(CD2) nearest neighbour estimator of Heagerty et al. [1]

The problems of the CD1 estimators motivated Heagerty et al. [1] to develop an alternative approach based on a bivariate survival function. This improved methodology uses the nearest neighbour estimator of the bivariate distribution of (X, T), introduced by Akritas [16]. As mentioned earlier, CD1 is not robust to marker-dependent censoring; however, censoring often depends on the marker. Thus, the independence of time-to-event and censoring time cannot be assumed and they are more likely independent conditionally on the marker. In this model-based approach, the probability of each individual is modelled for a case by 1 − S(t|X _i) and for a control by S(t|X _i) [13]. Akritas [16] proposed using the following model-based estimator for the conditional survival probability called the weighted Kaplan-Meier estimator and is defined by

where K_λn(X_j,X_i) is a kernel function that depends on a smoothing parameter λ _n. Akritas [16] uses a 0/1 nearest neighbour kernel, KλnXjXi=I−λn<F^XXi−F^XXj<λn where 2λ _n ∈ (0, 1) represents the percentage of individuals that are included in each neighbourhood (boundaries). The resulting sensitivity and specificity are defined by

where S^λnt=S^λn−∞,t. The above estimates of the sensitivity and specificity will produce ROC curve estimates that are invariant to monotone transformations of the marker. Both sensitivity and specificity are monotone and bounded in [0, 1]. Further, as contrast to CD1, this nonparametric method is efficient as a semi-parametric method and allows the censoring to depend on the marker space [16]. Heagerty et al. [1] used bootstrap resampling to estimate the confidence interval for this estimator. Motivated by the results gained by Akritas [16], Cai et al. [17], Hung and Chiang [2] and Hung and Chiang [18] discusses the asymptotic properties of CD2. They have established the usual n-consistency and asymptotic normality and concluded that bootstrap resampling techniques can be used to estimate the variances. In practice, it is suggested that the value for λ _n is chosen to be 𝒪(n^−⅓) [1]. Song and Zhou [19] extended the method to incorporate covariates other than those variables contained in the marker for constructing the ROC curves within this CD2 methodology. They have also explored their model by incorporating an ID mechanism.