  Motivated by a study of acute kidney injury, we consider the setting of
biomarker studies involving patients at multiple centers where the goal is to
develop a biomarker combination for diagnosis, prognosis, or screening. As
biomarker studies become larger, this type of data structure will be
encountered more frequently. In the presence of multiple centers, one way to
assess the predictive capacity of a given combination is to consider the
center-adjusted AUC (aAUC), a summary of the ability of the combination to
discriminate between cases and controls in each center. Rather than using a
general method, such as logistic regression, to construct the biomarker
combination, we propose directly maximizing the aAUC. Furthermore, it may be
desirable to have a biomarker combination with similar performance across
centers. To that end, we allow for penalization of the variability in the
center-specific AUCs. We demonstrate desirable asymptotic properties of the
resulting combinations. Simulations provide small-sample evidence that
maximizing the aAUC can lead to combinations with improved performance. We also
use simulated data to illustrate the utility of constructing combinations by
maximizing the aAUC while penalizing variability. Finally, we apply these
methods to data from the study of acute kidney injury.
