## 2 Materials and Methods
### Evaluation Metrics  
The evaluation of synthetic data quality has evolved significantly over time. However, a universal method for benchmarking the performance of synthetic data generation has not yet been established. Several recent studies have aimed to review the diversity of metrics used for synthetic data validation, especially for tabular data, which often receive less attention compared with synthetic image data. For example, ref. [25] concentrates on the evaluation of synthetic tabular data, highlighting the need for specialized metrics in this domain. Importantly, the choice of validation metric depends on the intended application of the synthetic data. Given that evaluation criteria vary based on the synthetic data's intended application, we select fidelity and utility metrics that align with the practical use of synthetic data in healthcare. Among the multiple synthetic data applications in the medical field, we aim to use synthetic data for machine learning model training when real data are insufficient due to privacy restrictions. The fidelity of synthetic data is a measure of how well the synthetic data represent the statistical properties of real data. To assess fidelity, we use the cumulative distribution function (CDF), following the approach proposed by [26]. The CDF measures the probability that a variable will take a value less than or equal to a given point, allowing the comparison of the probability distributions of real and synthetic datasets. The Kolmogorov-Smirnov (K-S) test, as in [27; 28; 29], is used to measure this similarity. The K-S test calculates a statistic based on the maximum absolute difference between the CDFs of the real and synthetic datasets. This maximum difference serves as the fidelity score, where a lower value indicates that the synthetic data distribution closely aligns with that of the real data, thereby demonstrating high fidelity. On the other side, a significant element of synthetic data quality is its utility, which is commonly assessed through the examination of the performance of machine learning models trained on synthetic data. This method of evaluation was initially proposed by [PERSON] et al. [30] under the framework \"Train on Synthetic, Test on Real\" (TSTR). In this approach, machine learning models are trained separately on real and synthetic data, and then tested on a common holdout dataset that has not been seen by either the models or the generative algorithm. This comparison of model performance between real and synthetic data has become a prevalent approach in the literature, as evidenced by [26; 31; 32; 33]. An alternative approach to evaluating data utility involves assessing the performance of models trained with augmented data (a combination of real and synthetic samples). Data augmentation increases the diversity and quantity of training data by incorporating synthetic samples [34]. This methodology has proven especially valuable in the health domain, addressing critical challenges such as class imbalance by generating additional samples for underrepresented classes, improving model generalization by exposing models to a broader range of data and reducing overfitting, and mitigating bias by creating more representative datasets that include minority groups [35]. Class imbalance, in particular, has been shown to result in unfair decisions for minority classes, as highlighted in [36]. Consequently, a key utility metric involves comparing the performance of models trained exclusively on real data against those trained on augmented datasets. This approach has been successfully demonstrated in previous studies, such as [37], where augmented datasets were used to enhance model performance and fairness. A variety of machine learning models can be used for this evaluation. In this paper we focus on random forests, logistic regression, and support vector machine (SVM) networks. These models are trained with real, synthetic, and augmented data and then tested on real data.