Impact of tree choice in metagenomics differential abundance studies

class: title-slide, center, middle
count: false

# Impact of tree choice in metagenomics differential abundance studies

## Antoine Bichat
<div style = "margin-top: -30px"></div>

## September 13, 2019 - MIAT

### Work in collaboration with C. Ambroise, M. Mariadassou & J. Plassais

---
class: center, middle, inverse

# Context

---
## Microbiota

_Ecological community of microorganisms that reside in an environmental niche_

.footnote[📘 Gut: The Inside Story of Our Body's Most Underrated Organ  (Giulia Enders)]

.pull-left[

#### Some figures for human gut

* `$10^{\small{14}}$` bacterial cells in one gut...

* ... weighing 2 kg

* More than 1&#8239;500 different species

* More than 10 millions unique genes

]

.pull-right[

#### Proven associations

* Immune system

* Crohn's disease

* Vaginosis

* Diabete

* Tobacco

* Diet

* Antibiotics

* Birth mode

]

---
## Data - abundances of taxa

.footnote[📄 Ravel et al. (2011)]

```
 # A tibble: 122 x 395
 Taxa S001 S002 S003 S004 S005 S006 S007 S008 S009 S010
 <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
 1 Lactob… 2318 1388 1361 2256 88 1770 1490 119 2136 1790
 2 Prevot… 0 1 1 0 525 7 134 753 0 0
 3 Megasp… 0 1 0 0 402 0 4 102 0 0
 4 Sneath… 0 0 0 0 302 0 35 272 0 0
 5 Atopob… 0 1 0 0 84 0 12 54 0 0
 6 Strept… 0 0 3 0 0 0 138 4 0 2
 7 Dialis… 0 1 0 0 152 4 2 192 0 0
 8 Anaero… 0 1 3 2 0 9 12 13 0 0
 9 Pepton… 0 1 0 0 7 2 6 50 0 0
 10 Eggert… 0 0 0 0 2 0 0 7 0 0
 # … with 112 more rows, and 384 more variables
```

* Count data (or compositional) data

* Zero-inflated data

* Correlation between species

* Counts spanning several orders of magnitude: `$1 \rightarrow 10^{\small{8}}$`

---
## Data - taxonomy

.footnote[📄 Ravel et al. (2011)]

```
 # A tibble: 129 x 5
 Phylum Class Order Family Genus 
 <chr> <chr> <chr> <chr> <chr> 
 1 Actinobacteria Actinobacteria Actinomycetales Actinomycetaceae Actinobaculum 
 2 Actinobacteria Actinobacteria Actinomycetales Actinomycetaceae Actinomyces 
 3 Actinobacteria Actinobacteria Actinomycetales Actinomycetaceae Arcanobacterium
 4 Actinobacteria Actinobacteria Actinomycetales Actinomycetaceae Mobiluncus 
 5 Actinobacteria Actinobacteria Actinomycetales Actinomycetaceae Varibaculum 
 # … with 124 more rows
```

---
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## Data - taxonomy

.center[<img src="img/tree_tax.png"/ width="500">]

