  To assess the classification accuracy of a continuous diagnostic result, the
receiver operating characteristic (ROC) curve is commonly used in applications.
The partial area under the ROC curve (pAUC) is one of widely accepted summary
measures due to its generality and ease of probability interpretation. In the
field of life science, a direct extension of the pAUC into the time-to-event
setting can be used to measure the usefulness of a biomarker for disease
detection over time. Without using a trapezoidal rule, we propose nonparametric
estimators, which are easily computed and have closed-form expressions, for the
time-dependent pAUC. The asymptotic Gaussian processes of the estimators are
established and the estimated variance-covariance functions are provided, which
are essential in the construction of confidence intervals. The finite sample
performance of the proposed inference procedures are investigated through a
series of simulations. Our method is further applied to evaluate the
classification ability of CD4 cell counts on patient's survival time in the
AIDS Clinical Trials Group (ACTG) 175 study. In addition, the inferences can be
generalized to compare the time-dependent pAUCs between patients received the
prior antiretroviral therapy and those without it.
