Transforming cumulative hazard estimates

Time to event outcomes are often evaluated on the hazard scale, but interpreting hazards may be difficult. Recently, there has been concern in the causal inference literature that hazards actually have a built in selection-effect that prevents simple causal interpretations. This is even a problem in randomized controlled trials, where hazard ratios have become a standard measure of treatment effects. Modeling on the hazard scale is nevertheless convenient, e.g. to adjust for covariates. Using hazards for intermediate calculations may therefore be desirable. Here, we provide a generic method for transforming hazard estimates consistently to other scales at which these built in selection effects are avoided. The method is based on differential equations, and generalize a well known relation between the Nelson-Aalen and Kaplan-Meier estimators. Using the martingale central limit theorem we also find that covariances can be estimated consistently for a large class of estimators, thus allowing for rapid calculations of confidence intervals. Hence, given cumulative hazard estimates based on e.g. Aalen's additive hazard model, we can obtain many other parameters without much more effort. We present several examples and associated estimators. Coverage and convergence speed is explored using simulations, suggesting that reliable estimates can be obtained in real-life scenarios.