By matching demographic or even genetic conditions, randomized clinical trials have often assumed that patients in different arms have same baseline conditions. In reality, however, it is not possible to make sure that subjects are in an identical baseline health state at entry time, for example, with healthy infants at birth or young adolescents before they engage in damaging behaviors (such as smoking). As subjects age, their conditions tend to diverge because of varying exposures, behaviors, experiences, and the like, that may or may not be measured or recorded. Event times are often of interest in contexts where individuals enter a study at different ages and in different health conditions and risks. Bias for late entry data in survival analysis has been investigated by many. In this article, we extend the above considerations using first-hitting-time (FHT) based threshold regression (TR) models. Unlike the proportional hazards methodology which ignored patient's baseline information in fitting the model, the TR model has a regression for patient's initial health condition and hence is well suited for analyzing late entry data.