Censored functional data for incomplete follow-up studies.

Functional data analysis plays an increasingly important role in medical research because patients are followed over time. Thus, the measurements of a particular biomarker for each patient are often registered as curves. Hence, it is of interest to estimate the mean function under certain conditions as an average of the observed functional data over a given period. However, this is often difficult as this type of follow-up studies are confronted with the challenge of some individuals dropping-out before study completion. Therefore, for these individuals, only a partial functional observation is available. In this study, we propose an estimator for the functional mean when the functions may be censored from the right, and thus, only partly observed. Unlike sparse functional data, the censored curves are observed until some (random) time and this censoring time may depend on the trajectory of the functional observations. Our approach is model-free and fully nonparametric, although the proposed methods can also be incorporated into regression models. The use of the functional structure of the data distinguishes our approach from the longitudinal data approaches. In addition, in this study, we propose a bootstrap-based confidence band for the mean function, examine the estimation of the covariance function, and apply our new approach to functional principal component analysis. Employing an extensive simulation study, we demonstrate that our method outperforms the only two existing approaches. Furthermore, we apply our new estimator to a real data example on lung growth, measured by changes in pulmonary function for girls in the United States.