Sampled on outcome, still predictable: a weighted gradient boosting machine matches Cox on uranium miner cancer data

Studies that sample on outcome can now train survival prediction models directly instead of treating the sampled data as a simple random sample. Prior methodological work on these designs focused on parameter estimation and inference for proportional hazards, additive hazards, Cox and accelerated failure time models, while survival prediction received limited attention. Simulations cover exponential and Weibull failure times, linear and nonlinear covariate effects, and dimensions 10, 200 and 400, with 500 replications per scenario, compared against a Cox model under the same sampling design.

The method places sampling weights inside each boosting round, so the model optimizes toward the target cohort rather than the sampled dataset. The weights follow an existing weight function for this sampling design, and each boosting round computes a per-covariate update by weighted least squares, scaled by a learning rate. Hyperparameters are chosen by minimizing a cross-validated weighted empirical integrated Brier risk, with the learning rate fixed at 0.1 and the number of iterations searched over a grid from 50 to 2000.

On the uranium miner cancer data, the weighted boosting machine and the Cox model gave nearly identical predictions. That application is relatively low-dimensional, and the covariate-risk relationships appear well approximated by the Cox model's linear log-hazard assumption, so comparable performance is expected. The full cohort holds 16,434 miners with 2,330 cancers, 1,444 of the type of interest, and a subcohort of 1,839; with a 60:40 train-test split the test C-index was 0.803 versus 0.801.

The method targets survival prediction under general failure-time outcome-dependent sampling, for epidemiological and biomedical cohorts where covariate collection is cost-limited and predictions must generalize to the target cohort. In the simulations the subcohort and supplemental cases were drawn by Bernoulli sampling, matching the assumptions behind the convergence result; the authors also ran a comparison with fixed-size sampling and found prediction performance nearly indistinguishable between the two settings. The authors propose extending the work to time-varying covariates, other outcome-dependent sampling designs, and reinforcement learning for survival data.

The theoretical result holds under regularity conditions, including Bernoulli sampling of the subcohort and of supplemental cases within each stratum, and a uniformly consistent censoring distribution estimate. In the real-data application the two models predicted almost identically, which the authors attribute to the relatively low dimensionality and the adequacy of the linear effect assumption there, so that application does not show the boosting machine's relative advantage. The authors also note that excessive boosting iterations can overfit and degrade prediction accuracy.