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\noindent Statistica Sinica {\bf 13}(2003), 593-612

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\centerline{\large\bf AUC-BASED TESTS FOR NONPARAMETRIC FUNCTIONS}
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\centerline{\large\bf WITH LONGITUDINAL DATA}
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\author{Yanqing Sun and Hulin Wu}
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\centerline{\it University of North Carolina and
University of Rochester}
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\begin{quotation}
\noindent {\it Abstract:}
   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.\par
\vspace{9pt}
\noindent {\it Key words and phrases:}
Censoring, confidence bands, fixed and random designs,  nonparametric maximum
deviation tests, nonparametric mixed-effects, one and two-sample problems.\par
\end{quotation}
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