Maximizing an ROC-type measure via linear combinations of biomarkers

Abstract

The receiver operating characteristic curve is a popular tool to describe and compare the diagnostic accuracy of biomarkers when the binary-scale gold standard is available. There are, however, many examples of diagnostic tests whose gold standards are continuous. Hence, several extensions are proposed to evaluate the diagnostic potential of biomarkers when the gold standard is continuous-scale. In practice, there may exist more than one biomarkers and diagnostic accuracy can be improved by combining multiple biomarkers. In this paper, an explicit form of diagnostic accuracy index and the corresponding linear combination that maximizes the diagnostic accuracy are derived under elliptical distribution assumption. Simulations are conducted to evaluate the performance of the diagnostic accuracy and the optimal linear combination under different distribution assumptions. The methodology is applied to a behavioral intervention study for families of youth with type 1 diabetes.

Appendices

Appendix 1: Generating data from a multivariate lognormal distribution

We use the following scheme to generate a random sample from a multivariate lognormal distribution.

• (1). Generate a p-dimensional random vector $(Z_1,Z_2,\ldots Z_p)$ from a multivariate normal distribution with mean $\mathbf {\mu }=(\mu _1,..\mu _p)$ and variance-covariance matrix ${\varvec{\Sigma }}=(\sigma _{ij})p\times p$.

• (2). Transform the p-dimensional random vector to a new one $(X_1,X_2,\ldots ,X_{p-1},Y)$, where

  $$\begin{aligned} X_i&= {} e^{Z_i}, i=1,2,\ldots ,p-1; \\ Y&= {} e^{Z_p}. \end{aligned}$$

The vector $(X_1,X_2,\ldots ,X_{p-1},Y)$ is then a p-dimensional lognormal vector with mean $\mathbf {\mu _L}=e^{\mathbf {\mu }+\frac{1}{2}diag({\varvec{\Sigma }})}$ and the variance-covariance matrix

$$\begin{aligned} {\varvec{\Sigma _L}}=\begin{bmatrix} \varSigma _{11}&\varSigma _{12}&\cdots&\varSigma _{1p} \\ \varSigma _{21}&\varSigma _{22}&\cdots&\varSigma _{2p} \\ \vdots&\ddots&\vdots \\ \varSigma _{p1}&\varSigma _{p2}&\cdots&\varSigma _{pp} \end{bmatrix}, \end{aligned}$$

where $diag({\varvec{\Sigma }})$ is a vector of diagonal elements in ${\varvec{\Sigma }}$, and $\varSigma _{ij}=e^{\mu _i+\mu _j+\frac{1}{2}(\sigma _{ii}+\sigma _{jj})}(e^{\sigma _{ij}}-1),i,j=1,\ldots ,p.$

Appendix 2: Generating data from a multivariate gamma distribution

We propose a simple scheme to generate a data set from a multivariate gamma distribution. The algorithm proceeds as follows.

• (1). Generate $p+1$ independent random observations, $G_0,G_1,\ldots G_p$, from gamma distributions, such that

  $$\begin{aligned} G_i \sim Gamma(\alpha _i,\beta ) , i=0,1,\ldots ,p. \end{aligned}$$

• (2). Generate a p-dimensional random vector $(X_1,X_2,\ldots ,X_{p-1},Y)$, where

  $$\begin{aligned} X_i&= {} G_0+G_i, i=1,2,\ldots ,p-1; \\ Y&= {} G_0+G_1+\cdots +G_p. \end{aligned}$$

The vector $(X_1,X_2,\ldots ,X_{p-1},Y)$ is then a p-dimensional gamma vector, where the marginal distribution of each $X_i$ is $Gamma(\alpha _0+\alpha _i,\beta )$ and the marginal distribution of Y is $Gamma(\sum _{i=0}^p \alpha _i , \beta )$. The variance-covariance matrix is

$$\begin{aligned} {\varvec{\Sigma }}=\begin{bmatrix} \sigma ^2_1&\sigma ^2_x&\cdots&\sigma ^2_x&\sigma ^2_{1} \\ \sigma ^2_x&\sigma ^2_2&\cdots&\sigma ^2_x&\sigma ^2_{2} \\ \vdots&\ddots&\vdots \\ \sigma ^2_x&\cdots&\sigma ^2_x&\sigma ^2_{p-1}&\sigma ^2_{(p-1)}\\ \sigma ^2_{1}&\sigma ^2_{2}&\cdots&\sigma ^2_{(p-1)}&\sigma ^2_y \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned} \sigma ^2_i&= {} \frac{\alpha _0+\alpha _i}{\beta ^2},i=1,\ldots ,p-1 \\ \sigma ^2_x&= {} \frac{\alpha _0}{\beta ^2},\\ \sigma ^2_y &= {} \frac{\sum _{i=0}^p \alpha _i}{\beta ^2}. \end{aligned}$$

This is a special case of multivariate gamma distributions, which is used for our simulation.