Data description is the first step for understanding the nature of the problem at hand. Usually, it is a simple task that does not require any particular assumption. However, the interpretation of the used descriptive measures can be a source of confusion and misunderstanding. The incidence rate is the quotient between the number of observed events and the sum of time that the studied population was at risk of having this event (person-time). Despite this apparently simple definition, its interpretation is not free of complexity. In this piece of research, we revisit the incidence rate estimator under right-censorship. We analyze the effect that the censoring time distribution can have on the observed results, and its relevance in the comparison of two or more incidence rates. We propose a solution for limiting the impact that the data collection process can have on the results of the hypothesis testing. We explore the finite-sample behavior of the considered estimators from Monte Carlo simulations. Two examples based on synthetic data illustrate the considered problem. The R code and data used are provided as Supplementary Material.

Dinse (Biometrics, 38:417-431, 1982) provides a special type of right-censored and masked competing risks data and proposes a non-parametric maximum likelihood estimator (NPMLE) and a pseudo MLE of the joint distribution function [Formula: see text] with such data. However, their asymptotic properties have not been studied so far. Under the extention of either the conditional masking probability (CMP) model or the random partition masking (RPM) model (Yu and Li, J Nonparametr Stat 24:753-764, 2012), we show that (1) Dinse\'s estimators are consistent if [Formula: see text] takes on finitely many values and each point in the support set of [Formula: see text] can be observed; (2) if the failure time is continuous, the NPMLE is not uniquely determined, and the standard approach (which puts weights only on one element in each observed set) leads to an inconsistent NPMLE; (3) in general, Dinse\'s estimators are not consistent even under the discrete assumption; (4) we construct a consistent NPMLE. The consistency is given under a new model called dependent masking and right-censoring model. The CMP model and the RPM model are indeed special cases of the new model. We compare our estimator to Dinse\'s estimators through simulation and real data. Simulation study indicates that the consistent NPMLE is a good approximation to the underlying distribution for moderate sample sizes.