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.

BACKGROUND: The follow-up rate, a standard index of the completeness of follow-up, is important for assessing the validity of a cohort study. A common method for estimating the follow-up rate, the \"Percentage Method\", defined as the fraction of all enrollees who developed the event of interest or had complete follow-up, can severely underestimate the degree of follow-up. Alternatively, the median follow-up time does not indicate the completeness of follow-up, and the reverse Kaplan-Meier based method and Clark\'s Completeness Index (CCI) also have limitations.
METHODS: We propose a new definition for the follow-up rate, the Person-Time Follow-up Rate (PTFR), which is the observed person-time divided by total person-time assuming no dropouts. The PTFR cannot be calculated directly since the event times for dropouts are not observed. Therefore, two estimation methods are proposed: a formal person-time method (FPT) in which the expected total follow-up time is calculated using the event rate estimated from the observed data, and a simplified person-time method (SPT) that avoids estimation of the event rate by assigning full follow-up time to all events. Simulations were conducted to measure the accuracy of each method, and each method was applied to a prostate cancer recurrence study dataset.
RESULTS: Simulation results showed that the FPT has the highest accuracy overall. In most situations, the computationally simpler SPT and CCI methods are only slightly biased. When applied to a retrospective cohort study of cancer recurrence, the FPT, CCI and SPT showed substantially greater 5-year follow-up than the Percentage Method (92%, 92% and 93% vs 68%).
CONCLUSIONS: The Person-time methods correct a systematic error in the standard Percentage Method for calculating follow-up rates. The easy to use SPT and CCI methods can be used in tandem to obtain an accurate and tight interval for PTFR. However, the FPT is recommended when event rates and dropout rates are high.

Person-time incidence rates are frequently used in medical research. However, standard estimation theory for this measure of event occurrence is based on the assumption of independent and identically distributed (iid) exponential event times, which implies that the hazard function remains constant over time. Under this assumption and assuming independent censoring, observed person-time incidence rate is the maximum-likelihood estimator of the constant hazard, and asymptotic variance of the log rate can be estimated consistently by the inverse of the number of events. However, in many practical applications, the assumption of constant hazard is not very plausible. In the present paper, an average rate parameter is defined as the ratio of expected event count to the expected total time at risk. This rate parameter is equal to the hazard function under constant hazard. For inference about the average rate parameter, an asymptotically robust variance estimator of the log rate is proposed. Given some very general conditions, the robust variance estimator is consistent under arbitrary iid event times, and is also consistent or asymptotically conservative when event times are independent but nonidentically distributed. In contrast, the standard maximum-likelihood estimator may become anticonservative under nonconstant hazard, producing confidence intervals with less-than-nominal asymptotic coverage. These results are derived analytically and illustrated with simulations. The two estimators are also compared in five datasets from oncology studies.