Biomarker assessment in ROC curve analysis using the length of the curve as an index of diagnostic accuracy: the binormal model framework

Abstract

In receiver operating characteristic (ROC) curve analysis, the area under the curve (AUC) is undoubtedly the most widely used index of diagnostic accuracy for the assessment of the utility of a biomarker or for the comparison of competing biomarkers. Along with the AUC, the maximum of the Youden index, J, is often used both as an index of diagnostic accuracy and as a tool useful for the estimation of an optimal cutoff point that can be used for diagnostic purposes based on the biomarker under consideration. In this work, we study the utility of the length of the binormal model-based ROC curve (LoC) as an index of diagnostic accuracy for biomarker evaluation. Estimation procedures for LoC, described in this article, are based either on normality assumptions or on the same assumptions after a Box–Cox transformation to normality. Two simulation studies are considered. In the first, the estimation procedures for LoC are compared in terms of bias and root mean squared error, while in the second one, the performance of LoC is compared with approaches based on AUC and J, both for the case of the assessment of a single biomarker and for the comparison of two biomarkers, in a parametric framework. We provide an interpretation for the proposed index and illustrate with an application on biomarkers from a colorectal cancer study.

Appendix

1.1 A. Derivation of Eq. (3)

Based on the standard formula for the length of a curve, for $y=\hbox {ROC}(p)$ the arc length is given by:

$$\begin{aligned} \int _{0}^{1}\sqrt{1+[\hbox {ROC}'(p)]^2}dp \end{aligned}$$

Therefore, Eq. (3) is obtained differentiating the binormal ROC curve, $\hbox {ROC}(p)=\Phi (a+b\Phi ^{-1}(p))$, which is given by

$$\begin{aligned} \hbox {ROC}'(p)= & {} \phi [a + b \Phi ^{-1}(p)]\times \frac{d}{dp}b \Phi ^{-1}(p) \\= & {} \frac{b\phi [a+b\Phi ^{-1}(p)]}{\phi [\Phi ^{-1}(p)]}=b\frac{e^{-\frac{1}{2}[a+b\Phi ^{-1}(p)]^2}}{e^{-\frac{1}{2}[\Phi ^{-1}(p)]^2}} \end{aligned}$$

where $\phi$ is the density function of a standard normal, $a =\left( \mu _{Y}-\mu _{X}\right) /\sigma _{Y}$ and $b =\dfrac{\sigma _{X}}{\sigma _{Y}}$.

The exponential function is always greater than zero, and hence, the fraction is always greater than zero. By definition, a standard deviation is greater than zero; therefore, $b=\sigma _1/\sigma _2 >0$. The product of two components that are always greater than zero must itself be greater than zero. So the slope of the binormal ROC curve is always positive for p in the interval (0, 1).

1.2 B. Estimation of L without the assumption of normality for the underlying distributions

For lognormal and gamma distributions, we obtain the ROC derivative from (5). However, for $X^{-1/3}\sim N\left( \mu _{X},\sigma _{X}^{2}\right)$ and $Y^{-1/3}\sim N\left( \mu _{Y},\sigma _{Y}^{2}\right)$, we have obtained the explicit formula:

Taking into account that

$$\begin{aligned} F_{X}(x)=\Phi \left( \frac{x^{-1/3}-\mu _{X}}{\sigma _{X}}\right) \end{aligned}$$

then

$$\begin{aligned} F_{X}^{-1}(1-t)=\left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}. \end{aligned}$$

Substituting this last expression in (5), we have

$$\begin{aligned} \hbox {ROC}^{\prime }\left( t\right)= & {} \frac{f_{Y}\left( \left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}\right) }{f_{X}\left( \left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}\right) }= \\= & {} \frac{\phi \left( \dfrac{\left( \left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}\right) ^{-1/3}-\mu _{Y}}{\sigma _{Y}} \right) }{3\sigma _{Y}\left( \left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}\right) ^{4/3}}\\&\quad \frac{3\sigma _{X}\left( \left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}\right) ^{4/3}}{\phi \left( \dfrac{\left( \left( \mu _{X}+\sigma _{X}\Phi ^{-1}\left( t\right) \right) ^{-3}\right) ^{-1/3}-\mu _{X}}{\sigma _{X}}\right) } \\= & {} b \frac{\phi \left( -a +b \Phi ^{-1}\left( t\right) \right) }{ \phi \left( \Phi ^{-1}\left( t\right) \right) }. \end{aligned}$$

1.3 B. R-code for the basic calculation of L