Microbiota

• Ecological community of microorganisms that resides in an environmental niche
• \(10^{14}\) bacteria in the gut among 1 500 species
• Associations with:
  • metabolism (diet, obesity, drug absorption, …)
  • diseases (IBD, allergies, diabete…)
  • behavior (smokers, antibiotics, C-section…)
  • environment (pet, water…)

Objectives

• Find which bacteria are differentially abundant between two or more groups
• Use a FDR multiple testing correction to prevent false positives (one test per bacteria)
• Incorporate hierarchical information to increase power
• Which tree?

Hierarchical False Discovery Rate

The z-scores \(\mathbf{z} = \Phi^{-1}(\mathbf{p})\) are smoothed using the following hierarchical model:

\[\mathbf{z} | \mathbb{\mu} \sim \mathcal{N}_n \left( \mathbb{\mu}, \sigma^2 \mathbf{I}_m \right) \qquad \mathbf{\mu} \sim \mathcal{N}_m\left(\gamma \mathbf{1} , \tau^2 \mathbf{C}_{\rho} \right)\]

where \(\mathbf{C}_{\rho} = \left(\text{exp} (−2\rho \mathbf{D}_{i,j} )\right)\) with \(\mathbf{D}\) the patristic distance matrix between taxa from the tree. By applying Bayes’s formula:

\[\mathbf{z} \sim \mathcal{N}_m \left(\gamma \mathbf{1},\tau^2 \mathbf{C}_{\rho} + \sigma^2 \mathbf{I}_m\right)\]

\[\mathbb{\mu}^* = \left(\mathbf{I}_m + \frac{\sigma_0^2}{\tau_0^2} \mathbf{C}_{\rho_0}^{-1}\right)^{-1}\left(\frac{\sigma_0^2}{\tau_0^2} \mathbf{C}_{\rho_0}^{-1}\gamma_0 \mathbf{1} + \mathbf{z}\right)\] Finally, a permutation-based FDR control is applied on \(\mathbb{\mu}^*\)