Resemblance between relatives

class: center, middle, inverse, title-slide

.title[
# Resemblance between relatives
]
.author[
### Jinliang Yang
]
.date[
### April 1, 2024
]

---

# Why we study resemblance?

Resemblance (=covariance) between related individuals.

__Heritability__ is the central concept in quantitative genetics
 - Proportion of variation due to additive genetic values (__Breeding values__)

`\begin{align*}
h^2 = \frac{\sigma^2_A}{\sigma^2_P}
\end{align*}`

#### Phenotypes (and `$\sigma^2_P$`) can be directly measured.

#### How about `$\sigma^2_A$` ?

- Estimates of `$\sigma^2_A$` require __known collections of relatives__.

---
# Types of Relatives

### Ancestral relatives

- Parent and offspring

- Grandparent and offspring

### Collateral relatives

- Half sibs: have one parent in common

- Full sibs: have both parents in common
- Twins

---

# Relatives (full sib families)

#### Full sibs

- have both parents in common

---
# Relatives (half sib families)

#### Half sibs

- have one parent in common

---
# Relatives (half sib families)

### A thought experiment

Body weight of newborn piglets
  - In the 1st half family `$O_{11}$` ... `$O_{3k}$`, mean `$=3 \pm 0.1$` lb 
  - In the 2nd half sib family `$O_{n1}$` ... `$O_{nk}$`, mean `$=4 \pm 0.1$` lb

---

# Two key observations

### A thought experiment

### Obs. One:
If trait variation has a significant genetic basis, the __closer the relatives__, the __more similar their phenotypic values__.

### Obs. Two:
The amount of __phenotypic resemblance__ among relatives for the trait provides an indication of the amount of __genetic variation__ for the trait.

---
# Resemblance and genetic covariance

###  Genetic covariance

- Resemblance between relatives has the genetics basis
- Genetic covariances arise because __two related individuals are more likely to share alleles__ than are two unrelated individuals

### Identical by descent (IBD)

- Sharing alleles means having alleles that are __identical by descent (IBD)__. 
- If two alleles are __IBD__, then both copies of alleles can be traced back to a single copy in a recent common ancestor.

---
# Genetic covariance between relatives

### Identify types of IBD?
- No allele IBD
- one allele IBD
- two alleles IBD

---

# Identical by descent (IBD)

If allele `$x_i$` carried by individual X is IBD to allele `$y_k$` in Y, then the covariance due to this allele is:

`\begin{align*}
Cov(\alpha_i, \alpha_k) & =  E[(\alpha_i - \mu_{\alpha})(\alpha_k - \mu_{\alpha})] \\
& = E[(\alpha_i - \mu_{\alpha})^2] \\
& = \sigma_{\alpha}^2
\end{align*}`

Because `$\alpha_i = \alpha_k$` if alleles `$x_i$` and `$y_k$` are IBD.

---
# Additive genetic covariance

The additive genetic covariance between relatives is generated through individuals sharing average effects of alleles.

Alleles in individuals `$X (x_ix_j)$` and `$Y (y_ky_l)$` can be IBD through four possible events:
  
--

`\begin{align*}
& x_i \equiv y_k \\
& x_i \equiv y_l \\
& x_j \equiv y_k \\
& x_j \equiv y_l \\
\end{align*}`

`\begin{align*}
Cov_\alpha(X, Y) = & P(x_i \equiv y_k)Cov(\alpha_i, \alpha_k) + P(x_i \equiv y_l)Cov(\alpha_i, \alpha_l) \\
 & + P(x_j \equiv y_k)Cov(\alpha_j, \alpha_k) + P(x_j \equiv y_l)Cov(\alpha_j, \alpha_l) \\
 = & 4f_{XY}\sigma_\alpha^2 \\
 = & 2f_{XY}\sigma_A^2 \\
\end{align*}`

Because `$\sigma_A^2 = \sigma_{\alpha_i}^2 + \sigma_{\alpha_j}^2 = 2\sigma_\alpha^2$` and `$\alpha_i = \alpha_j$` when alleles `$i$` and `$j$` are IBD.

---
# Parent-offspring

Recall that the __coefficient of co-ancestry ( `$f_{XY}$` from Ch.5)__ between a non-inbred parent and non-inbred offspring is 1/4. 
- Thus, the coefficient for `$\sigma_A^2$` is 1/2.

Consider the convariance between parent (P) and offspring (O),

- If we assume the two alleles of the parent are unrelated,

- Then all offspring can't share a common dominance deviation ( `$\sigma^2_D =0$` ).

Therefore,

`\begin{align*}
Cov(P, O) = \frac{1}{2}\sigma_A^2
\end{align*}`

---
# Parent-offspring

### From Breeding value

- Parent genotypic value: `$G = A + D$`.

