Cut-point identification in biomarkers for a censored failure time outcome

AIMS The use of a continuous biomarker X in clinical research often requires the definition of a cut-point (c) for patient classification. In the presence of binary outcome, methods based on the receiver operating characteristic (ROC) curve such as: i) the Youden index, i.e. the maximum difference between sensitivity and the complement to one of specificity at c of X, ii) the concordance probability, i.e. the maximum probability of being below or above c of X for any random pair of non-diseased and disease subjects, and iii) the c of X corresponding to the point closest-to-(0,1) corner in the ROC plane, are commonly and indistinctly used. The extension of these methods to the case of censored failure time outcome is not a trivial task: our aim is to extend these methods and to investigate their performance through simulations. METHODS Let Ti=min(Zi,Ci) be the observed time, where Zi is time to event (disease) and Ci the censoring time, δi the censoring indicator (δi=1 if Ti=Zi and δi=0 if Ti=Ci) and Xi the biomarker value for subject i. In this setting, the sensitivity (SE) and the complement to one of specificity (SP) at cut-point c of X are SE(c)=P(X>c|Zi≤τ) and 1-SP(c)=P(X>c|Zi>τ), where τ is a pre-defined reference time point of clinical interest. Since in the presence of censoring the counts of subject with Zi≤τ and with Zi>τ are not known, SE(c) and SP(c) cannot be straightforwardly estimated by proportions. However, by applying the Bayes’ theorem, these quantities can be rewritten in terms of survival probabilities and proportions of patients having the biomarker value above and below c. The performance of the three aforementioned criteria to identify the cut-point is evaluated in terms of bias and mean square error (MSE). RESULTS The relative bias of the investigated methods is small on all levels of classification accuracy of the biomarker, and it increases as the censoring fraction increases. Results on MSEs show that the point closest-to-(0,1) corner in the ROC plane and concordance probability methods have better performance than the Youden index. The MSE is inversely related to sample size and it increases as the censoring fraction increases. The performance of all methods improves with increasing biomarker classification accuracy. The methods are applied in a study on a molecular biomarker in acute lymphoblastic leukemia, with similar results. CONCLUSION Our study shows that the ROC based methods for defining a cut-point of a continuous biomarker can be extended to censored data. The point closest-to-(0,1) corner approach has the best performance. However, given the lack of clinical meaning of its objective function, the calculation of the Youden index or concordance probability associated to the cut-point identified through the closest-to-(0,1) corner approach could be used to ease interpretability of the classification accuracy of the biomarker.