Tang Liansheng, Du Pang, Wu Chengqing
Department of Statistics, George Mason University, Fairfax, VA 22030, USA.
J Stat Plan Inference. 2010 Nov 1;140(11):3540-3551. doi: 10.1016/j.jspi.2010.05.026.
Receiver operating characteristic (ROC) curve, plotting true positive rates against false positive rates as threshold varies, is an important tool for evaluating biomarkers in diagnostic medicine studies. By definition, ROC curve is monotone increasing from 0 to 1 and is invariant to any monotone transformation of test results. And it is often a curve with certain level of smoothness when test results from the diseased and non-diseased subjects follow continuous distributions. Most existing ROC curve estimation methods do not guarantee all of these properties. One of the exceptions is Du and Tang (2009) which applies certain monotone spline regression procedure to empirical ROC estimates. However, their method does not consider the inherent correlations between empirical ROC estimates. This makes the derivation of the asymptotic properties very difficult. In this paper we propose a penalized weighted least square estimation method, which incorporates the covariance between empirical ROC estimates as a weight matrix. The resulting estimator satisfies all the aforementioned properties, and we show that it is also consistent. Then a resampling approach is used to extend our method for comparisons of two or more diagnostic tests. Our simulations show a significantly improved performance over the existing method, especially for steep ROC curves. We then apply the proposed method to a cancer diagnostic study that compares several newly developed diagnostic biomarkers to a traditional one.
受试者工作特征（ROC）曲线是诊断医学研究中评估生物标志物的重要工具，它绘制了随着阈值变化的真阳性率与假阳性率。根据定义，ROC曲线从0到1单调递增，并且对测试结果的任何单调变换都是不变的。当患病和未患病受试者的测试结果遵循连续分布时，它通常是一条具有一定平滑度的曲线。大多数现有的ROC曲线估计方法并不能保证所有这些特性。其中一个例外是Du和Tang（2009），他们将某种单调样条回归程序应用于经验ROC估计。然而，他们的方法没有考虑经验ROC估计之间的内在相关性。这使得渐近性质的推导非常困难。在本文中，我们提出了一种惩罚加权最小二乘估计方法，该方法将经验ROC估计之间的协方差纳入权重矩阵。由此得到的估计器满足上述所有特性，并且我们证明它也是一致的。然后使用重采样方法扩展我们的方法，用于比较两个或更多的诊断测试。我们的模拟结果表明，与现有方法相比，性能有显著提高，特别是对于陡峭的ROC曲线。然后，我们将所提出的方法应用于一项癌症诊断研究，该研究比较了几种新开发的诊断生物标志物与一种传统生物标志物。
