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<!DOCTYPE html>
<html>
<head>
<title>Heritability</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<link rel="stylesheet" href="fonts/quadon/quadon.css">
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<link rel="stylesheet" href="slides_style_i.css">
<script type="text/javascript" src="assets/plotly/plotly-latest.min.js"></script>
</head>
<body>
<textarea id="source">
<!-- TODO add slide numbers & maybe slide name -->
### Heritablity of Human Structural Connectomes
Jaewon Chung
---
name:talk
### Outline
- [Background](#defn)
- [Measuring Heritability](#statistics)
- [Discussion](#disc)
---
name:defn
### Outline
- Background
- [Measuring Heritability](#statistics)
- [Discussion](#disc)
---
### 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?
---
### 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
---
### Graph = Adjacency matrix
---
### 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
---
name:statistics
### Outline
- [Background](#defn)
- Measuring Heritability
- [Discussion](#disc)
---
### 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 $D^Y$
---
### 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>
</center>
---
### Distance between graphs: Step 2
- Distance = Frobenius norm of difference in latent positions
- $\delta_X(G, H) = ||\hat{X}R - \hat{Y}||_F$
<center></center>
High distance: pair of graphs are less similar (or more dissimilar)
---
### 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></center>
---
### Are the connectome and genetics distances independent?
--
<br><br><br><br>
<center>No (reject null)</center>
---
### Distribution of distances are ordered
<center>
</center>
---
### 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
---
### What about confounders?
- Studies show anatomical features are heritable
- e.g. brain volume
- Prior results explained by anatomy?
---
### 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>
</center>
---
### 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>
---
### 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>
</center>
---
### 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?]
---
### 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>
<img src="images/funding/nsf_fpo.png" STYLE="HEIGHT:95px;"/>
<img src="images/funding/nih_fpo.png" STYLE="HEIGHT:95px;"/>
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<img src="images/funding/iarpa_fpo.jpg" STYLE="HEIGHT:95px;"/>
<img src="images/funding/KAVLI.jpg" STYLE="HEIGHT:95px;"/>
<img src="images/funding/schmidt.jpg" STYLE="HEIGHT:95px;"/>
</textarea>
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