- Offspring (half the breeding value of the parents from __Ch.7__): `$G=1/2A$`

`\begin{align*}
Cov(P, O) & = Cov(A + D, \frac{1}{2}A) \\
& = \frac{1}{2}Cov(A, A) + \frac{1}{2}Cov(A, D) \\
 & = \frac{1}{2}\sigma_A^2 \\
\end{align*}`

Because `$Cov(A, D) = 0$` from __Ch.8__.

---
# Parent-offspring

| Genotype  | Freq      | Breeding Value | Dominance Deviation  | Genotypic value | Offspring ( `$\mu_G$` ) |
| :-------: | :-------: | :-----------: |  :-------: | :-------: | :-------: | :-------: |
| `$A_1A_1$`  | `$p^2$`     | `$2q\alpha$`    |   `$-2q^2d$`   |  `$2q(\alpha - qd)$` | ? |
| `$A_1A_2$`  | `$2pq$`     | `$(q-p)\alpha$` |  `$2pqd$`   | `$(q-p)\alpha + 2pqd$`  | ? |
| `$A_2A_2$`  | `$q^2$`     | `$-2p\alpha$`   |  `$-2p^2d$`   |  `$-2p(\alpha + pd)$` | ? |

---
# Parent-offspring

| Genotype  | Freq      | Breeding Value | Dominance Deviation  | Genotypic value | Offspring ( `$\mu_G$` ) |
| :-------: | :-------: | :-----------: |  :-------: | :-------: | :-------: | :-------: |
| `$A_1A_1$`  | `$p^2$`     | `$2q\alpha$`    |   `$-2q^2d$`   |  `$2q(\alpha - qd)$` | `$q\alpha$` |
| `$A_1A_2$`  | `$2pq$`     | `$(q-p)\alpha$` |  `$2pqd$`   | `$(q-p)\alpha + 2pqd$`  | `$1/2(q-p)\alpha$` |
| `$A_2A_2$`  | `$q^2$`     | `$-2p\alpha$`   |  `$-2p^2d$`   |  `$-2p(\alpha + pd)$` | `$-p\alpha$` |

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

where,

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

--
$$
`\begin{aligned}
E(PO) = & p^2 \times 2q(\alpha-qd) \times q\alpha + 2pq \times ((q-p)\alpha+2pqd) \times 1/2(q-p)\alpha \\
        & + q^2 \times (-2p(\alpha+pd)) \times (-p\alpha) \\
      = & [2p^2q^2\alpha^2 - 2p^2q^3d\alpha] + [pq\alpha^2(q^2-2pq+p^2) + 2p^2q^2d\alpha(q-p)] \\
        & + [2p^2q^2\alpha^2 + 2p^3q^2d\alpha] \\
      = & pq\alpha^2(2pq + q^2 -2pq +p^2 + 2pq) + 2p^2q^2d\alpha(-q+q-p+p) \\
      = & pq\alpha^2
\end{aligned}`
$$
---
# Parent-offspring

| Genotype  | Freq      | Breeding Value | Dominance Deviation  | Genotypic value | Offspring ( `$\mu_G$` ) |
| :-------: | :-------: | :-----------: |  :-------: | :-------: | :-------: | :-------: |
| `$A_1A_1$`  | `$p^2$`     | `$2q\alpha$`    |   `$-2q^2d$`   |  `$2q(\alpha - qd)$` | `$q\alpha$` |
| `$A_1A_2$`  | `$2pq$`     | `$(q-p)\alpha$` |  `$2pqd$`   | `$(q-p)\alpha + 2pqd$`  | `$1/2(q-p)\alpha$` |
| `$A_2A_2$`  | `$q^2$`     | `$-2p\alpha$`   |  `$-2p^2d$`   |  `$-2p(\alpha + pd)$` | `$-p\alpha$` |

$$
`\begin{aligned}
Cov(P, O) & = E(PO) - E(P)E(O) \\
\end{aligned}`
$$

--
$$
`\begin{aligned}
E(PO) = & pq\alpha^2 \\
E(O) = & 0
\end{aligned}`
$$
--
Therefore,

$$
`\begin{aligned}
Cov(P, O) & = E(PO) - E(P)E(O) \\
          & = pq\alpha^2 \\
          & = 1/2 V_A
\end{aligned}`
$$
Because `$V_A = 2pq\alpha^2$`.

---
# Offspring and mid-parent

The covariance of the mean of the offspring and the mean of both parents (commonly called the __mid-parent__)

Let `$P$` and `$P'$` be the values of the two parents,
therefore `$\bar{P}=1/2(P + P')$`

$$
`\begin{aligned}
Cov(\bar{P}, O) & = Cov(1/2(P + P'), O) \\
          & = 1/2(Cov(P,O) + Cov(P', O))\\
          & = 1/2 V_A
\end{aligned}`
$$
If `$P$` and `$P'$` have the same variance.

