A novel estimator for the two-way partial AUC

Abstract Background The two-way partial AUC has been recently proposed as a way to directly quantify partial area under the ROC curve with simultaneous restrictions on the sensitivity and specificity ranges of diagnostic tests or classifiers. The metric, as originally implemented in the R package, is estimated using a nonparametric estimator based on a trimmed Mann-Whitney U-statistic, which becomes computationally expensive in large sample sizes. (Its computational complexity is of order $$O(n_x n_y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>y</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$n_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> and $$n_y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>y</mml:mi> </mml:msub> </mml:math> represent the number of positive and negative cases, respectively). This is problematic since the statistical methodology for comparing estimates generated from alternative diagnostic tests/classifiers relies on bootstrapping resampling and requires repeated computations of the estimator on a large number of bootstrap samples. Methods By leveraging the graphical and probabilistic representations of the AUC, partial AUCs, and two-way partial AUC, we derive a novel estimator for the two-way partial AUC, which can be directly computed from the output of any software able to compute AUC and partial AUCs. We implemented our estimator using the computationally efficient R package, which leverages a nonparametric approach using the trapezoidal rule for the computation of AUC and partial AUC scores. (Its computational complexity is of order $$O(n \log n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , where $$n = n_x + n_y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>y</mml:mi> </mml:msub> </mml:mrow> </mml:math> .). We compare the empirical bias and computation time of the proposed estimator against the original estimator provided in the package in a series of simulation studies and on two real datasets. Results Our estimator tended to be less biased than the original estimator based on the trimmed Mann-Whitney U-statistic across all experiments (and showed considerably less bias in the experiments based on small sample sizes). But, most importantly, because the computational complexity of the proposed estimator is of order $$O(n \log n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , rather than $$O(n_x n_y)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>y</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , it is much faster to compute when sample sizes are large. Conclusions The proposed estimator provides an improvement for the computation of two-way partial AUC, and allows the comparison of diagnostic tests/machine learning classifiers in large datasets where repeated computations of the original estimator on bootstrap samples become too expensive to compute.