Nonparametric estimation of the ROC curve based on smoothed empirical distribution functions

Alicja Jokiel-Rokita,Michał Pulit

doi:10.1007/s11222-012-9340-x

Abstract

The receiver operating characteristic (ROC) curve is a graphical representation of the relationship between false positive and true positive rates. It is a widely used statistical tool for describing the accuracy of a diagnostic test. In this paper we propose a new nonparametric ROC curve estimator based on the smoothed empirical distribution functions. We prove its strong consistency and perform a simulation study to compare it with some other popular nonparametric estimators of the ROC curve. We also apply the proposed method to a real data set.

Highlights

The receiver operating characteristic (ROC) curve is commonly used to describe the accuracy of a medical or another diagnostic test, which classifies individuals into “healthy” and “diseased” categories
We prove that proposed estimator converges uniformly to the theoretical ROC curve, almost surely
The kernel estimator performs even worse than the empirical ROC curve, which is probably caused by the largest bias related to this estimator

Summary

Introduction

The receiver operating characteristic (ROC) curve is commonly used to describe the accuracy of a medical or another diagnostic test, which classifies individuals into “healthy” and “diseased” categories. Suppose that the independent real random variables X and Y denote the test score from healthy and diseased patients, respectively, and for a given cutoff point c, the test result is positive if it is greater than c. Let F and G be completely unknown distribution functions. There exist many different methods of estimating the ROC curve, but most of them are based on parametric or semiparametric models Pepe 2000; Qin and Zhang 2003; Davidov and Nov 2012). Yn) be simple independent random samples from the healthy and diseased populations, respectively. There exist several methods of estimating nonparametrically the ROC curve from such data. The commonly used nonparametric estimator is the empirical ROC curve of the form

Objectives

Results

Conclusion