Estimation of smooth ROC curves for biomarkers with limits of detection

Abstract

Protein biomarkers found in plasma are commonly used for cancer screening and early detection. Measurements obtained by such markers are often based on different assays that may not support detection of accurate measurements due to a limit of detection. The ROC curve is the most popular statistical tool for the evaluation of a continuous biomarker. However, in situations where limits of detection exist, the empirical ROC curve fails to provide a valid estimate for the whole spectrum of the false positive rate (FPR). Hence, crucial information regarding the performance of the marker in high sensitivity and/or high specificity values is not revealed. In this paper, we address this problem and propose methods for constructing ROC curve estimates for all possible FPR values. We explore flexible parametric methods, transformations to normality, and robust kernel-based and spline-based approaches. We evaluate our methods though simulations and illustrate them in colorectal and pancreatic cancer data.

Keywords: Box-Cox; ROC; biomarker; cancer; censoring; classification; early detection; generalized gamma; kernels; limit of detection; spline.

Figure 1

Simulated data set of two normal distributions generated to attain true AUC=0.8 and expected level of censoring 50% due to a lower LOD. Left panel: Box-Cox (BC) and kernel(BC) based ROCs along with the true and the naive empirical. Right panel: Generalized gamma (GG), kernel(GG), and spline based ROCs along with the true and the naive empirical.

Figure 2

Figure (2a): ROC curves for CEA that refers to colon cancer. AUCs for the Box-Cox, Kernel(BC), HCNS (spline) and naive empirical are 0.699, 0.686, 0.744 and 0.6566 respectively. Figure (2b): ROC curves for CA19-9 that refers to pancreatic cancer. AUCs for the Box-Cox, Kernel(BC), GG, kernel(GG), spline and naive empirical are 0.8577, 0.8474, 0.8657, 0.8569, 0.8627, 0.8428. The corresponding AUC for the empirical when using all available data is 0.8651. We note that even though it seems so, the GG based ROC estimate does not exhibit a step on the right panel. It simply has a very steep curvature close to TPR = 0.7.