\section{Introduction}

High-throughput amplicon sequencing (16S rRNA) and shotgun metagenomics have transformed the study of host-associated microbial communities. A typical analysis pipeline produces a table of read counts per taxon per sample, normalized to relative abundances (proportions summing to 1) before statistical analysis. Thousands of studies have used proportional data to identify taxa associated with conditions ranging from obesity to inflammatory bowel disease (Turnbaugh et al., 2009; Qin et al., 2012; Halfvarson et al., 2017).

The fundamental problem with this approach was identified long before microbiome research existed. Aitchison (1986) demonstrated that compositional data --- vectors of positive components summing to a constant --- occupy a constrained sample space (the simplex) in which standard multivariate methods produce unreliable results. The unit-sum constraint induces negative correlations among components: if one proportion increases, at least one other must decrease, regardless of biological reality. For a $D$-part composition, the average pairwise correlation among raw proportions is constrained to be $-1/(D-1)$, a mathematical identity independent of true biological relationships.

Gloor et al. (2017) reminded the microbiome community that sequencing data are compositional and that this is "not optional." They recommended the centered log-ratio (CLR) transformation, $\text{clr}(\mathbf{x})_i = \ln(x_i / g(\mathbf{x}))$ where $g(\mathbf{x})$ is the geometric mean, as a general-purpose approach. Quinn et al. (2018) extended this with practical software. Friedman and Alm (2012) developed SparCC for compositional correlation inference. Lin and Peddada (2020) introduced ANCOM-BC with built-in bias correction. Lovell et al. (2015) proposed proportionality as a compositionally valid alternative to correlation. Yet a manual survey we conducted of 60 gut microbiome papers published in high-impact journals between 2018 and 2023 found that 38 (63%) analyzed relative abundances without any log-ratio transformation. The question we address is whether ignoring compositionality materially changes published biological conclusions.

\section{Related Work}

\subsection{Compositional Data Analysis Foundations}

Aitchison (1982, 1986) showed that the appropriate sample space for compositional data is the simplex $\mathcal{S}^D$ and that operations require log-ratio coordinates. Three transformations have gained use: additive log-ratio (ALR), centered log-ratio (CLR), and isometric log-ratio (ILR). Egozcue et al. (2003) formalized the Aitchison geometry and showed CLR coordinates preserve distances in the Aitchison metric. In the microbiome context, Gloor et al. (2017) demonstrated through simulation that Pearson and Spearman correlations on raw proportions produce false positive rates exceeding 50% when true correlations are zero. Silverman et al. (2017) introduced PhILR, using phylogenetic trees to construct ILR coordinates.

\subsection{Differential Abundance and Compositionality}

The compositionality problem affects differential abundance testing. A taxon appearing enriched in disease may be unchanged in absolute abundance; the apparent enrichment could be an artifact of another taxon decreasing. Mandal et al. (2015) introduced ANCOM to address this by testing ratios rather than individual proportions. Lin and Peddada (2020) extended this with ANCOM-BC. Nearing et al. (2022) benchmarked 14 differential abundance methods using mock communities, finding that methods ignoring compositionality (DESeq2, edgeR on proportions) had false positive rates 3 to 8 times higher than compositionally aware methods (ANCOM-BC, ALDEx2).

\section{Methodology}

\subsection{Dataset Selection}

We selected 12 published datasets meeting four criteria: (1) the study reported at least one significant taxon-disease association; (2) the analysis used relative abundances without log-ratio transformation; (3) raw count data were publicly available; (4) the study had been cited at least 50 times. The datasets spanned obesity (3), type 2 diabetes (2), inflammatory bowel disease (3), colorectal cancer (2), and Clostridioides difficile infection (2). Sample sizes ranged from 20 to 531. Detected taxa ranged from 18 to 487 genera. All used 16S rRNA amplicon sequencing.

\subsection{Replication and CLR Transformation}

For each dataset, we first replicated the original analysis using the reported statistical test (Wilcoxon rank-sum, Spearman correlation, linear regression, or DESeq2 on proportions). In 11 of 12 cases, we reproduced the original direction and significance.

We then applied the CLR transformation:

$$\text{clr}(\mathbf{x})$$i = \ln\left(\frac{x_i}{g(\mathbf{x})}\right), \quad g(\mathbf{x}) = \left(\prod{j=1}^D x_j\right)^{1/D}

We addressed zeros using three strategies: (1) pseudocount replacement (0.5 added to all zeros); (2) multiplicative replacement following Martin-Fernandez et al. (2003) with $\delta = 0.65/D^2$; (3) Bayesian replacement via the zCompositions R package (Palarea-Albaladejo and Martin-Fernandez, 2015). We report Strategy 1 results and note concordance with the others.

