Non-parametric inference on calibration of predicted risks

Moderate calibration, the expected event probability among observations with predicted probability $\pi$ being equal to $\pi$, is a desired property of risk prediction models. Current graphical and numerical techniques for evaluating moderate calibration of clinical prediction models are mostly based on smoothing or grouping the data. As well, there is no widely accepted inferential method for the null hypothesis that a model is moderately calibrated. In this work, we discuss recently-developed, and propose novel, methods for the assessment of moderate calibration for binary responses. The methods are based on the limiting distributions of functions of standardized partial sums of prediction errors converging to the corresponding laws of Brownian motion. The novel method relies on well-known properties of the Brownian bridge which enables joint inference on mean and moderate calibration, leading to a unified 'bridge' test for detecting miscalibration. Simulation studies indicate that the bridge test is more powerful, often substantially, than the alternative test. As a case study we consider a prediction model for short-term mortality after a heart attack. Moderate calibration can be assessed without requiring arbitrary grouping of data or using methods that require tuning of parameters. We suggest graphical presentation of the partial sum curves and reporting the strength of evidence indicated by the proposed methods when examining model calibration.