Using a Monotonic Density Ratio Model to Find the Asymptotically Optimal Combination of Multiple Diagnostic Tests

With the advent of new technology, new biomarker studies have become essential in cancer research. To achieve optimal sensitivity and specificity, one needs to combine different diagnostic tests. The celebrated Neyman–Pearson lemma enables us to use the density ratio to optimally combine different diagnostic tests. In this article, we propose a semiparametric model by directly modeling the density ratio between the diseased and nondiseased population as an unspecified monotonic nondecreasing function of a linear or nonlinear combination of multiple diagnostic tests. This method is appealing in that it is not necessary to assume separate models for the diseased and nondiseased populations. Further, the proposed method provides an asymptotically optimal way to combine multiple test results. We use a pool-adjacent-violation-algorithm to find the semiparametric maximum likelihood estimate of the receiver operating characteristic (ROC) curve. Using modern empirical process theory we show cubic root n consistency for the ROC curve and the underlying Euclidean parameter estimation. Extensive simulations show that the proposed method outperforms its competitors. We apply the method to two real-data applications. Supplementary materials for this article are available online.