Genetic components of variance

class: center, middle, inverse, title-slide

# Genetic components of variance
### Jinliang Yang
### Oct. 17th, 2018

---

# Genotypic value

| Genotype  | Freq.   | Value   | Freq. `$\times$` Val.   | 
| :-------: |: ------- :| :-------: | :-------:  | 
| `$A_1A_1$`  | `$p^2$`   | `$+a$`    |  `$p^2a$`        |
| `$A_1A_2$`  | `$2pq$`   | `$d$`     | `$2pqd$`           |
| `$A_2A_2$`  | `$q^2$`   | `$-a$`    | `$-q^2a$`          |
|           |         | Sum =   | `$a(p-q) + 2pqd$`  |

### Parents pass on alleles, not genotypes.

---
# Average Effect

### Definition of average effect:
The average effect of `$A_1$` is the **mean deviation** from the **population mean** of individuals which received that allele from one parent, and the alleles received from the other parents being at random.

According to Falconer & Mackey `Table 7.2`:

`\begin{align*}
\alpha_1 = q(a + d(q-p))
\end{align*}`

`\begin{align*}
\alpha_2 = -p(a + d(q-p))
\end{align*}`

### Allele substitution effect

`\begin{align*}
\alpha = & \alpha_1 - \alpha_2 \\
       = & a + d(q-p)
\end{align*}`

---
# Breeding value (A)

Breeding value is the "value" of an individual as a parent. Note that breeding value is population specific.

### Operational definition

The deviation of the value of an individual's offspring from the population mean.

### Conceptual definition

Sum of the average effects of the alleles an individual carries.

`\begin{align*}
BV =  \sum_{i=1}^k\sum_{j=1}^2
\end{align*}`

Where summation occurs across the number of loci (k) and the two alleles present at each locus.

---

# Dominance deviation (D)

We can examine the failure of the additive value to reflect the genotypic value as a deviation --- **a dominance deviation**.

### G = A + D

| Genotype  | Genotypic Value | Value as deviated from mean      | Breeding Value  | Dominance Deviation  | 
| :-------: | :-------: | :-----------: | :-----------: | :-------: | :-------: | 
| `$A_1A_1$`  | `$a$`   |  `$2q(\alpha - qd)$`   | `$2q\alpha$`    |   `$-2q^2d$`   |  
| `$A_1A_2$`  | `$d$`   |   `$(q-p)\alpha + 2pqd$`  | `$(q-p)\alpha$` |    `$2pqd$`   |  
| `$A_2A_2$`  | `$-a$`  |   `$-2p(\alpha + pd)$`  | `$-2p\alpha$`   |  `$-2p^2d$`   |

---

# Graphical Representation

```r
a = 1; d = 3/4*a
q = 1/5; p = 1 - q

alpha <- a + d*(q - p)
a1a1 <- 2*alpha*q
a1a2 <- (q-p)*alpha
a2a2 <- -2*p*alpha

plot(c(0, 1, 2), c(-a, d, a), xlab="Genotype",ylab="", cex.lab=1.5, xaxt="n", pch=17, cex=2, col="red"); 
axis(1, at=c(0, 1, 2), labels=c("A2A2", "A1A2", "A1A1")); mtext("Breeding Value", side = 4, line = 1, cex=1.5, col="blue"); mtext("Genotypic Value", side = 2, line = 2, cex=1.5, col="red")
points(c(0, 1, 2), c(a2a2, a1a2, a1a1), cex=2, col="blue")
lines(c(0, 1, 2), c(a2a2, a1a2, a1a1), lwd=2, col="blue")
```

---
# Expectation and Variance

### Expectation `$E(X)$`:  **a measure of the mean value of a variable**

`\begin{align*}
E[X] &= \mu_X \\
&= \sum\limits_{i=1}^kf(x_i)Pr(X = x_i)
\end{align*}`

### Variance `$Var(X)$`: **a measure of the spread of a variable**
`\begin{align*}
Var(X) & = \sigma_X^2 \\
       & = E[X^2] - E[X]^2 \\
\end{align*}`

---
# Example

Number of leaves per plant in a F1 population (N=25) crossed from two inbred lines of tobacco:

```
## Warning: package 'knitr' was built under R version 3.4.3
```

```
## Warning: package 'kableExtra' was built under R version 3.4.4
```

