Relationship Matrix

class: center, middle, inverse, title-slide

.title[
# Relationship Matrix
]
.author[
### Jinliang Yang
]
.date[
### Mar. 1, 2024
]

---

# Individual Inbreeding: `\(F_X\)`

`\begin{align*}
F_{X} = (\frac{1}{2})^n(1 + F_{A})
\end{align*}`

- Where `\(n\)` is the number of individuals in the path from the individual’s sire (dad), through the common ancestor, to the dam (mom).

- If multiple common ancestors, must sum the individual estimates.

------------

# Coancestry (Kinship): `\(f_{XY}\)`

Coancestry between X and Y depends on relationship among parents

`\begin{align*}
f_{XY} & = Pr(x \equiv y) \\
& = Pr(a 	\equiv c) + Pr(a 	\equiv d)  + Pr(b 	\equiv c)  + Pr(b 	\equiv d ) \\
& = \frac{1}{4}(f_{AC} + f_{AD} + f_{BC}+ f_{BD})
\end{align*}`

---
# How about the relationships for all individuals?

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

.pull-right[
|  | B  | C | E | A | D | F |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
| B  | __ `\(d_B\)`__ |  x | x | x  | x | x  |  
| C  |   | __ `\(d_C\)`__  | x | x  | x |  x |  
| E    |  |   | __ `\(d_E\)`__ | x  | x | x  |  
| A    |  |   |  | __ `\(d_A\)`__  | x | x  |  
| D    |  |   |  |   | __ `\(d_D\)`__ |  x |  
| F    |  |   |  |   |  | __ `\(d_F\)`__  |  
]

When complete the table will contain:

- All additive relationships among individuals (determined at the intersection of each row and column)

- All inbreeding coefficients (on the diagonal of the table).

---
# The Tabular Method of Relationships

The method will permit you to easily calculate the _additive_ relationship and _inbreeding coefficient_ among all individuals in a given pedigree.

- The method requires individuals to be __sorted chronologically__

- Those individuals without known parents are assumed to not be inbred and unrelated.

# Relationship vs Coancestry

|  | Coancestry   | Relationship    | 
| :-------: | : ------ : | :-------: | 
| selfing | 0.5 | 1 | 
| Parent-offspring  | 0.25    | 0.5    | 
| Full-siblings     | 0.25    | 0.5    |
| Half-siblings     | 0.125   | 0.25    |

---

# The Tabular Method of Relationships

Example: Consider the following pedigree of Individuals A through F

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

.pull-right[
| Individual | Father   | Mother    | 
| :-------: | : ------ : | :-------: | 
| B  | NA    | NA    | 
| C  | NA    | NA    |
| E    | NA   | NA   |
| A    | B   | C    |
| D    | B   | E    |
| F    | A   | D    |
- NA means data not available.
]

This simple pedigree will let us calculate the relationship among all individuals. 
- Note: some inbreeding is present.

---
# The Tabular Method of Relationships

#### Step 1/9. Determine which individuals will go into the table. 
Place them in chronological order (oldest to youngest).

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

.pull-right[
- Use birth dates!

- If you don’t have birth dates, individuals "higher" in the pedigree are older than those "lower" in the pedigree.

- Here the list could be __{B,C,E,A,D,F}__ 
  
  - or an equivalent could be __{C,E,B,D,A,F}__.
]

---
# The Tabular Method of Relationships

#### Step 2/9. Write the ID into the table.
Write the ID of each individual in order of age across the top of the table (the columns) and then down the side of the table.

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

|  | B  | C | E | A | D | F |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
| B  |  |   |  |   |  |   |  
| C  |   |   |  |   |  |   |  
| E    |  |   |  |   |  |   |  
| A    |  |   |  |   |  |   |  
| D    |  |   |  |   |  |   |  
| F    |  |   |  |   |  |   |

---
# The Tabular Method of Relationships

#### Step 3/9. Write the name or number of the parents of an individual above that individual’s name – if known.

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  |  |   |  |   |  |   |  
| C  |   |   |  |   |  |   |  
| E    |  |   |  |   |  |   |  
| A    |  |   |  |   |  |   |  
| D    |  |   |  |   |  |   |  
| F    |  |   |  |   |  |   |

---
# The Tabular Method of Relationships

#### Step 4/9. Put a 1.0 in each diagonal cell.

- This is an individual’s basic relationship to itself ( `\(2 \times f_{XX}\)`), unless inbred (which we calculate later).

- For those individuals in the base population (the oldest animals with no known parents) enter a zero (0.0) in the off-diagonal elements between them. 
  - For our example, individuals B, C, and E, are the base population, and the table would look like this.

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 |   |  |   |  
| C  |   |  1 | 0 |   |  |   |  
| E    |  |  | 1 |   |  |   |  
| A    |  |   |  |  1 |  |   |  
| D    |  |   |  |   | 1 |   |  
| F    |  |   |  |   |  |  1 |

---
# The Tabular Method of Relationships

#### Step 5/9. Now, go to the upper diagonal.

- This is zero for individuals which have parents that are unknown.

