Estimating  the receiver operating characteristic (ROC) curve has been an important problem in  diagnostic medicine, biometric recognition and others. In a variety of applications,  the data are always collected under two or more naturally ordered experimental conditions. So it is  natural  to assume a stochastic ordering for the observations under different experimental conditions. More importantly,  statistical inference incorporating such stochastic ordering condition is expected to improve efficiency. Clustered and correlated data occur when multiple measurements are gleaned from the same subject which makes the estimation of ROC curves more  complicated. Although there are some methods designed for the estimation of ROC curves from clustered data, but how to impose natural ordering on the estimation of ROC curves has not been studied yet.  In this article, we propose an ordered-restricted estimator for the ROC curve,  the area under the curve, and the partial area under the curve  to accommodate clustered and correlated data structure. We derive the asymptotic properties of the proposed order-restricted estimators and theoretically show that they possess lower mean-squared errors than the existing  estimators. Simulation studies are carried out to demonstrate better performance of the newly proposed estimators over existing methods for finite samples.