Flexible aggregation of compositional predictors with shared effects for microbiome association analysis

Problem Statement
Microbiome data analysis faces significant challenges due to the high-dimensional, sparse, and compositional nature of the data. Compositional constraints require that relative abundances sum to one, inducing dependence among taxa and necessitating regression approaches based on log-ratios (specifically the log-contrast model where coefficients sum to zero). Furthermore, microbiome datasets often contain numerous rare features (observed in few subjects), which are typically filtered out or aggregated based on fixed external phylogenetic trees. However, existing methods have limitations: penalty-based approaches (e.g., Lasso) provide point estimates that fail to capture uncertainty and may have high false positive rates; Bayesian methods often rely on external phylogenetic trees for feature aggregation, which may not accurately reflect functional similarity regarding the clinical outcome; and fixed-tree aggregation assumes evolutionary proximity dictates effect similarity, a premise the authors argue is often invalid for specific host-microbe associations.

Methodology: BRACE
The authors propose Bayesian Regression with Agglomerated Compositional Effects (BRACE), a novel framework designed for high-dimensional compositional regression that simultaneously performs variable selection, data-adaptive clustering, and enforces the log-contrast constraint. The method is built on a hierarchical Bayesian model with three key components:

1. Spike-and-Cluster Prior: The model introduces binary activity indicators ($s_j$) for each predictor to perform variable selection (spike component). For the active set, an Ewens partition prior (derived from the Chinese Restaurant Process) is applied to group active coefficients into clusters. This allows taxa with similar effects on the outcome to share a common regression coefficient, facilitating data-adaptive aggregation without relying on external trees.
2. Constrained Gaussian Prior: To satisfy the compositional constraint $\mathbf{1}^\top \beta = 0$, the model employs a projection-based constrained Gaussian prior on the cluster-level effects ($\theta$). Specifically, an unconstrained Gaussian vector is projected onto the hyperplane defined by the cluster sizes ($w^\top \theta = 0$). This ensures that the induced regression coefficients $\beta = Z\theta$ strictly satisfy the sum-to-zero constraint within the valid parameter space.
3. Posterior Inference: The authors develop a Gibbs sampler to perform posterior inference. A key technical innovation is the efficient sampling of cluster labels and effects while respecting the hyperplane constraint. This involves integrating out the cluster effects to obtain a collapsed marginal likelihood for label updates and using a fast algorithm for simulating from hyperplane-truncated multivariate normal distributions.

Key Results
The performance of BRACE was evaluated through extensive simulation studies and a real-world application:

• Simulation Studies:

  • Scenario 1 (Structured Signal): In controlled settings with structured signal and dependence, BRACE consistently achieved the lowest or near-lowest prediction error and $L_2$ loss compared to competitors (Lasso CLR, Lasso Comp, BAZE, BCGLM). It demonstrated superior ability to recover true cluster structures (high Adjusted Rand Index) and maintained well-calibrated predictive credible intervals.
  • Scenario 2 (Semi-synthetic ORIGINS Data): Mimicking the complexity of real oral microbiome data, BRACE outperformed all benchmark methods across varying signal-to-noise ratios. Notably, at high signal levels, BRACE achieved perfect variable selection (zero false positives and zero false negatives), a property not shared by competing methods.
  • Variable Selection: The authors introduced a practical null criterion combining posterior inclusion probability with a magnitude filter ($Pr(|\beta_j| > \epsilon | y) > 0.90$) to control the False Discovery Rate (FDR) effectively in correlated designs.

• Real Data Application (ORIGINS Study):

  • Applied to the Oral Infections, Glucose Intolerance, and Insulin Resistance Study, BRACE identified 17 features grouped into four nonzero clusters associated with fasting insulin levels.
  • The method identified specific taxa (e.g., Tannerella forsythia, Prevotella species) with positive associations to insulin resistance.
  • Crucially, the data-adaptive clusters did not strictly follow phylogenetic proximity. For instance, closely related species exhibited opposing effect directions, and phylogenetically distant taxa were grouped together due to shared functional effects. This confirmed the authors' hypothesis that phylogenetic trees alone are insufficient for guiding association analysis in this context.
  • Functional analysis (via PICRUSt2) revealed that the identified clusters corresponded to distinct functional profiles, supporting the biological relevance of the data-adaptive groupings.

Significance and Claims
The paper claims that BRACE represents the first Bayesian method for high-dimensional compositional regression that offers flexible, data-adaptive feature aggregation and selection without relying on fixed external phylogenetic structures. Its primary contributions are:

1. Methodological Innovation: The integration of the Ewens partition prior with a projection-based constrained Gaussian prior allows for the enforcement of the log-contrast constraint directly on the regression coefficients while enabling flexible clustering of effects.
2. Superior Performance: BRACE demonstrates superior prediction accuracy, dimension reduction, and feature selection capabilities compared to existing state-of-the-art methods, particularly in scenarios involving rare features and shared effects.
3. Interpretability: By grouping taxa with similar effects, the method yields more interpretable models and provides a mechanism to handle rare features by collapsing them into denser, more robust groupings.

Limitations
The authors acknowledge that the computational cost of the collapsed Gibbs sampler can be high, particularly when the number of inferred clusters is large relative to the number of predictors, due to repeated matrix inversions. They suggest that future work could focus on developing approximate inference strategies (e.g., variational Bayes) to improve scalability. Additionally, while the method handles zeros via standard pseudocount preprocessing, it does not explicitly model zero-inflation within the likelihood structure.