First, we propose a two-sample empirical likelihood method for comparing two correlated receiver operating characteristic (ROC) curves. We consider the log empirical likelihood ratio of the difference between areas under the ROC curves (AUC). By using empirical likelihood estimators of distribution functions, we derive the asymptotic distribution of the newly proposed statistic and so solve out the empirical likelihood confidence intervals of the difference between two correlated AUCs. The proposed empirical likelihood ratio statistic converges in distribution to a scaled chi-squared random variable. Simulation results show that the proposed empirical likelihood method has better finite-sample performance than other competitors. Consequently we can conclude that the proposed empirical likelihood confidence interval has a better performance than non-parametric Wald confidence intervals proposed by Delong et al. (1988).
Second, we take the consideration of a density ratio model, which concerned the relationship between two unknown distributions. We proposed a semi-parametric estimator of log empirical likelihood ratio of the difference between two correlated AUCs, and solve out the semi-parametric empirical likelihood confidence intervals for making inference between two correlated ROC curves. In this approach, we estimate the unknown distribution functions semi-parametrically by maximum likelihood estimation (MLE) based on the exponential expression given by the density ration model. This approach yield more accuracy than original non-parametric approaches.
Finally, we proposed an R package to implement the algorithm proposed in previous two topics. Additionally, this package also gives the nonparametric and semi-parametric estimatiors of a single ROC curve for two-level diagnostic test, or ROC surface for multi-level diagnostic test, including variance estimatiors. Besides numerical results, the plots of estimated distribution functions, estimated ROC curves or surfaces are given by default parameters. Especially, if one run the package on a Windows system, there is an interaction feature to find the estimated sensitivity (SE) and specificity (SP), defined, respectively, as the probability that a truly diseased subject has a positive test result and the probability that a truly nondiseased subject has a negative test result. One can get the estimated values of SE and SP by simply clicking the pop-up ROC curve or surface figure. The function is developed as S3 functional in R, and one can easily download the package from my home page and load it in R enviroment, then run logicROC function to get expected results. Detail parameter information is given in "help".