Characterization of the Npmple of the Disease Onset Distribution Function for a Survival-sacrifice Model

In carcinogenicity experiments with animals where the tumor is not palpable it is common to observe only the time of death of the animal, the cause of death (the tumor or another independent cause, as sacrifice) and whether the tumor was present at the time of death. These last two indicator variables are evaluated after an autopsy. Defining the non-negative variables T1 (time of tumor onset), T2 (time of death from the tumor) and C (time of death from an unrelated cause), we observe (Y,∆1,∆2), where Y = min {T2, C}, ∆1 = 1{T1≤C}, and ∆2 = 1{T2≤C}, T1 and T2 have a joint distribution function F such that P (T1 ≤ T2) = 1, and are independent of C. Some authors call this model a “survival-sacrifice model”. The interest here is to estimate the marginal distribution functions F1 and F2 of T1 and T2, respectively (since F is not identifiable). One possible way of doing that is by using a consistent estimator F̂2 for F2 (Kaplan-Meier, for example) and then plugging it in the loglikelihood to obtain F̂1, the nonparametric maximum pseudo likelihood estimator (NPMPLE) of F1. A characterization theorem of F̂1 is stated here and an algorithm for its calculation is presented.