A Comparison of Frequentist RCS, Bayesian RCS, and SHAP for Estimating Risk Inflection Points and Crossing Points of Biomarkers

Abstract Objective Continuous biomarkers often have a non-linear dose-response with the outcome so the need is to identify two key points on the fitted curve. The inflection point is the point at which the risk curve changes direction and the crossing point is the point at which the effect changes from protective to harmful. This is a very common requirement in clinical and epidemiological work but there is still no validated and practical method for estimating these points. Methods Monte Carlo simulation was used to compare frequentist RCS, Bayesian RCS and SHAP in terms of their ability to estimate inflection and crossing points. A range of dose response shapes, levels of collinearity and outcome types were considered and the accuracy and robustness of the three methods were assessed. Results Bayesian MCMC is best for crossing points. For the nonlinear monotone and symmetric U-shaped cases the bias is around ± 0.03 (a lot better than the − 0.23 to + 0.39 seen with the frequentist RCS). If the crossing point is very far from the exposure mean or on a sparse branch however the prior shrinkage in the Bayesian approach can make it less accurate in which case the frequentist method is superior. For inflection points the frequentist RCS is better (bias of -0.06 versus − 0.32 for the Bayesian in the asymmetric U case). INLA is not reliable for inflection points or the right root due to coefficient distortion so Bayesian estimation should use MCMC. SHAP has a built in bias of around − 0.18 to -0.24 which doesn't go away with tuning so it is only of use for exploratory screening. RCS is more precise when the data do a good job of covering the shape but no method can get a crossing point on a sparse branch. The approach is illustrated in two real examples: serum phosphate in CRRT patients and age in ectopic pregnancy patients. Conclusion use Bayesian MCMC for crossing points and frequentist RCS for inflection points. If the crossing point is far from the exposure mean or on a sparse branch then use frequentist RCS. SHAP should be used purely for exploratory screening.