See _P149 of F&M_ for the algebraic reduction.

---
# The genetic covariance for half sibs?

---
# Genetic covariance for half sibs?

`\begin{align*}
Cov(1/2A, 1/2A) & = 1/4V_A\\
\end{align*}`

| Genotype  | Freq      | Breeding Value | Dominance Deviation  | Genotypic value | Offspring ( `$\mu_G$` ) |
| :-------: | :-------: | :-----------: |  :-------: | :-------: | :-------: | :-------: |
| `$A_1A_1$`  | `$p^2$`     | `$2q\alpha$`    |   `$-2q^2d$`   |  `$2q(\alpha - qd)$` | `$q\alpha$` |
| `$A_1A_2$`  | `$2pq$`     | `$(q-p)\alpha$` |  `$2pqd$`   | `$(q-p)\alpha + 2pqd$`  | `$1/2(q-p)\alpha$` |
| `$A_2A_2$`  | `$q^2$`     | `$-2p\alpha$`   |  `$-2p^2d$`   |  `$-2p(\alpha + pd)$` | `$-p\alpha$` |

`\begin{align*}
Cov_{HS} = & p^2(q\alpha)^2 + 2pq \times 1/4(q-p)^2\alpha^2  + q^2 \times p^2\alpha^2 \\
=& pq\alpha^2[pq + 1/2(q-p)^2 + pq] \\
=& pq\alpha^2[1/2(p+q)^2] \\
=& 1/2pq\alpha^2 \\
= & 1/4V_A
\end{align*}`

---
# Additive genetic covariance for full sibs?

- Additive value for full sibs:
`\begin{align*}
G_{o1} & = \frac{1}{2}A + \frac{1}{2}A'\\
G_{o2} & = \frac{1}{2}A + \frac{1}{2}A'\\
\end{align*}`

- The additive genetic covariance for full sibs:
`\begin{align*}
Cov(G_{o1}, G_{o2}) = & Cov(\frac{1}{2}A + \frac{1}{2}A', \frac{1}{2}A + \frac{1}{2}A') \\
= & Var(\frac{1}{2}(A + A')) \\
= & \frac{1}{4}(\sigma_A^2 + \sigma^2_{A'}) \\
= & \frac{1}{2}\sigma_A^2 \\
\end{align*}`

---
# Dominance genetic covariance for full sibs?

- Among the progeny, only __four possible genotypes__, each with a frequency of 1/4.

- `$A_1A_3$`, `$A_1A_4$`, `$A_2A_3$`, `$A_2A_4$`
  
--

- Let the first sib has any of these genotypes. The 2nd has the same genotype is 1/4.
  - Thus one-quarter of all sib-pairs have the same genotype and consequently the same dominance deviation, `$D$`.
  
--

- Thus, the cross-product of the dominance deviation is `$\sigma_D^2$`, times frequency 1/4 = `$1/4\sigma_D^2$`

---
# Genetic covariance for full sibs?

### The genetic covariance of full sibs is therefore,

`\begin{align*}
Cov_{FS} = \frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2
\end{align*}`

### Covariance of half sibs

`\begin{align*}
Cov_{HS} = \frac{1}{4}\sigma_A^2
\end{align*}`

In principle, dominance variance can be calculated using full sibs and half sibs:

`\begin{align*}
Cov_{FS} - 2Cov_{HS} = & \frac{1}{2}\sigma_A^2 + \frac{1}{4}\sigma_D^2 - 2 \times \frac{1}{4}\sigma_A^2 \\
= & \frac{1}{4}\sigma_D^2
\end{align*}`

---
# Twins

### Monozygotic (identical) twins

$$
Cov_{MZ} = V_G
$$

### Dizygotic (fraternal) twins
- Not identical twins
- = Full sibs

---
# Summary

| Relationship  |       | r (of `$\sigma^2_A$`) | u (of `$\sigma^2_D$`)  | 
| :-------: | :-------: | :-----------: | :-------: | :-------: | 
| Identical twins |  | 1 | 1|
| First degree  | Parent-offspring   | 1/2    |  0  |  
| Second degree  | Half sibs    | 1/4 |  0 |  
|                | Full sibs     | 1/2 |  1/4 |

--
### General framework for calculating resemblance

Alleles shared between relatives that are __identical by descent (IBD)__ contribute to the covariance between relatives.

For example, if `$\sigma_A^2 = 0$`, meaning no variation in breeding values exists in the population for the trait of interest, then shared average allelic effects will not contribute to resemblance.

`$u \neq 0$` only if the relatives having the same genotype through two alleles IBD.