overlapping sets of people. We use an emerging technology based on cryptographic techniques called secure multi-party computation (MPC), and apply it to perform privacy-preserving survival analysis on vertically-distributed data by means of the Cox proportional hazards (CPH) model. Both MPC and CPH are explained. Methods We use a Newton-Raphson solver to securely train the CPH model with MPC, jointly with all data holders, without revealing any sensitive data. In order to securely compute the log-partial likelihood in each iteration, we run into several technical challenges to preserve the efficiency and security of our solution. To tackle these technical challenges, we generalize a cryptographic protocol for securely computing the inverse of the Hessian matrix and develop a new method for securely computing exponentiations. A theoretical complexity estimate is given to get insight into the computational and communication effort that is needed. Results Our secure solution is implemented in a setting with three different machines, each presenting a different data holder, which can communicate through the internet. The MPyC platform is used for implementing this privacy-preserving solution to obtain the CPH model. We test the accuracy and computation time of our methods on three standard benchmark survival datasets. We identify future work to make our solution more efficient. Conclusions Our secure solution is comparable with the standard, non-secure solver in terms of accuracy and convergence speed. The computation time is considerably larger, although the theoretical complexity is still cubic in the number of covariates and quadratic in the number of subjects. We conclude that this is a promising way of performing parametric survival analysis on vertically-distributed medical data, while realising high level of security and privacy.