Statistical methods for the quantitative genetic analysis of high-throughput phenotyping data

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# Statistical methods for the quantitative genetic analysis of high-throughput phenotyping data
## Structural equation model GWAS <br /> The 64th RBras and 18th SEAGRO Meeting
### Gota Morota <br /><a href="http://morotalab.org/" class="uri">http://morotalab.org/</a> <br />
### 2019/8/2

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# Multivariate traits
![](MTM1.png)

- Multi-trait GWAS (MTM-GWAS)

- Structural equation model GWAS (SEM-GWAS)

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# MTM-GWAS
<img src="MTMGWAS.png" height="406px" width="650px"/>

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# Multiple-trait GWAS (MTM-GWAS)
<img src="MTM.png" height="506px" width="750px"/>

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# Interrelationships among traits 
![](MTM2.png)

1. Biological prior 
    - previous experiments
    - literature

2. Infer from the data

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# Phenotypic,  genetic, and residual networks
![](MTM3.png)

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# Structural equation model GWAS (SEM-GWAS)
Direct effect of SNP on Trait 4

![](SEM1.png)

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# SEM-GWAS
Indirect effect of SNP on Trait 4 through Trait 1 (1)

![](SEM2.png)

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# SEM-GWAS
Indirect effect of SNP on Trait 4 through Trait 2 (2)

![](SEM3.png)

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# SEM-GWAS
Indirect effect of SNP on Trait 4 through Trait 2 & 3 (3)

![](SEM4.png)

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# SEM-GWAS
Indirect effect of SNP on Trait 4 through Trait 3 (4)

![](SEM5.png)

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# SEM-GWAS vs. MTM-GWAS
Effect of SNP on Trait 4

`\begin{align*}
     \text{Effect of SNP (MTM-GWAS)} &\approx \text{Overall effect (SEM-GWAS)}\\
     &= \text{Direct effect (SEM-GWAS)} \\
     &+ \text{Indirect effect (SEM-GWAS)} 
\end{align*}`

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# How to fit SEM-GWAS?

1. Fit MTM and decompose phenotypes into genetics and residuals

2. Infer the network structure of traits

3. Fit SEM-GWAS given the network structure

Example 1: Chicken data (n = 1,351),  Momen et al. (2018)

- breast muscle (BM)

- body weight (BW)

- hen-house egg production (HHP)

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# Inferred network structure (1)
<img src="chickenSEM1.png" height="276px" width="600px"/>

Structual equation model

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# Inferred network structure (2)
<img src="chickenSEM1.png" height="276px" width="600px"/>

Structual equation model

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# Inferred network structure (3)
<img src="chickenSEM1.png" height="276px" width="600px"/>

Decomposition of SNP effects 
<img src="chickenSEM2.png" height="176px" width="500px"/>

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# Manhattan plots of BW (overall, direct, and indirect SNP effects)
<img src="chickenSEM6.png" height="426px" width="560px"/>

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# More details about Chicken SEM-GWAS
<img src="frontiers.png" height="400px" width="630px"/>

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# Rice data

- Drought data
- Rice diversity panel
    - Projected shoot area
    - Root biomass
    - Water use efficiency
    - Water use
  
Inferred trait network structure
<img src="riceSEM1.png" height="326px" width="460px"/>

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# Inferred trait network structure
<img src="riceSEM3.png" height="426px" width="700px"/>

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# Structural equation model GWAS (SEM-GWAS)
`\begin{align*}
\mathbf{y} =\boldsymbol{\Lambda}  \mathbf{y} + \mathbf{ws} + \mathbf{Zg} + \boldsymbol{\epsilon}
\end{align*}`

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# Structural equation coefficients
<img src="riceSEM4.png" height="506px" width="700px"/>

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# Projected shoot area

`\begin{align*}
\text{Direct}_{s_j \rightarrow y_{1_{\text{PSA}}}} &= s_{j(y_{1_{\text{PSA}}})} \\
\text{Total}_{s_j \rightarrow y_{1_{\text{PSA}}}} &= \text{Direct}_{s_j \rightarrow y_{1_{\text{PSA}}}}\\
&= s_{j(y_{1_{\text{PSA}}})}
\end{align*}`