\subsection{Classification of Outcomes}

For each dataset, we classified the CLR reanalysis outcome:

\textbf{Reversal:} Direction of association changed sign with statistical significance ($p < 0.05$) in the CLR result.

\textbf{Loss of significance:} Direction unchanged but CLR $p$-value exceeded 0.05.

\textbf{Concordant:} Direction and significance preserved under CLR.

\subsection{Mathematical Analysis of Reversal Conditions}

Consider two groups (disease and healthy) with compositions $\mathbf{x}^{(D)}$ and $\mathbf{x}^{(H)}$. The CLR difference for taxon $i$ is:

$$\Delta_i^{\text{CLR}} = \overline{\ln x_i^{(D)}} - \overline{\ln x_i^{(H)}} - \left(\overline{\ln g(\mathbf{x}^{(D)})} - \overline{\ln g(\mathbf{x}^{(H)})}\right)$$

Reversal occurs when the geometric mean difference dominates the taxon-specific difference --- when a taxon appears enriched only because overall microbial load (geometric mean) is lower in disease, inflating all proportions. For a composition with $D$ parts and a dominant taxon at fraction $f$, the critical dominance fraction above which reversal is guaranteed for subordinate taxa with proportional difference $|\Delta_i^{\text{raw}}|$ is:

$$f^* = 1 - (D-1) \cdot e^{-D \cdot |\Delta_i^{\text{raw}}| / (D-1)}$$

For $D = 20$ and $|\Delta_i^{\text{raw}}| = 0.01$, $f^* \approx 0.37$: if any taxon exceeds 37% of the community and shifts between groups, subordinate associations reverse.

\subsection{Stratification by Data Characteristics}

To test the prediction that low-diversity communities are more susceptible to compositional artifacts, we stratified the 12 datasets by three characteristics:

\textbf{Taxonomic diversity:} Low ($D < 50$ genera), medium ($50 \leq D \leq 200$), high ($D > 200$). These bins were chosen because the compositional bias $-1/(D-1)$ transitions from substantial ($-0.020$ at $D = 50$) to negligible ($-0.003$ at $D = 300$) across this range.

\textbf{Dominance:} The proportion of reads captured by the most abundant genus in the disease group. We categorized datasets as high-dominance (top genus $> 30$ of reads) or low-dominance ($\leq 30$). The 30% threshold was chosen based on the critical dominance fraction $f^$f∗ derived above: for most values of $D$ and $|\Delta_i^{\text{raw}}|$ in our datasets, $f^$ falls between 0.25 and 0.45.

\textbf{Sample size:} Small ($n < 50$), medium ($50 \leq n \leq 200$), large ($n > 200$). We included this stratification to test whether sample size modulates the compositionality problem. Because compositionality induces a systematic bias rather than random noise, we predicted that sample size would have little effect on reversal rates.

For each stratification, we computed the reversal rate (reversals / datasets in stratum) and the combined artifact rate (reversals + losses / datasets in stratum).

\subsection{Sensitivity Analyses}

We conducted three sensitivity analyses to verify robustness:

\textbf{ALR instead of CLR.} We repeated all 12 reanalyses using the additive log-ratio transformation with the most abundant taxon as reference denominator:

$$\text{alr}(\mathbf{x})$$i = \ln\left(\frac{x_i}{x{\text{ref}}}\right), \quad i \neq \text{ref}

ALR results were concordant with CLR in 11 of 12 datasets. The single discordance occurred in Dataset D04, where ALR showed loss of significance but CLR showed reversal, attributable to the choice of reference taxon (Bacteroides, which was itself differentially abundant between groups).

\textbf{Filtering threshold.} We varied the minimum prevalence threshold for taxon inclusion from 5% to 30% of samples. Stricter filtering reduces $D$, which increases compositional bias in raw proportions but also changes the geometric mean. Our qualitative conclusions (5 reversals, 3 losses) were stable across all thresholds.

\textbf{Rarefaction depth.} For the 8 datasets where rarefied count tables were available at multiple depths, we tested whether rarefaction depth affected reversal classification. It did not in any case, confirming that reversal is driven by compositional structure rather than sequencing depth.

\subsection{Comparison with ANCOM-BC}

As additional validation, we reanalyzed all 12 datasets using ANCOM-BC (Lin and Peddada, 2020), a method designed specifically for compositional differential abundance testing. ANCOM-BC estimates and removes sample-specific compositional bias through a linear model on the log-transformed observed abundances. The ANCOM-BC structural model assumes that observed abundances $O_{ij}$ relate to absolute abundances $A_{ij}$ via a sample-specific bias $b_j$: $\mathbb{E}[\ln O_{ij}] = \ln A_{ij} + b_j$. Removing $b_j$ eliminates the compositional artifact.

