## 2 Materials and Methods
### Detection of Outliers in Relative Abundance Variations  
Differential gene expression analyses such as DESeq2 and edgeR assume some invariable features, like the HKG expression level. However, these assumptions are probably inappropriate in highly variable microbial communities. The variation of the microbial community can change constantly with unexpected factors such as inter-individual differences, and it would be unrealistic to assume many invariable components. We developed a new computational framework called BRaVO, based on the relative variation for each component (Figure 1). The source code implementing this framework is available at [[https://github.com/omics-tools/BRaVO](https://github.com/omics-tools/BRaVO)]([https://github.com/omics-tools/BRaVO](https://github.com/omics-tools/BRaVO)). Our approach relatively evaluates the compositional variation based on the abundance. However, the standard deviation (or variance) is an unsuitable statistic to compare features with different component ratios. For example, the variance values of predominant microbial species tend to be much higher than those of minority microbial species. Thus, our framework uses the magnitude of the coefficient of variation as a measure of the relative taxonomic abundance change. In this study, we refer to the magnitude as the relative variation (\(RV\)). \(RV\) given by the following expression:  
\[RV=\frac{\sqrt{\frac{1}{n}\sum_{i=0}^{n}(x_{i}-\overline{x})^{2}}}{|\overline{ x}|} \tag{6}\]  
where \(x\) is the CLR-transformed count data for each of the features (OTUs), \(n\) is the number of samples, and \(\overline{x}\) is the mean value of \(x\). Our method finds significant outliers for relative taxonomic composition changes, based on the Z-score. Here, we performed robust Z-score calculations without assuming normality in the distribution of \(RV\). A robust Z-score is defined by the following expression:  
\[robustZ=\frac{RV_{i}-RV_{m}}{NIQR} \tag{7}\]  
where \(RV_{i}\) is the magnitude of the coefficient of variation of each feature (OTU) and \(RV_{m}\) is the median \(RV\) in all features (OTUs) within the same experimental conditions. \(NIQR\) is the normalized interquartile range (\(IQR\); 25-75 percentile range) associated with the standard normal distribution. The standard normal distribution with a mean of 0 and a standard deviation of 1 is given by the following probability density function:  
\[\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^{2}}{2}} \tag{8}\]  
Therefore, \(NIQR\) is provided by the following expression:  
\[NIQR=\frac{IQR}{\varphi(0.75)-\varphi(0.25)} \tag{9}\]  
These outliers are statistically evaluated based on p-values derived from the robust Z-scores using RV values. To considers the adjustment to the type I error (false positive), our framework also performs the Benjamini and Hochberg procedure [32]. This procedure adjusts \(p\)-values based on the false discovery rate (FDR). The FDR was used as the threshold level \(\alpha\) to detect outliers with significant compositional changes. In this study, we set the threshold level \(\alpha\) as 0.05 for this procedure. However, some microbes may have compositional changes that are independent of the intervention. Their compositional changes may be caused by not the intervention but individual differences or sampling biases. To distinguish them from microbial species with compositional changes induced by the intervention, we describe them as potentially variable features. Similar to edgeR and DESeq2, BRaVO also analyzes samples by labeling them in some groups like before the intervention (e.g., PRE) and after the intervention (e.g., POST). Our framework considers microbialspecies with common compositional changes independent of grouping as potentially variable features. These potentially variable microbes are detected as features with shared variation among different experimental conditions, such as before and after the dietary intervention.