Finite sample and asymptotic distributions of a statistic for sufficient follow-up in cure models

The existence of immune or cured individuals in a population and whether there is sufficient follow-up in a sample of censored observations on their lifetimes to be confident of their presence are questions of major importance in medical survival analysis. Here we give a detailed analysis of a statistic designed to test for sufficient follow-up in a sample. Assuming an i.i.d. censoring model, we obtain exact finite-sample and asymptotic distributions for the statistic, and use these to calculate the power of a test based on it. A particularly useful finding is that the asymptotic distribution of the test statistic is parameter-free in the null case when follow-up is insufficient. The methods are illustrated with application to a glioma cancer dataset.