---
class: center, middle, inverse

# Differential abundance studies

---
## Evaluation of binary classifiers

<div class="tabwid"><table style='border-collapse:collapse;'><tbody><tr><td colspan="2" rowspan="2" style="width:54px;height:36px;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;"></td><td colspan="2" style="width:54px;height:36px;background-color:transparent;vertical-align: middle;border-bottom: 2px solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">True condition</td></tr><tr><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(190, 190, 190, 0.20);vertical-align: middle;border-bottom: 1px solid rgba(0, 0, 0, 1.00);border-top: 2px solid rgba(0, 0, 0, 1.00);border-left: 2px solid rgba(0, 0, 0, 1.00);border-right: 1px dashed rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">Condition positive</td><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(190, 190, 190, 0.20);vertical-align: middle;border-bottom: 1px solid rgba(0, 0, 0, 1.00);border-top: 2px solid rgba(0, 0, 0, 1.00);border-left: 1px dashed rgba(0, 0, 0, 1.00);border-right: 2px solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">Condition negative</td></tr><tr><td rowspan="2" style="width:54px;height:36px;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 2px solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">Prediction</td><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(190, 190, 190, 0.20);vertical-align: middle;border-bottom: 1px dashed rgba(0, 0, 0, 1.00);border-top: 2px solid rgba(0, 0, 0, 1.00);border-left: 2px solid rgba(0, 0, 0, 1.00);border-right: 1px solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">Predicted positive</td><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(0, 255, 0, 0.50);vertical-align: middle;border-bottom: 1px dashed rgba(0, 0, 0, 1.00);border-top: 1px solid rgba(0, 0, 0, 1.00);border-left: 1px solid rgba(0, 0, 0, 1.00);border-right: 1px dashed rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">TP</td><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(255, 0, 0, 0.50);vertical-align: middle;border-bottom: 1px dashed rgba(0, 0, 0, 1.00);border-top: 1px solid rgba(0, 0, 0, 1.00);border-left: 1px dashed rgba(0, 0, 0, 1.00);border-right: 2px solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">FP</td></tr><tr><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(190, 190, 190, 0.20);vertical-align: middle;border-bottom: 2px solid rgba(0, 0, 0, 1.00);border-top: 1px dashed rgba(0, 0, 0, 1.00);border-left: 2px solid rgba(0, 0, 0, 1.00);border-right: 1px solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">Predicted negative</td><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(255, 0, 0, 0.50);vertical-align: middle;border-bottom: 2px solid rgba(0, 0, 0, 1.00);border-top: 1px dashed rgba(0, 0, 0, 1.00);border-left: 1px solid rgba(0, 0, 0, 1.00);border-right: 1px dashed rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">FN</td><td style="width:54px;height:36px;background-clip: padding-box;background-color:rgba(0, 255, 0, 0.50);vertical-align: middle;border-bottom: 2px solid rgba(0, 0, 0, 1.00);border-top: 1px dashed rgba(0, 0, 0, 1.00);border-left: 1px dashed rgba(0, 0, 0, 1.00);border-right: 2px solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;"><p style="margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:2px;padding-top:2px;padding-left:5px;padding-right:5px;background-color:transparent;">TN</td></tr></tbody></table></div>

The true condition is usually unknown for real dataset.

The prediction is ususally determined by comparing the _p_-value to `$\alpha = 0.05$`.
--

`$$\text{TPR} = \frac{\text{TP}}{\left(\text{TP} + \text{FN}\right) \wedge 1 }$$`

`$$\text{ACC} = \frac{\text{TP} + \text{TN}}{\text{TP} + \text{TN} + \text{FP} + \text{FN}}$$`

`$$\text{FDR} = \frac{\text{FP}}{\left(\text{TP} + \text{FP}\right) \wedge 1 }$$`

---
## Statistical issue

Univariate tests on hundred of taxa

Need for a multiple testing controling procedure!

.footnote[📄 Philippot et al. (2010)]

.pull-left[
A hierarchy is available

<img src="img/coherence.png" width="400">
]

.pull-right[
Can we use it to do it better?
]

---
count: false
## Statistical issue

.footnote[📄 Philippot et al. (2010)]

Univariate tests on hundred of taxa

Need for a multiple testing controling procedure!

.pull-left[
A hierarchy is available

<img src="img/coherence.png" width="400">
]

.pull-right[

Can we use it to do it better?

* Hierarchical FDR

* _z_-scores smoothing
]

---
## Hierarchical FDR

.footnote[📄 Yekutieli (2008)]

This procedure inscreases statistical power by lessening the number of test to do with a descending method:

* Test the family `$\mathcal{T}_0$`

* If node `$t$` is rejected, test `$\mathcal{T}_t = \left\{H_i \mid \text{Par}(i) = t\right\}$` with a BH procedure at level `$q$`

.center[
 <img src="img/tree_hyp.png" width="420">
]

---
count: false
## Hierarchical FDR

.footnote[📄 Yekutieli (2008)]

This procedure inscreases statistical power by lessening the number of test to do with a descending method:

* Test the family `$\mathcal{T}_0$`

* If node `$t$` is rejected, test `$\mathcal{T}_t = \left\{H_i \mid \text{Par}(i) = t\right\}$` with a BH procedure at level `$q$`

.center[
 <img src="img/tree_hyp_1.png" width="420">
]

---
count: false

## Hierarchical FDR

.footnote[📄 Yekutieli (2008)]

This procedure inscreases statistical power by lessening the number of test to do with a descending method:

