Heritability

### Heritablity of Human Structural Connectomes

![:scale 40%](images/neurodata_blue.png)

Jaewon Chung

---
name:talk

### Outline

- [Background](#defn)
- [Measuring Heritability](#statistics)
- [Discussion](#disc)

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name:defn

### Outline

- Background
- [Measuring Heritability](#statistics)
- [Discussion](#disc)

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### What is Heritability?

- .ye[Heritability]: phenotypic variations due to genetic variations
- understand effects of genes (+environment) on brain circuitry
- understand neurologic diseases

Question: Are the connectivity patterns in human brains heritable?

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### What is a graph?
(aka networks or connectomes)

- Vertex = a region of interest
- Edges = connectivity measure between a pair of vertices
- Diffusion MRI = # of estimated neuronal fibers

![:scale 75%](images/herit/graph.png)

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### Graph = Adjacency matrix

![:scale 100%](images/herit/representations.png)

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### What data will we be using?

- Human Connectome Project dataset
- Identical twins (monozygotic), fraternal twins (dizygotic), siblings
  - $N\approx 1200$ individuals recruited
- Most have diffusion and functional MRI

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name:statistics

### Outline 
- [Background](#defn)
- Measuring Heritability
- [Discussion](#disc)

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### Hypothesis testing: distance correlation (DCorr)
- Tests whether $X$ and $Y$ are independent. 
- Key idea: measures correlation between distance matrices $D^X$ and $D^Y$
  - $D_{ij}^X = \delta_X(x_i, x_j), D_ij^Y = \delta_Y(y_i, y_j)$

<br><br>
Are differences in pairs of connectomes indepedent of genetics?
- Need to compute $D^X$ and $D^Y$

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### Distance between graphs: Step 1
- Compute adj. spectral embedding (ASE) on graphs $G, H$
  - Embeddings = latent positions
  - ASE$(G) = \hat{X}$, ASE$(H) = \hat{Y}$
    - $\hat{X}, \hat{Y}\in\mathbb{R}^{N\times d}$

<center>
![:scale 80%](images/herit/ase.png)
</center>

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### Distance between graphs: Step 2
- Distance = Frobenius norm of difference in latent positions
  - $\delta_X(G, H) = ||\hat{X}R - \hat{Y}||_F$

<center>![:scale 80%](images/herit/compute_distance.png)</center>

High distance: pair of graphs are less similar (or more dissimilar)

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### What is genetic distance?
- Encode via labels
  - $\delta_Y(y_i, y_j) = 0$ if monozygotic (or self)
  - $\delta_Y(y_i, y_j) = 1$ if dizygotic/sibling
  - $\delta_Y(y_i, y_j) = 2$ if unrelated

<center>![:scale 35%](images/herit/genetic_distance.png)</center>

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### Are the connectome and genetics distances independent?

--
<br><br><br><br>
<center>No (reject null)</center>

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### Distribution of distances are ordered

<center>
  ![:scale 50%](images/herit/distances.png)
</center>

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### Mann-Whitney test

<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>P-value</th>

</tr>
  </thead>
  <tbody>
    <tr>
      <th>MZ < DZ</th>
      <td>1.647243e-02</td>

</tr>
    <tr>
      <th>MZ < Sibling</th>
      <td>2.302542e-08</td>

</tr>
    <tr>
      <th>MZ < Unrelated</th>
      <td>9.732951e-19</td>

</tr>
    <tr>
      <th>DZ < Sibling</th>
      <td>1.042699e-02</td>

</tr>
    <tr>
      <th>DZ < Unrelated</th>
      <td>1.154470e-08</td>
 
    </tr>
    <tr>
      <th>Sibling < Unrelated</th>
      <td>1.060595e-13</td>

</tr>
  </tbody>
</table>

Caveat: samples are not independent
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### What about confounders?

- Studies show anatomical features are heritable
  - e.g. brain volume

- Prior results explained by anatomy?

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### Are the covariates themselves heritable?
- Features: brain volume, four measurements of diffusivity
- Covariate distance: $\delta_Z(z_i, z_j) = ||z_i - z_j||_F$
- DCorr(covariate distance, genetic distances) = <font color="red">reject null</font>

<center>
  ![:scale 40%](images/herit/covariate_dist.png)
</center>

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### Mann-Whitney test

<table border="2" class="table">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>P-values</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>MZ < DZ</th>
      <td>1.074841e-02</td>
    </tr>
    <tr>
      <th>MZ < Sibling</th>
      <td>3.092601e-04</td>
    </tr>
    <tr>
      <th>MZ < Unrelated</th>
      <td>1.462330e-08</td>
    </tr>
    <tr>
      <th>DZ < Sibling</th>
      <td>3.345203e-01</td>
    </tr>
    <tr>
      <th>DZ < Unrelated</th>
      <td>4.825626e-03</td>
    </tr>
    <tr>
      <th>Sibling < Unrelated</th>
      <td>3.549662e-06</td>
    </tr>
  </tbody>
</table>

---

### Conditional Distance Correlation (CDCorr)

- Same as DCorr, but we now test $X \perp Y | Z$
- CDCorr(connectome distances, genetic distances | covariate distances) = <font color="red">reject null</font>

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### Finding “non-heritable” Induced Subgraph
- Subgraph = graph formed from subset of vertices
- Motivation: heritability explained by small subset of vertices?
- Repeat CDCorr for each vertex, discard vertices with small p-value

<center>
  ![:scale 40%](images/herit/subgraph.png)
</center>

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### Mann-Whitney Test
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>P-value</th>
    </tr>

</thead>
  <tbody>
    <tr>
      <th>MZ < DZ</th>
      <td>4.237065e-01</td>

</tr>
    <tr>
      <th>MZ < Sibling</th>
      <td>3.990682e-03</td>

</tr>
    <tr>
      <th>MZ < Unrelated</th>
      <td>9.618273e-07</td>

</tr>
    <tr>
      <th>DZ < Sibling</th>
      <td>6.148544e-03</td>

</tr>
    <tr>
      <th>DZ < Unrelated</th>
      <td>1.987379e-06</td>

</tr>
    <tr>
      <th>Sibling < Unrelated</th>
      <td>7.779698e-06</td>
    </tr>
  </tbody>
</table>
---
class: middle

.center[questions?]
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### Acknowledgements

<div class="small-container">
  <img src="faces/jovo.png"/>
  <div class="centered">Josh Vogelstein</div>
</div>
<div class="small-container">
  <img src="faces/cep.png" />
  <div class="centered">Carey Priebe</div>
</div>
<div class="small-container">
  <img src="faces/ebridge.jpg" />
  <div class="centered">Eric</div>
</div>
<div class="small-container">
  <img src="faces/jesus.jpg"/>
  <div class="centered">Jesus</div>
</div>
<div class="small-container">
  <img src="faces/jayanta.jpg"/>
  <div class="centered">Jayanta</div>
</div>
<div class="small-container">
  <img src="faces/pedigo.jpg"/>
  <div class="centered">Ben</div>
</div>
<div class="small-container">
  <img src="faces/loftus.jpg"/>
  <div class="centered">Alex</div>
</div>
<div class="small-container">
  <img src="faces/ross.jpg"/>
  <div class="centered">Ross</div>
</div>