# Correlation Measures in Metagenomic Data: The Blessing of Dimensionality
## 3 Results
### Comparison Between Correlation-Based Methods on Real Data  
In order to assess whether the conclusions drawn from our simulations are consistent with real data, we compared different correlation-based methods on a subset of 51 samples from the HMP2 dataset, selecting the subject 69-001 in healthy condition. Correlations were computed both at the OTU level (171 variables after filtering) and at the phylum level (7 aggregated taxa). We considered Pearson correlation after L1 normalization (Pearson + L1), proportionality (Rho), and SparCC, using Pearson correlation on CLR-transformed data (Pearson + CLR) as a reference. Rho is based on the concept of proportionality and offers a scale-invariant alternative to correlation, aiming to mitigate the impact of the constant-sum constraint in compositional data. SparCC, instead, estimates correlations from log-ratio transformed data under the assumption of sparsity in the underlying structure and is the only method among those considered that explicitly models and corrects compositional effects. At the OTU level, both Rho and SparCC showed near-perfect agreement with Pearson + CLR (R = 0.99, Figure 3B,C), confirming the consistency among these methods in high-dimensional settings. In contrast, Pearson + L1 displayed a more dispersed pattern (R = 0.81, Figure 3A). This divergence is likely driven by uneven taxa-wise heterogeneity, with an average within-sample diversity \(\underline{P}\) of 0.68, consistent with the patterns observed  
Figure 2: Different behaviors of spurious correlations for L1 and CLR transformations: (**A**) heatmap of MAE as a function of the dataset diversity \(\underline{P}\) and dimensionality D for L1 normalization (**top**) and CLR normalization (**bottom**) in \(log_{10}\) scale. (**B**) Scatter plot illustrating the MAE for CLR normalization on correlation as a function of dimensionality (error bars are negligible). in the simulation results (Figure 2A) where L1-based correlations are strongly affected by sample diversity. At the phylum level, the differences among methods become more pronounced. Rho remains closely aligned with Pearson + CLR (R = 0.98, Figure 3E), while Pearson + L1 fails to reconstruct any meaningful pattern (R = 0.15, Figure 3D). This failure is likely due to the combination of low dimensionality and reduced within-sample diversity, which averages around 0.5 across phyla. SparCC also diverges more markedly from Pearson + CLR (R = 0.38, Figure 3F), as it actively applies corrections for compositionality that become more relevant in such low-dimensional settings. These results are coherent with our simulation findings, which demonstrated that compositional biases are amplified at low dimensionality, where transformations alone are insufficient to fully mitigate distortion. Conversely, at higher dimensionality (e.g., the OTU-level case), the impact of such corrections is minimal, explaining the near-complete overlap among methods. Overall, these results validate our simulation findings: marginal correlations, combined with appropriate transformations such as CLR, yield stable and interpretable estimates in high-dimensional microbiome data. Additionally, they clearly illustrate how  
Figure 3: Comparison of correlation-based methods on real microbiome data (subject 69-001, HMP2). Panels (**A–C**) refer to OTU-level data (171 variables), while panels (**D–F**) refer to phylum-level data (\(\mathcal{T}\) variables). Each panel shows the pairwise correlation weights from a given method (Pearson + L1, Rho, SparCC) plotted against those from Pearson + CLR, used here as a reference. At high dimensionality (**A–C**), Rho and SparCC show near-perfect agreement with Pearson + CLR (R = 0.99), while Pearson + L1 deviates more substantially (R = 0.81), in line with the compositional bias patterns observed in simulations.