Evaluation of Repeated Biomarkers: Non-parametric Comparison of Areas under the Receiver Operating Curve Between Correlated Groups Using an Optimal Weighting Scheme

Receiver Operating Characteristic (ROC) curves are often used to evaluate the prognostic performance of a continuous biomarker. In a previous research, a non-parametric ROC approach was introduced to compare two biomarkers with repeated measurements. An asymptotically normal statistic, which contains the subject-specific weights, was developed to estimate the areas under the ROC curve of biomarkers. Although two weighting schemes were suggested to be optimal when the within subject correlation is 1 or 0 by the previous study, the universal optimal weight was not determined. We modify this asymptotical statistic to compare AUCs between two correlated groups and propose a solution to weight optimization in non-parametric AUCs comparison to improve the efficiency of the estimator. It is demonstrated how the Lagrange multiplier can be used as a strategy for finding the weights which minimize the variance function subject to constraints. We show substantial gains of efficiency by using the novel weighting scheme when the correlation within group is high, the correlation between groups is high, and/or the disease incidence is small, which is the case for many longitudinal matched case-control studies. An illustrative example is presented to apply the proposed methodology to a thyroid function dataset. Simulation results suggest that the optimal weight performs well with a sample size as small as 50 per group