## 3 Results
The shape and statistical properties of the generated data were evaluated for both the R and T groups in all cases (Figure 5). Figure 5 shows that the generated data did not exactly match the normal distribution of the original data and had a central tendency of about 100, especially when the encoder and decoder were limited to two hidden layers (Figure 5a). The findings indicate that there was a noteworthy enhancement in outcomes when the encoder and decoder were equipped with three hidden layers (Figure 5b). This particular arrangement facilitated a more effective capture of the shape and mean of the original dataset. This held true in cases where both the encoder and decoder had four hidden layers (Figure 5c). In the next step, a statistical analysis was conducted to compare the properties of the generated datasets, which were obtained from subsamples of different sizes ranging from 10 to 90, with those of the original dataset. The assessment of equivalence (or non-equivalence) was explored for this objective. The research included exploration with diverse autoencoder configurations and data variability. Specifically, coefficient of variation values of 10%, 20%, and 40% were employed. Furthermore, the investigation examined the effects of distinct activation functions, specifically \"softplus\" and linear, on the hidden layer of the convolutional neural networks. The aforementioned procedure was implemented multiple times utilizing Monte Carlo simulations and the percentage of equivalence acceptance (i.e., the probability to reject the null hypothesis) was counted. Figure 6 illustrates the probability of accepting equivalence under the TOST hypothesis. The diagram depicts the diverse subsample levels, alongside the two groups characterized by the three coefficients of variation values (10%, 20%, and 40%) and the two discrete activation functions employed for the hidden layers. Figure 6 demonstrates the trend of equivalence acceptance as the subsample size proportion increased (from left to right) in the case of CVs 10% and 20% (Figure 6a,b, respectively). On the contrary, when the CV was equal to 40%, the probability of accepting equivalence rose with an increase in the subsample size (Figure 6c). This attribute could be observed in both the R and T groups. Ultimately, it seems that the probability of rejecting the null hypothesis (namely, declaring equivalence) is higher for the \"softplus\" when compared to the linear activation function. Figure 5: Distribution of the generated data for both the R and T groups using variational autoencoders with 2 (**a**), 3 (**b**), and 4 (**c**) hidden layers for the encoder and the decoder. In a subsequent step, T-R group comparisons were carried out. The datasets of the reference group, including the original, generated, and subsampled datasets, were compared to their corresponding counterparts in the test group (Figure 7). Several subsample sizes were assessed, spanning from 10% to 90%, with intervals of 10%. This statistical analysis aimed to investigate equivalence and was performed across multiple CV levels. It should be mentioned that these comparisons were exclusively carried out in the case in which the activation function of the hidden layers was \"softplus\". Figure 7 reveals that for low/medium CV values (10% and 20%), the probability of showing equivalence was quite high for all datasets. Especially for the original data (Figure 7a), where the probability is 100% since both the T and R groups were assumed to exhibit identical average performances at the endpoint. In the case of the subsampled (Figure 7b) and generated (Figure 7c) datasets, the probability of accepting equivalence was found to be low only in the cases using a very small part of the original dataset (e.g., 10% or 20%), and it increased to almost 100% acceptance when portions larger than 30% of this original sample size were used. When the high variability of the data was considered (CV = 40%), the observed performance was as expected.