Smoothed empirical likelihood for the Youden index

Abstract

For a continuous scale biomarker of binary disease status, the Youden index is a frequently used measurement of diagnostic accuracy in the context of the receiver operating characteristic curve and provides an optimal threshold for making diagnosis. The majority of existing inference methods for the Youden index are either parametric or bootstrap based. In the current paper, the empirical likelihood method for the Youden index is derived via defining novel smoothed estimating equations, and Wilks’ theorem for the empirical likelihood ratio statistic is established. Extensive simulation studies suggest that the chi-square calibrated empirical likelihood interval estimators are robust to model assumptions, enjoy computational efficiency and perform better than the bootstrap procedure almost uniformly across a variety of scenarios in terms of coverage probabilities.

Introduction

In the past few decades, the receiver operating characteristic (ROC) curve has become a standard statistical tool to evaluate the discriminatory ability of a diagnostic test to separate diseased subjects from non-diseased subjects. For a diagnostic test with continuous scale, sensitivity (true positive rate) and specificity (true negative rate) are inversely related, in the sense that the increase of the one is accompanied with the decrease of the other as the cutoff point moves along the real number line. The ROC curve is the plot of sensitivity against 1-specificity at all possible threshold points. For comprehensive reviews of ROC analysis, see Shapiro (1999), Zhou et al. (2009), Pepe (2004), and Zou et al. (2010).

Even though the area under the ROC curve (AUC) has been widely used for measuring the accuracy of a diagnostic test, the Youden index has its unique advantage in advising an optimal cutoff for the clinicians to make diagnosis. The Youden index, firstly proposed by Youden (1950), is defined as the maximum of the sum of sensitivity and specificity minus one. The cutoff point, where the maximum is achieved, provides an optimal threshold for the clinicians to use the diagnostic test for classification if equal weight is placed on sensitivity and specificity. The possible values of the Youden index range from 0 to 1 with 0 indicating no discriminatory ability and 1 indicating perfect diagnostic accuracy. Graphically, the Youden index is the maximum vertical distance between the ROC curve and the diagonal chance line.

A variety of approaches have been developed for the inference of the Youden index. Fluss et al. (2005) provided several estimators for the Youden index and the associated cutoff points based upon normal assumption, empirical distribution function or kernel smoothing. For interval estimation of the Youden index, Schisterman and Perkins (2007) developed parametric methods under either normal or gamma assumption. Lai et al. (2012) made some improvement by utilizing a generalized variable approach. Even though monotonic transformation, such as Box–Cox transformation, can be applied, these parametric interval estimators may still be not satisfactory given the departures from distribution assumptions, particularly when the diseased and non-diseased populations are not from the same family of distributions. Nonparametric interval estimators of the Youden index are mainly developed from bootstrap. Based upon the Youden index estimators derived from different methods, including estimators under normal assumption, estimators from delta method, and estimators derived from empirical distribution function and kernel density estimation, a range of commonly used bootstrap methods, such as percentile, normal approximation and bias correction and acceleration (BCa) adjustment, have been considered by Faraggi (2003), Fluss et al. (2005) and Schisterman and Perkins (2007), respectively. More recently, Zhou and Qin (2012) proposed an adjusted bootstrap procedure via an approximate method for interval estimation of a single proportion introduced by Agresti and Coull (1998). Via extensive simulation studies, the authors suggested that their modified bootstrap method was comparable to parametric methods when distribution assumption holds unless the Youden index is close to upper boundary ($J\ge 0.90$) and their methods outperformed the previously developed bootstrap methods.

In this paper, we aim to propose a novel interval estimator of the Youden index via the empirical likelihood. Empirical likelihood, formally proposed by Owen, 1988, Owen, 1990, is an appealing nonparametric method with many desirable features such as automatic determination of the shape of the confidence regions by data, straightforward incorporation of side information and being Bartlett correctable in many cases; see Owen (2001) for a comprehensive review. Claeskens et al. (2003) derived empirical likelihood confidence regions for ROC curves over a certain range of specificity values. Molanes-López and Letón (2011) proposed an empirical likelihood approach for the Youden index and its associated optimal cutoff point from a quantile function point of view. As it is not ready to profile out the nuisance parameter, the authors had to propose a fairly complicated two-cycle bootstrap procedure and the resulting interval estimator seemed to be over-conservative as suggested by simulation studies. In this paper, we develop empirical likelihood based upon novel estimation equations using kernel smoothing methods.

The rest of the paper is organized as follows: In Section  2, the novel empirical likelihood method is introduced. We also establish the asymptotic properties of the empirical likelihood ratio statistic and discuss potential computation algorithms. In Section  3, we evaluate the empirical performance of our method through extensive simulation studies. We illustrate the proposed method in Section  4 via the application to a published data set. We draw conclusions and make discussions in Section  5.