* Test the family `$\mathcal{T}_0$`

* If node `$t$` is rejected, test `$\mathcal{T}_t = \left\{H_i \mid \text{Par}(i) = t\right\}$` with a BH procedure at level `$q$`

.center[
 <img src="img/tree_hyp_2.png" width="420">
]

---
count: false
## Hierarchical FDR

.footnote[📄 Yekutieli (2008)]

This procedure inscreases statistical power by lessening the number of test to do with a descending method:

* Test the family `$\mathcal{T}_0$`

* If node `$t$` is rejected, test `$\mathcal{T}_t = \left\{H_i \mid \text{Par}(i) = t\right\}$` with a BH procedure at level `$q$`

.center[
 <img src="img/tree_hyp_3.png" width="420">
]

.center[
__This procedure controls the FDR at level__

`$\large{1.44 \times q \times \frac{\#\text{discoveries } +\text{ } \#\text{families tested}}{\#\text{discoveries } +\text{ } 1}}$`
]

---
## z-scores smoothing

.footnote[📄 Xiao et al. (2017)]

Denote by `$\mathbf{z}$` the vector of observed _z_-scores and `$\mathbf{\mu}$` the vector of "true" _z_-scores

Assume that 
`$\mathbf{z} | \mathbb{\mu} \sim \mathcal{N}_n \left( \mathbb{\mu}, \sigma^2 \mathbf{I}_m \right)$` and `$\mathbf{\mu} \sim \mathcal{N}_m\left(\gamma \mathbf{1} , \tau^2 \mathbf{C}_{\rho} \right)$`

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

and Bayes formula gives 
 
`$$\mathbb{\mu}^* = \left(\mathbf{I}_m + \frac{\sigma_0^2}{\tau_0^2} \mathbf{C}_{\rho_0}^{-1}\right)^{-1}\left(\frac{\sigma_0^2}{\tau_0^2} \mathbf{C}_{\rho_0}^{-1}\gamma_0 \mathbf{1} + \mathbf{z}\right)$$`

with `$\sigma_0$`, `$\tau_0$`, `$\rho_0$` and `$\gamma_0$` hypermarameters

After smoothing, a multiple testing correction could be done on smoothed values

---
## Which tree?

#### Taxonomy? Phylogeny?

* Proxy for correlations at high-level niches

* Not so much for low-level niches?

* Not available everytime

---
class: nologo
count: false
## Which tree?

#### Taxonomy? Phylogeny?

* Proxy for correlations at high-level niches

* Not so much for low-level niches?

* Not available everytime

#### Correlation tree?

* Actual correlation between taxa

* Computed from data using pairwise correlation

.center[
<img src="img/tree_cor.png" width="350">
]

---
class: center, middle, inverse

# Comparison of trees

---
## Billera-Holmes-Vogtamnn distance

.footnote[📄 Billera, Holmes & Vogtmann (2001)]

The BHV distance is the length of the unique shortest path between the trees on treespace

.pull-left[
 .center[
<img src="img/dBHV_2.png" height=300/>
 ]
]

.pull-right[
 .center[
 <img src="img/dBHV_4.png" height=300/>
 ]
]

---
## Quantifying distance between trees

* __trees of primary interest__

* correlation tree on original data
 
 * taxonomy
 
--

* __what is the confident region for the correlation tree?__

* correlation trees on boostrapped data (resampling on samples)

* __are trees significantly closer than two random trees?__

* trees created by random shuffling of correlation tree tip labels
 
 * trees created by random shuffling of taxonomy tip labels

We compute all pairwise distances between these trees

---
## Random shuffling

.center[<img src="img/shuffling.png", height="500">]

---
## Dataset

.footnote[📄 Ravel et al. (2011)]

* Vaginal microbiome of non pregnant women sequenced by 16S

* 40 different genera after filtering (~&#8239;30&#8239;%)

.pull-left[ 
 <img src="img/tree_tax.png">
 .center[Taxonomy]
]

.pull-right[
 <img src="img/tree_cor.png">
 .center[Correlation tree] 
]

---
## Pairwise distances

.pull-left[
.center[<img src="img/prez_ravel_comp_box.png" width="650">]
]

.pull-right[
.center[<img src="img/prez_ravel_comp_pcoa.png" width="650">]
]

---
count: false
## Pairwise distances

.center[<img src="img/comp_cohorts-bhv.jpg" width="800">]

😀 The correlation tree is different from the taxonomy

---
class: center, middle, inverse

# Evaluation of hFDR

---
## Chlamydiae dataset

.footnote[📄 Caporaso et al. (2011), Sankaran & Holmes (2014)]

* Small subset of the `GlobalPatterns` dataset narrowed to Chlamydiae phylum

* 21 different OTUs

* 26 samples representing 9 very different environments: soil, ocean, feces, skin...

