## 1 Introduction  
Binary classification (BC) is a very common machine learning (ML) task where each case is assigned to one of two possible outcome groups, called classes: the event class, e.g., \"yes\", \"disease\", and \"high risk\", or non-event class, e.g., \"no\", \"no disease\", and \"low risk\" (Table S1 provides a glossary of technical terms) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. BC is widely used in various domains, such as fraud detection, spam filtering, finance, medical screening, clinical medicine, public health (PH), epidemiology, and social and technological sciences, where predictive models support risk prediction and stratification, early disease detection, targeted prevention strategies, and operational decision-making. Examples include identifyingpatients at high risk of hospital readmission, death, or other adverse outcomes; targeting preventive interventions toward high-risk subgroups; detecting fraudulent transactions or spam messages. Performance of the predictive model depends critically on the appropriate selection of variables, making variable selection (VS) a key step in developing prediction models [14]. It aims to identify a subset of variables that ensures high prediction accuracy and model simplicity, while avoiding overfitting, noise, and redundancy. This step can become particularly challenging in real-world tasks with high-dimensional datasets, collinearity between variables or weakly informative variables, and where class imbalances can additionally influence both model performance and evaluation. As imbalanced datasets are prevalent in real-world applications, VS is particularly important in these scenarios. Although various approaches have been proposed to address this issue, an imbalanced setting still poses a significant challenge in both model fitting and evaluation [15; 16; 17; 18; 19; 20; 21; 22; 23; 24]. The predictive accuracy for the minority (usually event) class is typically of greater clinical or practical relevance, such as in rare diseases research, medical screening, and spam detection. Therefore, a model's bias towards the majority (usually non-event) class can lead to serious consequences of false-negative results. In PH and clinical settings, where prediction tools inform resource allocation or treatment decisions, understanding prediction behaviors in both outcome classes is essential. A common VS strategy is adding a candidate variable to an existing prediction model [1; 25; 26]. Traditional evaluation metrics, such as the difference in the area under the receiver operating characteristic (ROC) curve (\(\Delta\)AUC), the likelihood-ratio test (LRT), the difference in the Brier score (\(\Delta\)BS), the Brier skill score (BSS), the net reclassification index (NRI), the difference in the F1-score (\(\Delta\)F1-score), or the difference in a Matthews correlation coefficient (\(\Delta\)MCC), are often used to assess the added predictive value of a candidate variable [25; 26; 27; 28; 29; 30]. These methods provide global assessments of model performance and do not indicate whether a variable improves predictions for events, non-events, or both classes. Also, the visualization of results is often limited. These limitations become even more pronounced in imbalanced BC, where traditional methods often underperform and may fail to detect significant effects [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. This underscores the need for tools that not only evaluate model performance globally but also provide granular insights into class-specific behavior, especially when clinical outcomes depend on minority-class accuracy. To address these limitations, we developed the U-smile method for evaluating the usefulness of a new variable in a BC model [6; 7]. The U-smile method quantifies prediction changes with three families of interpretable coefficients and provides a standardized graphical summary in the form of the U-smile plot. This approach enables a detailed assessment of prediction improvement and worsening, separately in each outcome class.