Functional data are common in cancer studies and other biomedical research, such as biomarkers measured over time in cancer experiments and other clinical trials, growth curves, hormone profiles, circadian rhythms in biological signals and drug activities. Although much work has been done on functional models for independent data, extensions to incorporate complex designs and correlations are still very preliminary. The first specific aim of this application is to develop general functional models using smoothing splines that can incorporate complex designs and allow flexible nonparametric between-curve random effects. Another long-existing problem for functional models is the heavy computational demand. Except in very simple cases, most of the current estimation procedures need to invert large dimensional matrices. This prevents applications to large data sets. In this application, we will develop O(N) sequential estimation procedures for general functional models by modifications of the Kalman filtering and fixed interval smoothing. Serial measurements have become a natural part of patient monitoring and medical diagnosis. In monitoring and predicting a patient-specific outcome based on laboratory tests or other biomarkers, we can obtain more accurate predictions by borrowing the strength from the existing patient population profiles over time. In medical diagnosis, we can gain efficiency by using the up-to-date cumulative information and compare the individual profile with the existing group profiles. In this application, we will develop dynamic patient monitoring and diagnostic methods, in which flexible functional models will be used to model both the population and individual profiles. With the proposed sequential estimation procedures, these methods can be efficiently calculated and implemented in a real time setting, which leads to rapid medical interventions. Most current statistical inference procedures rely on the distributional assumptions, such as the normality assumption. When the distribution is multimodal, it is often difficult to make parametric assumptions, and therefore nonparametric density estimation methods are needed. In this application, we will develop general density models and their associated inference procedures, and apply these methods to accessible biomedical data sets.
