Guo, Guang (1993). Event-History Analysis for Left-Truncated Data. Sociological Methodology, 23
A subject is left-truncated when it comes under observation after having been exposed to the risk of an event for some time. Left-truncated subjects tend to have lower risks at shorter durations than those in a normal sample because high-risk subjects tend to experience the event and drop out before reaching the point at which observation begins. When start times are unknown, left truncation is practically intractable unless the hazard rate is constant or all left-truncated subjects are discarded. When start times are known, left-truncated data can be handled by the conditional likelihood approach. The critical information on the start time of a left-truncated subject can be frequently obtained from human subjects. The concentration of covariate information in the observation period in a longitudinal social survey is well-suited for the conditional approach. We show that a piece-wise exponential hazard model and a discrete-time model based on the conditional likelihood approach can be readily estimated by extant packages like SAS. We also describe an alternative conditional partial likelihood approach. An empirical example of marital dissolution in the United States is provided.