Heritability: precision of estimation

class: center, middle, inverse, title-slide

.title[
# Heritability: precision of estimation
]
.author[
### Jinliang Yang
]
.date[
### April 8, 2024
]

---

# Variance componenets in a sib design

| Source        |    df     |  Sums of Squares      | MS      | E(MS) |
| :------:      | :-------: | :--------------------:|:------: | :---------------: |
| Sires         | s-1       |  `\(dn\sum\limits_{i=1}^s(\bar{p}_i - \bar{p})^2\)` | `\(MS_s\)` | `\(= \sigma_w^2 + n\sigma_d^2 + dn\sigma_s^2\)`   |
| Dams (Sires)  | s(d-1)    |  `\(n\sum\limits_{i=1}^s\sum\limits_{j=1}^d(\bar{p}_{ij} - \bar{p}_i)^2\)` | `\(MS_d\)`  | `\(= \sigma_w^2 + n\sigma_d^2\)` | 
| Sibs (Dams)   | sd(n-1)   |  `\(\sum\limits_{i=1}^s\sum\limits_{j=1}^d\sum\limits_{k=1}^n(p_{ijk} - \bar{p}_{ij})^2\)`  | `\(MS_w\)`       | `\(= \sigma_w^2\)` |

Under the assumption that individuals are random members of the same population:

#### Phenotypic variance

Phenotypic variance is the sum ( `\(\sigma_T^2\)` ) of the three observational components:

`\begin{align*}
 V_P & = \sigma_T^2 \\
& = \sigma^2_s + \sigma^2_d + \sigma^2_w\\
& = \sigma^2_A + \sigma^2_D + \sigma^2_E\\
\end{align*}`

---
# Interpretation of variance components

### Among-sire component `\(\sigma_s^2\)`

This is the __variance between the means of half-sib families__

- A key concept in the __analysis of variance (ANOVA)__ is that:
  - The variance between groups is equal to the covariance within groups.
- It therefore estimates the phenotypic __covariance of half sibs__.

`\begin{align*}
& \sigma_s^2 = Cov(HS) = \frac{1}{4}\sigma_A^2 \\
\end{align*}`

---
# Interpretation of variance components

### Within-progeny component `\(\sigma_w^2\)`
- Another key concept in the __analysis of variance (ANOVA)__ is:

- The total variance = between-group variance + within-group variance

- It follows that the  __within-group variance = total variance - covariance of members of the groups__.

Within-dams are full-sibs, therefore,

`\begin{align*}
\sigma_w^2 &  = V_P - Cov(FS) \\
& = \sigma^2_A + \sigma^2_D + \sigma^2_E - (\frac{1}{2}\sigma^2_A + \frac{1}{4}\sigma^2_D) \\
& = \frac{1}{2}\sigma^2_A + \frac{3}{4}\sigma^2_D + \sigma^2_E
\end{align*}`

---
# Interpretation of variance components

### Among-dam component `\(\sigma_d^2\)`

`\begin{align*}
\sigma_d^2 &  = \sigma_P^2 - \sigma_s^2 - \sigma_w^2 \\
& = \sigma^2_A + \sigma^2_D + \sigma^2_E - \frac{1}{4}\sigma_A^2 - (\frac{1}{2}\sigma^2_A + \frac{3}{4}\sigma^2_D + \sigma^2_E) \\
& = \frac{1}{4}\sigma^2_A + \frac{1}{4}\sigma^2_D
\end{align*}`

Alternatively,

`\begin{align*}
\sigma_w^2 &  = V_p - Cov(FS) \\
\sigma_s^2 & = Cov(HS) \\
\sigma_d^2 & = \sigma_P^2 - \sigma_s^2 - \sigma_w^2 \\
& = V_p - Cov(HS) - V_p + Cov(FS) \\
& = Cov(FS) - Cov(HS)
\end{align*}`

---

# Summary of the variance components

---

# Summary of the variance components

| Observational  | Covariance and causal components  |  Estimated values  |
| :------------: | :-------: |  | 
| Sires | `\(\sigma_s^2 = Cov(HS)\)`  | `\(=\frac{1}{4}\sigma_A^2\)`    |
| Dams  | `\(\sigma_d^2 = Cov(FS) - Cov(HS)\)` | `\(=\frac{1}{4}\sigma_A^2 + \frac{1}{4}\sigma_D^2\)` | 
| Progeny | `\(\sigma_w^2 = V_P - Cov(FS)\)`  | `\(= \frac{1}{2}\sigma_A^2 +\frac{3}{4}\sigma_D^2\)` |
| Total | `\(\sigma_T^2 = V_P = \sigma_s^2 + \sigma_d^2 + \sigma_w^2\)` | `\(=\sigma_A^2 + \sigma_D^2 + \sigma_E^2\)` |
| Sires + Dams   | `\(\sigma_s^2 + \sigma_d^2 = Cov(FS)\)`       | `\(=\frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2\)` |

