The analysis of retrospectively ascertained data in the presence of reporting delays

Suppose the progress of a disease consists of two chronologically ordered events, termed the starting event and the failure event. In retrospective sampling, the sampling scheme under which observations in the data set are identified retrospectively, individuals who experienced the starting event but not the failure event are excluded and thus are truncated from the data set, and only those who experienced both the starting event and the failure event before a given time are observed. The problem of reporting delays arises when some of the failure events are not reported before the given time and thus the corresponding cases also are excluded from the data set. In survival studies failure time data sometimes are collected under the retrospective sampling scheme subject to reporting delays. This article explores nonparametric and semiparametric methods of dealing with this type of data. The results generalize some existing nonparametric and semiparametric methods for analyzing right-truncated data when reporting delays are absent. Estimation of the expected number of events is studied in detail, interpretation of the proposed estimates is discussed, and an analysis of the blood transfusion data from the Centers for Disease Control is presented.