## Method

* Find which bacteria are differentially abundant between environments

* Association using Fisher statistic (ANOVA)

* Correction with hierarchical FDR

---
## Abundances of detected species

.center[<img src="img/realdata_chlamydia_box.png" height=500/>]

---
## Representation of evidences on trees

.center[<img src="img/tree_mirors.png" height=500/>]

---
## But...

`$\alpha = 0.10$` is only the family-level FDR.

The _a posteriori_ global FDR is:

* `$\alpha' = 0.32$` for phylogenetic correction

* `$\alpha' = 0.324$` for correlation correction

A BH procedure at the same global FDR level leads to 15 discoveries (+5)

😀 Using correlation tree instead of taxonomy yields more results

☹️ Vanilla BH beats hFDR for a given level

---
## Chaillou Dataset

.footnote[📄 Chaillou et al. (2015)]

* Food-associated microbiota of processed meat and seafood products

* 97 different OTUs

* 80 samples across 8 different food type: beef, veal, salmon, shrimp...

---
## Abundances of detected species

.center[<img src="img/chaillou_abund.png" height=500/>]

---
## Impact of the test

.pull-left[

.center[
 Position of selected OTUs on the correlation tree
 <img src="img/chaillou_tree_abund1.jpeg" height=400/>
 ]
]

.pull-right[
OTUs in red are only detected by phylogeny
]

---
count: false
## Impact of the test

.pull-left[

.center[
 Position of selected OTUs on the correlation tree
 <img src="img/chaillou_tree_abund2.jpeg" height=400/>
 ]
]

.pull-right[
OTUs in red are only detected by phylogeny

All OTU in the clade are differentially abundant

Obvious unequal variances between groups

]

---
count: false
## Impact of the test

.pull-left[

.center[
 Position of selected OTUs on the correlation tree
 <img src="img/chaillou_tree_abund3.jpg" height=400/>
 ]
]

.pull-right[
OTUs in red are only detected by phylogeny

All OTU in the clade are differentially abundant

Obvious unequal variances between groups

KW test is more robust than F-test in this setting

☹️ Implemented tests are not appropriate to metagenomic data

]

---
class: center, middle, inverse

# Evaluation of z-scores smoothing

---
## Zeller dataset

.footnote[📄 Zeller et al. (2014)]

* Dataset from cancer study

* 119 different genera (after filtering)

* 199 samples: 42 adenoma, 91 carcinoma and 66 control

## Method

* Find which bacteria are differentially abundant between diseases

* Association using Kruskal-Wallis test

* Correction with hierarchical _p_-value smoothing

---
## Impact of the tree

.center[<img src="img/zeller-lines-bold.png" height=430/>]

😊 _z_-scores smoothing is slightly better than vanilla BH

☹️ All hierachies give highly similar results

---
class: center, middle, inverse

# Simulations

---
## Workflow

.footnote[📄 Brito et al. (2016)]

* Simulate DA taxa starting from an homogeneous dataset

* Correction with BH and hierarchical _p_-value smoothing

.center[<img src="img/workflow_simu.png" height=450/>]

---
## Change in z-scores

`$$\text{Mean z-smoothing} = |z_{\text{raw}} - z_{\text{adjusted}}|$$`

.center[<img src="img/simu_np_mean_zsmoothing.jpg" height=290/>]

☹️ Taxonomy behaves like random tree and has little impact

😩 In most cases, smoothing has absolutely no impact on the result

---
## Evaluation

.center[<img src="img/simu_np_lines_prez.png" height=390/>]

😀 Using correlation tree instead of taxonomy yields more results

☹️ Vanilla BH is better

🤔 Taxonomy is worst than random trees

---
class: center, middle, inverse

# Conclusions

---

## Conclusions

😀 Correlation tree and taxonomy are very different

😀 Replacing taxonomy with correlationn increases the TPR

☹️ Vanilla BH is more powerful than hFDR

☹️ Bayesian smoothing does not really depend on the tree for _z_-scores smoothing