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# Root biomass
`\begin{align*}
\text{Direct}_{s_j \rightarrow y_{2_{\text{RB}}}} &=s_{j(y_{2_{\text{RB}}})} \\
\text{Total}_{s_j \rightarrow y_{2_{\text{RB}}}} &= \text{Direct}_{s_j \rightarrow y_{2_{\text{RB}}}}\\
&= s_{j(y_{2_{\text{RB}}})}
\end{align*}`

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# Water use
`\begin{align*}
\text{Direct}_{s_j \rightarrow y_{3_{\text{WU}}}} &=s_{j(y_{3_{\text{WU}}})} \\
\text{Indirect(1)}_{s_j \rightarrow y_{3_{\text{WU}}}} &= \lambda_{13}s_{j(y_{1_{\text{PSA}}})} \\
\text{Indirect(2)}_{s_j \rightarrow y_{3_{\text{WU}}}} &= \lambda_{23}s_{j(y_{2_{\text{RB}}})} \\
\text{Total}_{s_j \rightarrow y_{2_{\text{WU}}}} &= \text{Direct}_{s_j \rightarrow y_{2_{\text{WU}}}} + \text{Indirect(1)}_{s_j \rightarrow y_{3_{\text{WU}}}} + \text{Indirect(2)}_{s_j \rightarrow y_{3_{\text{WU}}}}\\
&= s_{j(y_{3_{\text{WU}}})} + \lambda_{13}s_{j(y_{1_{\text{PSA}}})} + \lambda_{23}s_{j(y_{2_{\text{RB}}})}
\end{align*}`

---
# Water use efficiency
`\begin{align*}
\text{Direct}_{s_j \rightarrow y_{4_{\text{WUE}}}} &=s_{j(y_{4_{\text{WUE}}})} \\
\text{Indirect(1)}_{s_j \rightarrow y_{4_{\text{WUE}}}} &= \lambda_{14}s_{j(y_{1_{\text{PSA}}})} \\
\text{Indirect(2)}_{s_j \rightarrow y_{4_{\text{WUE}}}} &= \lambda_{34}s_{j(y_{3_{\text{WU}}})} \\
\text{Indirect(3)}_{s_j \rightarrow y_{4_{\text{WUE}}}} &= \lambda_{13}\lambda_{34}s_{j(y_{1_{\text{PSA}}})} \\
\text{Indirect(4)}_{s_j \rightarrow y_{4_{\text{WUE}}}} &= \lambda_{23}\lambda_{34}s_{j(y_{2_{\text{RB}}})} \\
\text{Total}_{S_j \rightarrow y_{4_{\text{WUE}}}} &= \text{Direct}_{s_j \rightarrow y_{4_{\text{WUE}}}} + \text{Indirect(1)}_{s_j \rightarrow y_{4_{\text{WUE}}}} \\
&+ \text{Indirect(2)}_{s_j \rightarrow y_{4_{\text{WUE}}}} + \text{Indirect(3)}_{s_j \rightarrow y_{4_{\text{WUE}}}} \\
&+ \text{Indirect(4)}_{s_j \rightarrow y_{4_{\text{WUE}}}}\\
&= s_{j(y_{4_{\text{WUE}}})} + \lambda_{14}s_{j(y_{1_{\text{PSA}}})} + \lambda_{34}s_{j(y_{3_{\text{WU}}})} \\
&+ \lambda_{13}\lambda_{34}s_{j(y_{1_{\text{PSA}}})} + \lambda_{23}\lambda_{34}s_{j(y_{2_{\text{RB}}})} 
\end{align*}`

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# Water use efficiency
<img src="riceSEM8.png" height="536px" width="700px"/>

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# Preprint
<img src="SEMpreprint.png" height="436px" width="650px"/>

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# Ongoing SEM-GWAS projects

- Italian Brown Swiss cattle (University of Padua)
    - Milk yield
    
    - Milk pH
    
    - Milk lactose
    
    - Somatic cell score
    
    - Non-casein N