Then, made 25 F2 plants:
<table>
 <thead>
  <tr>
   <th style="text-align:right;"> v1 </th>
   <th style="text-align:right;"> v2 </th>
   <th style="text-align:right;"> v3 </th>
   <th style="text-align:right;"> v4 </th>
   <th style="text-align:right;"> v5 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 16 </td>
   <td style="text-align:right;"> 16 </td>
   <td style="text-align:right;"> 20 </td>
   <td style="text-align:right;"> 21 </td>
   <td style="text-align:right;"> 14 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 20 </td>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:right;"> 13 </td>
   <td style="text-align:right;"> 18 </td>
   <td style="text-align:right;"> 17 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 19 </td>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:right;"> 15 </td>
   <td style="text-align:right;"> 13 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 17 </td>
   <td style="text-align:right;"> 15 </td>
   <td style="text-align:right;"> 15 </td>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:right;"> 15 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:right;"> 17 </td>
   <td style="text-align:right;"> 16 </td>
   <td style="text-align:right;"> 18 </td>
   <td style="text-align:right;"> 13 </td>
  </tr>
</tbody>
</table>

---
# Example

### Mean and variance of F1 and F2 populations?

```r
f1 <- as.vector(as.matrix(f1))
f2 <- as.vector(as.matrix(f2))
mean(f1)
```

```
## [1] 15.72
```

```r
mean(f2)
```

```
## [1] 15.84
```

```r
var(f1)
```

```
## [1] 1.46
```

```r
var(f2)
```

```
## [1] 5.973333
```

---
# Example

```r
library(ggplot2)
```

```
## Warning: package 'ggplot2' was built under R version 3.4.4
```

```r
df <- rbind(data.frame(number=f1, pop="F1"), data.frame(number=f2, pop="F2"))
ggplot(df, aes(x=pop, y=number, fill=pop)) +
  scale_fill_manual(values=c("#E69F00", "#56B4E9")) +
  geom_violin(trim=FALSE)
```

---
# Covariance

To quantify to what extent the two variables **co-vary**.

$$
`\begin{aligned}
Cov(X, Y) & = \sigma_{XY} \\
& = E([X - E(X)][Y - E(Y)]) \\
& = E(XY) - E(X)E(Y) \\
\end{aligned}`
$$

where,

$$
`\begin{aligned}
E(XY) = \sum_i \sum_j x_i y_j Pr(X = x_i, Y = y_j)
\end{aligned}`
$$

### The variance of a sum

`\begin{align*}
& Var(X+Y) = Var(X) + Var(Y) + 2Cov(X, Y) \\
& \sigma_{X+Y}^2 = \sigma_X^2 + \sigma_Y^2 + 2\sigma_{XY} \\
\end{align*}`

---
# P = G + E

`\begin{align*}
& Var(P) = Var(G+E) \\
& \sigma_{P}^2 = \sigma_{G+E}^2 = \sigma_G^2 + \sigma_E^2 + 2\sigma_{GE}
\end{align*}`

- `$\sigma_{G}^2$` is the variance of the genotypic effects
- `$\sigma_{E}^2$` is the variance of the environmental effects
- `$\sigma_{GE}$` is the covariance between genotypic effects and environmental effects. Normally, we assume `$\sigma_{GE} = 0$`

Therefore,
`\begin{align*}
& \sigma_{P}^2 = \sigma_G^2 + \sigma_E^2
\end{align*}`

---
# Example

### Estimate the degree of genetic determination in the F2 generation.
What assumptions have to be made to do this?

```r
var(f1)
```

```
## [1] 1.46
```

```r
var(f2)
```

```
## [1] 5.973333
```

---
# Genetic variance partitioning

The genotypic value could be partitioned into the breeding value and dominance deviation.

### Genotypic value:  `$G = A + D$`

`\begin{align*}
& \sigma_{G}^2 = \sigma_A^2 + \sigma_D^2
\end{align*}`

### Breeding Value:  `$A = \alpha_i + \alpha_j$`

`\begin{align*}
& \sigma_{A}^2 = \sigma_{\alpha_i}^2 + \sigma_{\alpha_i}^2
\end{align*}`

Therefore,

`\begin{align*}
& \sigma_{G}^2 = \sigma_{\alpha_i}^2 + \sigma_{\alpha_j}^2 + \sigma_{\delta_{ij}}^2
\end{align*}`