Discretizing multiple continuous predictors with U-shaped relationships with lnOR: introducing the recursive gradient scanning method in clinical and epidemiological research.

Background: Assuming a linear relationship between continuous predictors and outcomes in clinical prediction models is often inappropriate, as true linear relationships are rare, potentially resulting in biased estimates and inaccurate conclusions. Our research group addressed a single U-shaped independent variable before. Multiple U-shaped predictors can improve predictive accuracy by capturing nuanced relationships, but they also introduce challenges like increased complexity and potential overfitting. This study aims to extend the applicability of our previous research results to more common scenarios, thereby facilitating more comprehensive and practical investigations.

Methods: In this study, we proposed a novel approach called the Recursive Gradient Scanning Method (RGS) for discretizing multiple continuous variables that exhibit U-shaped relationships with the natural logarithm of the odds ratio (lnOR). The RGS method involves a two-step approach: first, it conducts fine screening from the 2.5th to 97.5th percentiles of the lnOR. Then, it utilizes an iterative process that compares AIC metrics to identify optimal categorical variables. We conducted a Monte Carlo simulation study to investigate the performance of the RGS method. Different correlation levels, sample sizes, missing rates, and symmetry levels of U-shaped relationships were considered in the simulation process. To compare the RGS method with other common approaches (such as median, Q₁-Q₃, minimum P-value method), we assessed both the predictive ability (e.g., AUC) and goodness of fit (e.g., AIC) of logistic regression models with variables discretized at different cut-points using a real dataset.

Results: Both simulation and empirical studies have consistently demonstrated the effectiveness of the RGS method. In simulation studies, the RGS method showed superior performance compared to other common discretization methods in discrimination ability and overall performance for logistic regression models across various U-shaped scenarios (with varying correlation levels, sample sizes, missing rates, and symmetry levels of U-shaped relationships). Similarly, empirical study showed that the optimal cut-points identified by RGS have superior clinical predictive power, as measured by metrics such as AUC, compared to other traditional methods.

Conclusions: The simulation and empirical study demonstrated that the RGS method outperformed other common discretization methods in terms of goodness of fit and predictive ability. However, in the future, we will focus on addressing challenges related to separation or missing binary responses, and we will require more data to validate our method.