Testing for a common subspace in compositional datasets with structural zeros

Technical Summary: Testing for a Common Subspace in Compositional Datasets with Structural Zeros

Problem Statement
The paper addresses a specific challenge in Compositional Data Analysis (CoDA): testing for the existence of a common principal subspace between two groups of compositional data when those groups exhibit different patterns of structural zeros. Structural zeros represent the genuine absence of a phenomenon (e.g., a bacterial phylum not present in a specific body site), distinct from counting or rounded zeros.

Standard CoDA methodologies, such as Principal Component Analysis (PCA) based on logratio transformations (clr, ilr), require strictly positive data. Consequently, datasets with structural zeros are typically split into subsamples based on their zero patterns, and separate PCAs are performed. However, this approach prevents direct comparison of the resulting principal subspaces because the transformed data reside in different dimensional vector spaces ($\mathbb{R}^{D-Q_1-1}$ vs. $\mathbb{R}^{D-Q_2-1}$). Existing methods for comparing principal subspaces (e.g., Flury's CPC model or Schott's test) assume data lie in the same Euclidean space, an assumption violated by structural zeros.

Methodology
The authors propose a statistical framework to test the null hypothesis ($H_0$) that two compositional subpopulations, $Y$ and $Z$, with distinct structural zero patterns, share a common principal component subspace of size $K$.

1. Geometric Framework:

   • The dataset is partitioned into two groups, $Y$ and $Z$, where $Y$ has $Q_1$ structural zeros and $Z$ has $Q_2$ structural zeros.
   • The authors apply the isometric logratio (ilr) transformation to the non-zero parts of each group. Crucially, they utilize a specific "backward pivot" coordinate system where the first $M-1$ components (where $M$ is the number of common non-zero parts) are unaffected by the structural zeros of the other group.
   • To compare the subspaces, the authors define linear canonical inclusion operators ($\iota_Y$ and $\iota_Z$) that map the eigenvectors from the reduced spaces ($\mathbb{R}^{D-Q_1-1}$ and $\mathbb{R}^{D-Q_2-1}$) into a common ambient space $\mathbb{R}^{D-1}$ by padding with zeros corresponding to the structural zero positions.

2. Test Statistic:

   • Inspired by Schott (1988), the authors define a test statistic $T$ based on the eigenvalues of the sample covariance matrices of the transformed data ($\hat{\Omega}_Y, \hat{\Omega}_Z$) and a pooled covariance matrix constructed via the inclusion operators.
   • $T = \sum_{i=1}^K \left( (n_y - 1)\hat{\alpha}_i + (n_z - 1)\hat{\beta}_i - \hat{\gamma}_i \right)$, where $\hat{\alpha}, \hat{\beta}$ are eigenvalues of the individual groups and $\hat{\gamma}$ are eigenvalues of the pooled matrix.

3. Inference Procedures:

   • Parametric Approach: Assuming the data follow a normal distribution on the simplex (logratio-normality), the authors derive an analytical approximation for the null distribution of $T$. This distribution is approximated by a scaled chi-squared distribution, $T \sim \frac{\sigma_T^2}{2\mu_T} \chi^2_{\frac{2\mu_T^2}{\sigma_T^2}}$, where parameters $\mu_T$ and $\sigma_T^2$ are functions of the population eigenvalues and eigenvectors. These are estimated using sample statistics.
   • Nonparametric Approach: A bootstrap procedure is introduced for cases where distributional assumptions do not hold. This involves rotating the datasets to align their principal subspaces under the null hypothesis, resampling with replacement, and computing the empirical p-value.

Key Results
The validity of the proposed methods was evaluated through simulation studies and a real-data application:

• Simulation Study:

  • Gaussian Data: Under logratio-normality, the parametric test (using Schott's formula) generally achieved type I error rates closest to the nominal 5% level and exhibited the highest power. The bootstrap test performed comparably.
  • Heavy-Tailed Data (Student's t): When data were sampled from multivariate Student's t-distributions (low degrees of freedom), the parametric Schott test became overly liberal (high type I error). In contrast, the bootstrap test maintained better control over type I error and demonstrated higher power than the estimated parametric version.
  • Uniform Data: The parametric tests were found to be too conservative (type I error < 0.05), while the bootstrap test maintained error rates closer to the nominal level.
  • Sample Size: The bootstrap procedure showed superior performance for small sample sizes compared to the estimated parametric test.

• Real-Data Application (Human Microbiome Project):

  • The method was applied to 16S rRNA sequencing data aggregated at the phylum level across different body sites.
  • Comparison 1 (Right Antecubital Fossa vs. Anterior Nares): The null hypothesis of a common subspace ($K=1, 2$) was not rejected. In this comparison, $Q_1=0$ (Right Antecubital Fossa has no structural zeros) and $Q_2=4$ (Anterior Nares has 4 structural zero phyla). The analysis identified Cyanobacteria and Fusobacteria as the primary contributors to this shared subspace, driven by their sparse presence/absence patterns.
  • Comparison 2 (Stool vs. Saliva): The null hypothesis was rejected for all tested $K$, confirming that these sites do not share a common principal subspace in their phylum-level composition.

Significance and Contributions
The paper claims to generalize the hypothesis testing framework for common principal subspaces (specifically extending Schott, 1988) to the complex setting of compositional data with structural zeros. Its primary contributions are:

1. Geometric Compatibility: The method allows for the comparison of principal subspaces across groups with different zero patterns while remaining fully compatible with Aitchison geometry and logratio analysis, avoiding the pitfalls of zero-replacement strategies.
2. Robust Inference: The introduction of a nonparametric bootstrap procedure provides a robust alternative when the assumption of logratio-normality is violated, a common issue in real-world microbiome and omics data.
3. Practical Utility: The application to microbiome data demonstrates the method's ability to identify shared biological characteristics (common subspaces) across different anatomical sites despite differing structural zero patterns, offering a tool for stratification and classification studies in high-throughput compositional datasets.

The authors conclude that while the parametric test is effective under normality, the bootstrap approach offers a more reliable and powerful alternative in the presence of heavy tails or small sample sizes, making it a valuable tool for analyzing complex compositional datasets.