Indiana University-Purdue University Indianapolis (IUPUI) / In many longitudinal studies, the time scales upon which we assess the primary outcomes are anchored by pre-specified events. However, these anchoring events are often not observable and they are randomly distributed with unknown distribution. Without direct observations of the anchoring events, the time scale used for analysis are not available, and analysts will not be able to use the traditional longitudinal models to describe the temporal changes as desired. Existing methods often make either ad hoc or strong assumptions on the anchoring events, which are unveri able and prone to biased estimation and invalid inference. Although not able to directly observe, researchers can often ascertain an interval that includes the unobserved anchoring events, i.e., the anchoring events are interval censored. In this research, we proposed a two-stage method to fit commonly used longitudinal models with interval censored anchoring events. In the first stage, we obtain an estimate of the anchoring events distribution by nonparametric method using the interval censored data; in the second stage, we obtain the parameter estimates as stochastic functionals of the estimated distribution. The construction of the stochastic functional depends on model settings. In this research, we considered two types of models. The first model was a distribution-free model, in which no parametric assumption was made on the distribution of the error term. The second model was likelihood based, which extended the classic mixed-effects models to the situation that the origin of the time scale for analysis was interval censored. For the purpose of large-sample statistical inference in both models, we studied the asymptotic properties of the proposed functional estimator using empirical process theory. Theoretically, our method provided a general approach to study semiparametric maximum pseudo-likelihood estimators in similar data situations. Finite sample performance of the proposed method were examined through simulation study. Algorithmically eff- cient algorithms for computing the parameter estimates were provided. We applied the proposed method to a real data analysis and obtained new findings that were incapable using traditional mixed-effects models. / 2 years