Confidence intervals for the cumulative incidence function via constrained NPMLE

The cumulative incidence function (CIF) displays key information in the competing risks setting, which is common in medical research. In this article, we introduce two new methods to compute non-parametric confidence intervals for the CIF. First, we introduce non-parametric profile-likelihood confidence intervals. The method builds on constrained non-parametric maximum likelihood estimation (NPMLE), for which we derive closed-form formulas. This method can be seen as an extension of that of Thomas and Grunkemeier (J Am Stat Assoc 70:865-871, 1975) to the competing risks setting, when the CIF is of interest instead of the survival function. Second, we build on constrained NPMLE to introduce constrained bootstrap confidence intervals. This extends an interesting approach introduced by Barber and Jennison (Biometrics 52:430-436, 1999) to the competing risks setting. A simulation study illustrates how these methods can perform as compared to benchmarks implemented in popular software. The results suggest that more accurate confidence intervals than usual Wald-type ones can be obtained in the case of small to moderate sample sizes and few observed events. An application to melanoma data is provided for illustration purpose.