\subsection{Software and Reproducibility}

All analyses were implemented in R 4.3 using the compositions package (van den Boogaart and Tolosana-Delgado, 2008) for CLR/ALR transformations, zCompositions (Palarea-Albaladejo and Martin-Fernandez, 2015) for Bayesian zero replacement, and ANCOM-BC via the ANCOMBC package (Lin and Peddada, 2020). Statistical tests (Wilcoxon rank-sum, Spearman correlation) used base R functions. Bonferroni correction was applied as $p_{\text{adj}} = \min(p \cdot D, 1)$ for each dataset independently.

\section{Results}

\subsection{Overall Outcomes}

\begin{table}[h] \caption{Reanalysis outcomes for 12 microbiome-disease datasets. "Raw dir." and "CLR dir." show the sign of the taxon-disease association (+ = enriched in disease). "Bonf. sig." indicates Bonferroni-corrected significance in the CLR analysis.} \begin{tabular}{llllllll} \hline Dataset & Disease & $D$ & $n$ & Raw dir. & CLR dir. & Outcome & Bonf. sig. \ \hline D01 & Obesity & 34 & 154 & + & $-$ & Reversal & Yes \ D02 & Obesity & 187 & 531 & + & + & Concordant & Yes \ D03 & Obesity & 42 & 20 & $-$ & + & Reversal & No \ D04 & T2D & 28 & 345 & + & $-$ & Reversal & Yes \ D05 & T2D & 256 & 174 & + & + & Concordant & Yes \ D06 & IBD & 41 & 93 & + & n.s. & Loss & No \ D07 & IBD & 312 & 122 & $-$ & $-$ & Concordant & Yes \ D08 & IBD & 18 & 66 & + & $-$ & Reversal & Yes \ D09 & CRC & 487 & 120 & $-$ & $-$ & Concordant & No \ D10 & CRC & 38 & 90 & + & $-$ & Reversal & No \ D11 & CDI & 73 & 338 & $-$ & n.s. & Loss & No \ D12 & CDI & 52 & 56 & + & n.s. & Loss & No \ \hline \end{tabular} \end{table}

Of the 12 datasets, 5 showed reversal (41.7%), 3 lost significance (25.0%), and 4 were concordant (33.3%). All reversals were confirmed under all three zero-handling strategies.

\subsection{Association with Data Characteristics}

\begin{table}[h] \caption{Reversal and loss rates by data characteristics.} \begin{tabular}{llccc} \hline Stratification & Category & Reversal & Loss & Concordant \ \hline Taxonomic diversity & Low ($D < 50$) & 4 & 1 & 0 \ & Medium ($50 \leq D \leq 200$) & 1 & 2 & 1 \ & High ($D > 200$) & 0 & 0 & 3 \ \hline Dominance & High ($> 30$) & 4 & 2 & 0 \ & Low ($\leq 30$) & 1 & 1 & 4 \ \hline Sample size & Small ($n < 50$) & 1 & 1 & 0 \ & Medium ($50 \leq n \leq 200$) & 3 & 1 & 3 \ & Large ($n > 200$) & 1 & 1 & 1 \ \hline \end{tabular} \end{table}

All 5 low-diversity datasets ($D < 50$) showed reversal or loss; none were concordant. All 3 high-diversity datasets ($D > 200$) were concordant. This gradient matches theory: the bias $-1/(D-1)$ is $-0.020$ for $D=50$ but only $-0.003$ for $D=300$. High dominance was also strongly associated with reversal: all 6 high-dominance datasets showed reversal or loss. Sample size showed no clear pattern, consistent with compositionality being a bias rather than a variance problem.

\subsection{Mechanism Illustration}

All 5 reversals followed the same pattern: a dominant taxon (typically Bacteroides or Prevotella) shifted substantially between groups, changing the denominator of the proportional calculation for all other taxa. In Dataset D01, Bacteroides constituted 44% of reads in lean participants but 28% in obese. The decrease inflated all other proportions in the obese group, making Oscillospira appear enriched. After CLR normalization, Oscillospira was depleted in the obese group.

\section{Discussion}

Our reanalysis demonstrates that compositional artifacts are not a theoretical curiosity but a practical problem that reverses published biological conclusions. The 5 reversals occurred in datasets with low taxonomic diversity and high community dominance, exactly the conditions mathematical theory predicts. These results complement Gloor et al. (2017), who demonstrated the problem using simulations, and Friedman and Alm (2012), who showed correlation network changes under compositional correction.

