  Results from multiple diagnostic tests are usually combined to improve the
overall diagnostic accuracy. For binary classification, maximization of the
empirical estimate of the area under the receiver operating characteristic
(ROC) curve is widely adopted to produce the optimal linear combination of
multiple biomarkers. In the presence of large number of biomarkers, this method
proves to be computationally expensive and difficult to implement since it
involves maximization of a discontinuous, non-smooth function for which
gradient-based methods cannot be used directly. Complexity of this problem
increases when the classification problem becomes multi-category. In this
article, we develop a linear combination method that maximizes a smooth
approximation of the empirical Hypervolume Under Manifolds (HUM) for
multi-category outcome. We approximate HUM by replacing the indicator function
with the sigmoid function or normal cumulative distribution function (CDF).
With the above smooth approximations, efficient gradient-based algorithms can
be employed to obtain better solution with less computing time. We show that
under some regularity conditions, the proposed method yields consistent
estimates of the coefficient parameters. We also derive the asymptotic
normality of the coefficient estimates. We conduct extensive simulations to
examine our methods. Under different simulation scenarios, the proposed methods
are compared with other existing methods and are shown to outperform them in
terms of diagnostic accuracy. The proposed method is illustrated using two real
medical data sets.