📦 `correlationtree`

📦 `yatah` + [![CRAN_Status_Badge](https://www.r-pkg.org/badges/version/yatah)](https://cran.r-project.org/package=yatah)

📦 `evabic`

---
class: center, middle, inverse

# Further work
# Evolution of z-scores on a tree

---
## Motivations

With _z_-scores smoothing procedure form Xiao, there is no mathematical explanatory model to describe the relation between _z_-scores

We will use proper mathematical formalism to model evolution of _z_-scores on a tree and propose an adapted tree-based smoothing

Similar work has been done by Paul Bastide

---
## Ornstein–Uhlenbeck process

An Ornstein-Uhlenbeck (OU) process is a Gaussian process that satisfies the SDE:

`$$\text{d}W_t = - \alpha (W_t - \beta) \text{d}t + \sigma \text{d}B_t.$$`

`$\beta$` is the optimum value and `$\alpha$` is the rate of convergence

.center[<img src="img/plot_OU.png" height=400/>]

---
## Stochastic process on a tree

.center[<img src="img/OU_tree1.png" height=200/>]
.center[<img src="img/OU_tree2.png" height=300/>]

.footnote[📄 Bastide et al. (2016)]

---
## Model

* Each tip `$i \in \{1, ..., m\}$` has an associated _p_-value `$p_i$` and thus an associated _z_-score `$z_i$`

* Under `$H_0$`, `$p_i \sim \mathcal{U}$` so that  `$z_i \sim \mathcal{N}(0, 1)$`

* Under `$H_1$`, `$p_i \leq \mathcal{U}$` so we can assume `$z_i \sim \mathcal{N}(\mu_i, 1)$` with `$\mu_i \leq 0$`

* With the tree structure, `$Z \sim \mathcal{N}_m\left(M, \Sigma \right)$` where `$\Sigma_{i, j} = \frac{\sigma^2}{2\alpha}\left(1 - e^{-2\alpha t_{i,j} }\right) e^{-\alpha d_{i,j}}$` is given by the tree

* Work remains to be done about estimating `$M \in \left\{\sum_{i=1}^{m}\mu_i e_i, \mu_i \leq 0\right\} = \mathcal{C}_m$`

---
## Decomposition of M

We denote

* `$n$` the number of internal nodes ( `$n = m-1$` for binaries trees)

* `$T \in \{0, 1\}^{m\times (n+m)}$` the tree incidence matrix _ie_ `$T_{i,j} = 1$` iff branch `$j$` lies on the path from the root to node `$i$`

* `$\Delta \in \mathcal{C}_{n+m}$` the vector of shifts on the branches

When a shift occurs on a branch, it is progagates to its descendant, thus

`$$M = T\Delta$$`

Then, if we impose a sparsity condition on `$\Delta$`

`$$\widehat{\Delta} \in \text{argmax}_{\Delta \in \mathcal{C}_{n+m}} \left(Z-T\Delta\right)^T\left(\Sigma\right)^{-1}\left(Z-T\Delta\right) +  \lambda \|\Delta\|_1$$`

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## References

* Blander, J. Magarian, et al. "Regulation of inflammation by microbiota interactions with the host." Nature immunology 18.8 (2017): 851.

* Morgan, Xochitl C., et al. “Dysfunction of the intestinal microbiome in inflammatory bowel disease and treatment.” Genome biology 13.9 (2012): R79.

* Ravel, Jacques, et al. “Vaginal microbiome of reproductive-age women.” Proceedings of the National Academy of Sciences 108.Supplement 1 (2011): 4680-4687.

* Qin, Junjie, et al. “A metagenome-wide association study of gut microbiota in type 2 diabetes.” Nature 490.7418 (2012): 55.

* Opstelten, Jorrit L., et al. "Gut microbial diversity is reduced in smokers with Crohn's disease." Inflammatory bowel diseases 22.9 (2016): 2070-2077.

* Bokulich, Nicholas A., et al. “Antibiotics, birth mode, and diet shape microbiome maturation during early life.” Science translational medicine 8.343 (2016): 343ra82-343ra82.

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