Evaluating risk prediction models: the Predictiveness curve and its geometric summaries

Risk prediction models are widely used, but their evaluation is dominated by discrimination-based measures such as the ROC curve and AUC, which provide limited insight into how risks are distributed across the population. We extend the predictiveness curve to risk prediction models, characterize its geometry, and derive three geometric summaries-the Pietra index, Gini index, and scaled Brier score. To obtain well-calibrated and stable risk estimates, we implement a three-step prediction post-processing procedure consisting of cross-validation, isotonic recalibration, and bootstrap averaging. We illustrate the approach using toy examples and a nationwide Taiwan cohort of 23 839 lung cancer patients, in whom a Cox model was used to estimate five-year fatality risk. The three indices captured complementary aspects of performance and supported purpose-specific model choice: Pietra for minimizing the gray zone, Gini for maximizing separation, and scaled Brier for increasing certainty. In the lung cancer application, the final post-processed model showed broad risk stratification: 25.2% of patients were very low risk (<10% fatality) and 50.2% high risk (>75% fatality), whereas only 4.8% were near the average risk. Overall, the predictiveness curve and its geometric summaries provide a population-oriented framework for transparent, purpose-specific evaluation of risk prediction models.