Boosting the concordance index for survival data--a unified framework to derive and evaluate biomarker combinations

Abstract

The development of molecular signatures for the prediction of time-to-event outcomes is a methodologically challenging task in bioinformatics and biostatistics. Although there are numerous approaches for the derivation of marker combinations and their evaluation, the underlying methodology often suffers from the problem that different optimization criteria are mixed during the feature selection, estimation and evaluation steps. This might result in marker combinations that are suboptimal regarding the evaluation criterion of interest. To address this issue, we propose a unified framework to derive and evaluate biomarker combinations. Our approach is based on the concordance index for time-to-event data, which is a non-parametric measure to quantify the discriminatory power of a prediction rule. Specifically, we propose a gradient boosting algorithm that results in linear biomarker combinations that are optimal with respect to a smoothed version of the concordance index. We investigate the performance of our algorithm in a large-scale simulation study and in two molecular data sets for the prediction of survival in breast cancer patients. Our numerical results show that the new approach is not only methodologically sound but can also lead to a higher discriminatory power than traditional approaches for the derivation of gene signatures.

Figure 1. Coefficient estimates for pre-selected markers obtained from 100 simulation runs.

The marker combinations were optimized via gradient boosting based on training samples of size (left) and (right). Boxplots represent the empirical distribution of the resulting coefficients. Only markers to had an actual effect on the survival time.

Figure 2. Simulation results for the discriminatory power obtained via the proposed -index boosting approach and competing Cox-based estimation schemes.

The marker combinations were optimized via the different approaches based on training samples of size (left) and (right). Boxplots represent the empirical distribution of the resulting on corresponding test samples. The dotted line corresponds to the discriminatory power resulting from the true combination of predictors with known coefficients.

Figure 3. Comparing the discriminatory power of biomarker combinations to predict the time to distant metastases resulting from the proposed -index boosting approach with competing estimation schemes.

The plot on the left refers to the Desmedt et al. data, whereas the plot on the right presents results from the van de Vijver et al. data. All biomarker combinations were optimized via the corresponding algorithms based on the same 100 learning samples. Boxplots represent the empirical distribution of the resulting on corresponding test samples. The dotted line corresponds to the median -index resulting from the new approach.