Penalized Regression Methods With Modified Cross-Validation and Bootstrap Tuning Produce Better Prediction Models

Abstract

Risk prediction models fitted using maximum likelihood estimation (MLE) are often overfitted resulting in predictions that are too extreme and a calibration slope (CS) less than 1. Penalized methods, such as Ridge and Lasso, have been suggested as a solution to this problem as they tend to shrink regression coefficients toward zero, resulting in predictions closer to the average. The amount of shrinkage is regulated by a tuning parameter, $\lambda ,$ commonly selected via cross-validation (“standard tuning”). Though penalized methods have been found to improve calibration on average, they often over-shrink and exhibit large variability in the selected $\lambda$ and hence the CS. This is a problem, particularly for small sample sizes, but also when using sample sizes recommended to control overfitting. We consider whether these problems are partly due to selecting $\lambda$ using cross-validation with “training” datasets of reduced size compared to the original development sample, resulting in an over-estimation of $\lambda$ and, hence, excessive shrinkage. We propose a modified cross-validation tuning method (“modified tuning”), which estimates $\lambda$ from a pseudo-development dataset obtained via bootstrapping from the original dataset, albeit of larger size, such that the resulting cross-validation training datasets are of the same size as the original dataset. Modified tuning can be easily implemented in standard software and is closely related to bootstrap selection of the tuning parameter (“bootstrap tuning”). We evaluated modified and bootstrap tuning for Ridge and Lasso in simulated and real data using recommended sample sizes, and sizes slightly lower and higher. They substantially improved the selection of $\lambda$, resulting in improved CS compared to the standard tuning method. They also improved predictions compared to MLE.