Given expression (1), the relation between the hazard λ~(t) of the recorded event (observed hazard) and the hazard λ(t) of the true event (true hazard) follows (cf. Additional file 2: Web Appendix):

where the weight w(t) is defined as

The observed hazard is thus the sum of the hazard of recording the true event and the false-positive rate. The sensitivity se accounts for the possibility of not recording the true event. The weight w(t) equals the ratio of the true survival probability and the survival probability observed in absence of false positives and adjusts for the fact that a true but unrecorded event may have already removed the subject from the study’s risk set before time t. It holds that w(t)1.

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