AUC-based tests for nonparametric functions with longitudinal data

Longitudinal data are very common in biomedical and clinical research, for example, CD4+ cell responses and viral load responses in AIDS clinical research. It is challenging to do inference for the whole trajectory of these longitudinal data if a parametric function is not available to model the trajectories. In this paper we develop an area-under-the-curve (AUC) based nonparametric method to compare the two groups of longitudinal data under both fixed and random designs. The proposed test does not involve any smoothing. The method is also applicable to one-sample problems. The test statistic is based on the maximum deviation of the weighted averages of AUCs between two groups. The weight functions are used to account for censored or early drop-out subjects. For both cases that the number of measurements per subject goes to infinity and is finite, we show that the test statistic processes converge weakly to Gaussian processes, where for the case of the number of measurements per subject going to infinity, a nonparametric mixed-effects model is considered. A Monte Carlo method is developed to generate the distribution of test statistics. Simulations show that the test is valid and promising. We applied the test to compare CD4+ responses over time between two treatment groups in an AIDS clinical trial.