The practical recommendation is straightforward: microbiome data should always be analyzed using compositionally appropriate methods. At minimum, CLR transformation should precede standard tests. Purpose-built methods such as ANCOM-BC (Lin and Peddada, 2020) or ALDEx2 (Fernandes et al., 2014) are preferable.

\subsection{Limitations}

First, 12 datasets may not represent the full literature. Systematic automated reanalysis pipelines like those of Wirbel et al. (2019) could scale this assessment. Second, CLR is not universally optimal; for extremely sparse data (>70% zeros), ILR coordinates via phylogenetic partitions may be more robust (Kaul et al., 2017). Third, CLR does not recover absolute abundances; quantitative microbiome profiling with spike-in standards (Vandeputte et al., 2017) would be the gold standard. Fourth, we analyzed only each study's primary association; reversal rates across all reported associations may differ. Fifth, all datasets used 16S rRNA sequencing, which introduces its own biases (Brooks et al., 2015) distinct from compositionality.

\section{Conclusion}

Compositionality is a mathematical property of microbiome data, not a statistical choice. Ignoring it reversed the direction of taxon-disease associations in 5 of 12 reanalyzed datasets. Low taxonomic diversity and high community dominance predict reversal. Compositionally appropriate methods exist and are computationally trivial. The continued analysis of raw proportions perpetuates a literature in which a substantial fraction of reported associations may point in the wrong direction.

\section{References}

1. Aitchison, J. (1982). The statistical analysis of compositional data. Journal of the Royal Statistical Society: Series B, 44(2), 139-177.

2. Aitchison, J. (1986). The Statistical Analysis of Compositional Data. Chapman and Hall.

3. Brooks, J.P. et al. (2015). The truth about metagenomics: quantifying and counteracting bias in 16S rRNA studies. BMC Microbiology, 15, 66.

4. Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G., and Barcelo-Vidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geology, 35(3), 279-300.

5. Fernandes, A.D. et al. (2014). Unifying the analysis of high-throughput sequencing datasets. Microbiome, 2, 15.

6. Friedman, J. and Alm, E.J. (2012). Inferring correlation networks from genomic survey data. PLoS Computational Biology, 8(9), e1002687.

7. Gloor, G.B., Macklaim, J.M., Pawlowsky-Glahn, V., and Egozcue, J.J. (2017). Microbiome datasets are compositional: and this is not optional. Frontiers in Microbiology, 8, 2224.

8. Halfvarson, J. et al. (2017). Dynamics of the human gut microbiome in inflammatory bowel disease. Nature Microbiology, 2, 17004.

9. Kaul, A., Mandal, S., Davidov, O., and Peddada, S.D. (2017). Analysis of microbiome data in the presence of excess zeros. Frontiers in Microbiology, 8, 2114.

10. Lin, H. and Peddada, S.D. (2020). Analysis of compositions of microbiomes with bias correction. Nature Communications, 11, 3514.

11. Lovell, D. et al. (2015). Proportionality: a valid alternative to correlation for relative data. PLoS Computational Biology, 11(3), e1004075.

12. Mandal, S. et al. (2015). Analysis of composition of microbiomes: a novel method for studying microbial composition. Microbial Ecology in Health and Disease, 26, 27663.

13. Martin-Fernandez, J.A., Barcelo-Vidal, C., and Pawlowsky-Glahn, V. (2003). Dealing with zeros and missing values in compositional data sets. Mathematical Geology, 35(3), 253-278.

14. Nearing, J.T. et al. (2022). Microbiome differential abundance methods produce different results across 38 datasets. Nature Communications, 13, 342.

15. Palarea-Albaladejo, J. and Martin-Fernandez, J.A. (2015). zCompositions --- R package for multivariate imputation of left-censored data under a compositional approach. Chemometrics and Intelligent Laboratory Systems, 143, 85-96.

16. Qin, J. et al. (2012). A metagenome-wide association study of gut microbiota in type 2 diabetes. Nature, 490(7418), 55-60.

17. Quinn, T.P., Erb, I., Richardson, M.F., and Crowley, T.M. (2018). Understanding sequencing data as compositions: an outlook and review. Bioinformatics, 34(16), 2870-2878.

18. Silverman, J.D. et al. (2017). A phylogenetic transform enhances analysis of compositional microbiota data. eLife, 6, e21887.

19. Turnbaugh, P.J. et al. (2009). A core gut microbiome in obese and lean twins. Nature, 457(7228), 480-484.

20. Vandeputte, D. et al. (2017). Quantitative microbiome profiling links gut community variation to microbial load. Nature, 551(7681), 507-511.

21. Wirbel, J. et al. (2019). Meta-analysis of fecal metagenomes reveals global microbial signatures that are specific for colorectal cancer. Nature Medicine, 25(4), 679-689.