#### Variance componenets in a sib design

| Source        |    df     |  E(MS) | Estimated values |
| :------:      | :----------: | :--------------------:| :------: |
| Sires         | s-1       | `\(\sigma_w^2 + n\sigma_d^2 + dn\sigma_s^2\)`   | `\(= \frac{1}{2}\sigma_A^2 +\frac{3}{4}\sigma_D^2 + \frac{n}{4}\sigma_A^2 + \frac{n}{4}\sigma_D^2 + \frac{dn}{4}\sigma_A^2\)` |
| Dams (Sires)  | s(d-1)    | `\(\sigma_w^2 + n\sigma_d^2\)` | `\(= \frac{1}{2}\sigma_A^2 +\frac{3}{4}\sigma_D^2 + \frac{n}{4}\sigma_A^2 + \frac{n}{4}\sigma_D^2\)` |
| Sibs (Dams)   | sd(n-1)   | `\(\sigma_w^2\)` | `\(= \frac{1}{2}\sigma_A^2 +\frac{3}{4}\sigma_D^2\)` |

---
# Example: a sib experiment

- A balanced experimental data collected from 100 sires each mated to 4 dams.
- Have body length records of 5 male and 5 female offspring from each dam.
- Get `\(h^2\)` for body length?

### __ANOVA__ table for body length:

| Source         |    Mean Sq    | Value  |
| :------------: | :-----------: | :-------: | 
| Between Sires  |    `\(MS_s\)`      | =7.83      | 
| Between Dams   |    `\(MS_d\)`     | =4.51     |
| Within progenies |  `\(MS_w\)`    | =1.27    |

---
# A sib experiment

#### __ANOVA__ table for body length:

| Source         |  Value    |    Mean Sq    |  Expectation of MS |
| :------------: | :-------: | :-------:     | :-------: |
| Between Sires  | 7.83      |    `\(MS_s\)`     | `\(= \sigma_w^2 + n\sigma_d^2 + dn\sigma_s^2\)` |
| Between Dams   | 4.51      |    `\(MS_d\)`     | `\(= \sigma_w^2 + n\sigma_d^2\)` | 
| Within progenies | 1.27    |    `\(MS_w\)`     | `\(= \sigma_w^2\)` |

First get `\(V_P\)`:  
`\begin{align*}
& \sigma_w^2 = MS_w = 1.27 \\
& \sigma_d^2 = (MS_d - \sigma_w^2)/n = (4.51-1.27)/10 = 0.324 \\
& \sigma_s^2 = (MS_s - MS_d)/(dn) = (7.83-4.51)/(4 \times 10) = 0.083 \\
& \sigma^2_P = \sigma_d^2 + \sigma_s^2 + \sigma_w^2 = 1.27 + 0.324 + 0.083 = 1.677 \\
\end{align*}`

Then get `\(V_A\)`:  
`\begin{align*}
& \sigma^2_s = \sigma_A^2/4\\
& \sigma_A^2 = 0.083 \times 4 = 0.332 \\
\end{align*}`

---
# A sib experiment

#### __ANOVA__ table for body length:

First get `\(V_P\)`:  
`\begin{align*}
& \sigma^2_P = \sigma_d^2 + \sigma_s^2 + \sigma_w^2 = 1.27 + 0.324 + 0.083 = 1.677 \\
\end{align*}`

Then get `\(V_A\)`:  
`\begin{align*}
& \sigma_A^2 = 0.083 \times 4 = 0.332 \\
\end{align*}`

--
Finally `\(h^2\)`
`\begin{align*}
& h^2 =  \frac{\sigma^2_A}{\sigma^2_P} =  0.332/1.677 = 0.198\\
\end{align*}`

---

# Precision and design

### Data collection must be practical

- time/cost/etc is usually the important determinant
 
--

### Be freedom from bias
  - Random mating
  - Absence of common environmental effects, e.g. maternal effects.

---

# Precision and design

If we want to design a balanced experiment:
- `\(s\)` sires each mated to `\(d\)` dams
- Each dam has `\(n\)` progenies.

#### Questions before experimental design?

1. Parent-offspring, half-sib, full-sib, or others?
1. How many families?
2. Numbers of progeny?
3. What if it is unbalanced?

---

# Sampling variance of `\(b\)`

F & M page 178, the sampling variance of the parent-offspring regression ( `\(b\)` ) is approximately:

`\begin{align*}
SV_b = \frac{k[1+(n-1)t]}{nN}
\end{align*}`

- `\(N\)` families (offspring and parents)
- `\(k\)` parents (1 or 2) for each family
- `\(n\)` offspring per family
- `\(t\)` the intra-class correlation between offspring in a family

### One parent

Sampling variance is minimal when `\(n=1\)`, i.e. `\((n-1)t=0\)`.

The most efficient design:
1. as many families as possible
2. measure only one offspring per family

---

# One parent

`\begin{align*}
& SV(h^2=2b) = 4/N \\
& s.e.(h^2) = 2/\sqrt{N} \\
\end{align*}`

```r
N=1:1000
se <- 2/sqrt(N)

plot(N, se, type="l", lwd=5, col="red")
```