.pull-left[
<div align="center">
<img src="tab_ped.png" width=350>
</div>
]

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | ?  | ? |  ? |  
| C  |    |  1 | 0 |   |  |   |  
| E    |   |   | 1 |   |  |   |  
| A    |   |   |  |  1 |  |   |  
| D    |   |   |  |   | 1 |   |  
| F    |   |   |  |   |  |  1 |

---
# The Tabular Method of Relationships

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | __?__  | ? |  ? |  
| C  |    |  1 | 0 |   |  |   |  
| E    |   |   | 1 |   |  |   |  
| A    |   |   |  |  1 |  |   |  
| D    |   |   |  |   | 1 |   |  
| F    |   |   |  |   |  |  1 |

#### Step 6/9. Start in the first row. Go to the first unknown element (in this example, between individual B and A). 
- Stay in the row, and find the element in this row for the first parent (B, the value is 1) and the second parent (C, the value is 0).

- Add these together (1 + 0 = 1) and divide by 2 (1⁄2). Enter this value in the table.

---
# The Tabular Method of Relationships

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | 1/2  | 1/2 |  __?__ |  
| C  |    |  1 | 0 |   |  |   |  
| E    |   |   | 1 |   |  |   |  
| A    |   |   |  |  1 |  |   |  
| D    |   |   |  |   | 1 |   |  
| F    |   |   |  |   |  |  1 |

#### Step 6/9. Start in the first row. Go to the first unknown element (in this example, between individual B and A. 
- Stay in the row, and find the element in this row for the first parent (B, the value is 1) and the second parent (C, the value is 0).

- Add these together (1 + 0 = 1) and divide by 2 (1⁄2). Enter this value in the table.

- Move to the next column in the same row, find parent 1 and parent 2, in that row. Add those values together and divide by 2. Continue until the end of the row. If a parent is unknown, put in a zero.

---
# The Tabular Method of Relationships

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | 1/2  | 1/2 |  1/2 |  
| C  |  0  |  1 | 0 |   |  |   |  
| E    |  0 | 0  | 1 |   |  |   |  
| A    | 1/2 |   |  |  1 |  |   |  
| D    | 1/2  |   |  |   | 1 |   |  
| F    | 1/2  |   |  |   |  |  1 |

#### Step 7/9. When the row is finished, copy the elements down the column.

#### Step 8/9. Continue to the next row. Complete Rules 6 and 7.

---
# The Tabular Method of Relationships

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | 1/2  | 1/2 |  1/2 |  
| C  |  0  |  1 | 0 | 1/2  | 0 | 1/4  |  
| E    |  0 | 0  | 1 | 0  | 1/2 |  1/4 |  
| A    | 1/2 | 1/2  | 0 |  1 | 1/4 | 5/8  |  
| D    | 1/2  | 0  | 1/2 | 1/4  | 1 | 5/8  |  
| F    | 1/2  | 1/4  | 1/4 | 5/8  | 5/8 |  1 |

#### Step 8/9. Continue to the next row. Complete Rules 6 and 7.

#### Step 9/9. Update the diagonal.

- To update the inbreeding coefficient in the diagonal, simply find the relationship between the two parents, divide by two, and add this to the 1.

---
# The Tabular Method of Relationships

In our example, the final table would look like this:

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | 1/2  | 1/2 |  1/2 |  
| C  | 0  |  1 | 0 | 1/2  | 0 | 1/4  |  
| E    | 0 | 0 | 1 | 0  | 1/2 | 1/4  |  
| A    | 1/2 | 1/2  | 0 |  1 | 1/4 | 5/8   |  
| D    | 1/2 | 0  | 1/2 | 1/4  | 1 | 5/8  |  
| F    | 1/2 | 1/4  | 1/4 | 5/8   | 5/8 |  __1 + 1/8__ |

### Additive relationship

To find the additive relationship between any pair of individuals simply go the appropriate row and column cell for that pair.

- For example, the additive relationship between individuals F and C is `\(\frac{1}{4}\)`.

---
# The Tabular Method of Relationships

In our example, the final table would look like this:

|  |   |  |  | B-C | B-E | A-D |
| :--: | : --: | :--: | :--: | :--: | :--: | :--: |  
|  | B  | C | E | A | D | F | 
| B  | 1 | 0  | 0 | 1/2  | 1/2 |  1/2 |  
| C  | 0  |  1 | 0 | 1/2  | 0 | 1/4  |  
| E    | 0 | 0 | 1 | 0  | 1/2 | 1/4  |  
| A    | 1/2 | 1/2  | 0 |  1 | 1/4 | 5/8   |  
| D    | 1/2 | 0  | 1/2 | 1/4  | 1 | 5/8  |  
| F    | 1/2 | 1/4  | 1/4 | 5/8   | 5/8 |  __1 + 1/8__ |

### Inbreeding coefficient

To find the inbreeding coefficient for any individual, simply go to the diagonal element for that individual and subtract 1. 
- The only inbred individual in this pedigree is F, with a value of `\(1+1/8 -1 = 1/8\)` 
- This is the half-siblings!

