## 2 Materials and Methods
### Monte Carlo Simulations  
The methodology employed for the generation of subjects was as follows: Initially, a sample of 100 subjects was generated for the reference (i.e., R) group through a random process, utilizing a normal distribution with a mean of \(\mu_{\text{R}}\) (i.e., the average endpoint value) and a standard deviation of \(\sigma_{\text{R}}\). Then, a random subsampling procedure was performed on the original R group, whereby a proportion (gradually decreasing from 90% to 10% with a step of 10%) of the data, termed \"subsample size\", was selected from the distribution. Subsequently, the subsample was utilized to train the VAE model, followed by sampling from the inferred latent distribution and generating a total of 100 virtual subjects for the R group. Similarly, the aforementioned procedure was also repeated for the test (i.e., T) group of subjects. In the case of the T group, random generation was based on a mean endpoint value of \(\mu_{\text{T}}\) and standard deviations (\(\sigma_{\text{T}}\)). The aforementioned procedure is schematically shown in Figure 2.  
Several ratios between the average endpoint values of the T and R groups were explored. In order to achieve this task, the mean endpoint value for R was set at 100, while for the T group, it was equal to 1.00\(\times\), 1.10\(\times\), 1.25\(\times\), and 1.50\(\times\) times that of R. In all cases, the coefficient of variation (CV) was equal between the T and R groups, and three different CV values were explored: 10% (low variability), 20% (medium variability), and 40% (high variability).  
According to the aforementioned procedure, three different types of samples were utilized in the simulations: (a) the original dataset, (b) the subsampled group, and (c) the regenerated group by using the VAE system. To find out how well the VAE approach works, statistical analyses were conducted to compare the original, subsampled, and made-up data within and between the R and T groups. In addition, comparisons were made between the T and R groups of each dataset, namely, the T vs. R of the original dataset, the T vs. R of the sub-sampled dataset, and the T vs. R of the VAE dataset.  
Figure 1: Visual representation of a variational autoencoder. The process of encoding involves compressing data from their original space to a latent space, while the decoding process involves decompressing the data. The methodology involves the utilization of neural networks as both an encoder and a decoder, with the aim of acquiring an optimal encoding-decoding scheme through an iterative optimization process. Variational autoencoders aim to establish mapping between the input data and a probability distribution across the latent space.  
After the generation of the virtual subjects, they were appropriately classified into two groups in order to construct a parallel clinical design [4]. Ln-transformation (Napierian) was applied to the generated values before proceeding to the statistical analysis. Since the data obtained from bioequivalence studies may not follow a normal distribution, the official approach to address this issue is to apply a natural logarithm transformation to the data before conducting statistical analyses [5, 6]. Thus, regardless of the distribution of the original data, In-transformation is always applied in the field of bioequivalence before statistical analysis [4, 5, 6]. After the analysis is completed, the results are transformed back to the original scale to interpret the findings in a clinically meaningful way, as we did in our study. The statistical analysis was performed using the typical bioequivalence criteria (90% confidence interval), and a decision regarding equivalence or not was made if the 90% confidence interval fell within the acceptance limits of 80.00-125.00% [5, 6]. The statistical assessment followed the principles of equivalence testing, namely, the two-one-sided test (TOST) procedure [5, 6]. The above-mentioned procedure was repeated several times (500) to allow for reliable estimates.