---

# Inbreeding coefficient for half-siblings

`\begin{align*}
F_{X=F} = f_{AD} & = \frac{1}{4}(f_{EB} + f_{EC} + f_{BB} + f_{BC}) \\
& = \frac{1}{4}(0 + 0 + \frac{1}{2} + 0) \\
& = \frac{1}{8}
\end{align*}`

---

# Genomic relationship

- SNPs give clues on family relationships
 -  For example two individuals sharing lots of genotypes are likely because they belong to the same family (or genetically related)

- Genomic relationships is usually more accurate than the one estimated from pedigree

### Genetic values = sum of SNP effects

`\begin{align*}
\mathbf{g} = \mathbf{Z}\mathbf{a}
\end{align*}`
Here we use matrix notation
- `\(\mathbf{g}\)`, a vector of genetic values
- `\(\mathbf{Z}\)`, genotype matrix
- `\(\mathbf{g}\)`, a vector of SNP effects
> [VanRaden (2008)](https://pubmed.ncbi.nlm.nih.gov/18946147/)

---

# Genomic relationship

The first way to calculate G matrix:

`\begin{align*}
\mathbf{G} = \frac{\mathbf{ZZ'}}{2\sum p_i(1-p_i)}
\end{align*}`

The 2nd way to calculate G matrix:

`\begin{align*}
\mathbf{G} = \mathbf{ZDZ'}
\end{align*}`

where D is diagonal with

`\begin{align*}
D_{ii} = \frac{1}{m[2p_i(1-p_i)]}
\end{align*}`

> [VanRaden (2008)](https://pubmed.ncbi.nlm.nih.gov/18946147/)

---

# Genomic relationship

### SNP-based coancestry

|  |   |  |  |
| :--: | : --: | :--: |
|  | Ind. A  | Ind. B | `\(f_{AB}\)` |
| Locus1  | AA | AT  | 0.5 | 
| Locus2  | CG  |  CG | 0.5 | 
| Locus 3  | TT | CC | 0 |

The probability that two alleles taken at random, one form each individual, is

`\begin{align*}
f_{AB} = \frac{\sum f_{l_i, l_j}}{L} = \frac{0.5 + 0.5}{3} = 0.33
\end{align*}`

---

# SNP-based coancestry

In more formal terms, if `\(g_{A_k}\)` and `\(g_{B_k}\)` is the frequency of an allele (i.e., the minor allele) in individuals A and B.

|  |   |  |  | |  |  |
| :--: | : --: | :--: |:--: |:--: |
|  | Ind. A  | Ind. B | `\(f_{l_{AB}}\)` | Minor allele | `\(g_{A_k}\)` | `\(g_{B_k}\)`
| Locus1  | AA | AT  | 1 | T| 0 | 0.5 | 
| Locus2  | CG  |  CG | 0.5 | C| 0.5 | 0.5|
| Locus 3  | TT | CC | 0 | T | 1 | 0 |

Note that if SNP coding is __0, 1, 2__, the `\(g\)`s here are half of `\(\mathbf{Z}\)` matrix
- 2 for homozygous minor alleles
- 1 for heterozygous
- 0 for homozygous major alleles

|  |   |  |  | |  |  |
| :--: | : --: | :--: |:--: |:--: |
|  | Ind. A  | Ind. B | `\(f_{l_{AB}}\)` | Minor allele | `\(g_{A_k}\)` | `\(g_{B_k}\)`
| Locus1  | 0 | 1  | 1 | T | 0 | 0.5 | 
| Locus2  | 1  |  1 | 0.5 | C| 0.5 | 0.5|
| Locus 3  | 2 | 0 | 0 | T | 1 | 0 |

---

# SNP-based coancestry

`\begin{align*}
f_{AB} & = \frac{(\sum g_{A_k}g_{B_k} + (1-g_{A_k})(1-g_{B_k}))}{L} \\
& = \frac{(0 \times 0.5 + 1 \times 0.5) + (0.5 \times 0.5 + 0.5 \times 0.5) + (1 \times 0 + 0 \times 1)}{3} \\
& = 0.33
\end{align*}`

```r
# row is the ind.s and col is the SNP loci
Z <- matrix(c(0,1,1,1,2,0), ncol=3)
Z
```

```
##      [,1] [,2] [,3]
## [1,]    0    1    2
## [2,]    1    1    0
```

---

# Genomic relationship

`\begin{align*}
\mathbf{G} = \mathbf{ZDZ'}
\end{align*}`

where D is diagonal with

`\begin{align*}
D_{ii} = \frac{1}{m[2p_i(1-p_i)]}
\end{align*}`

or simply the number of marker:

`\begin{align*}
D_{ii} = \frac{1}{m}
\end{align*}`

```r
Z <- matrix(c(0,1,1,1,2,0), ncol=3)
num <- Z %*% t(Z)

G <- (Z %*% t(Z)) / ncol(Z)
G
```

```
##           [,1]      [,2]
## [1,] 1.6666667 0.3333333
## [2,] 0.3333333 0.6666667
```