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For complicated pedigrees (mainly found in animal breeding) the method of Coancestry offers an easier option than path analysis for calculating the Coefficient of Inbreeding. Coancestries are especially useful under systems of very close inbreeding where the number of inheritance pathways may be so great that identifying every one becomes too laborious. The Coefficient of Inbreeding of an individual (F) is equal to the Coancestry (f) between the two parents. In Figure 18:

F_{X} = f_{PQ}

f_{PQ = }1/4(f_{AC} + f_{AD} + f_{BC} + f_{BD})

Supplementary Rules for Overlapping Generations

f_{PC} = 1/2(f_{AC} + f_{BC})

f_{PD} = 1/2(f_{AD }+ f_{BD})

Also f_{PQ } = 1/2(f_{PC} + f_{PD})

__Rules for Calculating Coancestries of Early Generations__

Assume that all coancestries of the first generation are 0. The coancestries of the next generation will depend on the relationship between the prospective parents:

__Selfing__(Monoecious plants and hermaphrodites)- Offspring and Parent
- Full Sibs

f_{AA} = 1/2(1 + F_{A})

If F_{A} is unknown or assumed to be 0 then f_{AA} reduces to 1/2. The coancestry f_{AA} is used again below in calculating coancestries between other closely-related parents.

Since these are from different generations the supplementary rule is used:

f_{PA} = 1/2(f_{AB} + f_{AA})

If A and B are not related and A is not inbred, then f_{AB} = 0 and f_{ AA} = 1/2 . The coancestry f_{PA} then reduces to 1/4.

The basic rule applies here, but the pedigree needs to be redrawn to find the formula.

Figure 19 Modified Diagram for Full Sibs

Since f_{AB }= f_{BA} then:

f_{PQ} = 1/4(2f_{AB} + f_{AA} + f_{BB})

With no previous inbreeding or genetic relationship between A and B this reduces to f_{PQ} = 1/4.

d) __Half Sibs__

Redrafting the pedigree as for full sibs:

Figure 20 Modified Diagram for Half Sibs

Applying the basic rule to this diagram we have:

f_{PQ} = 1/4(f_{AB }+ f_{AC} + f_{BB} + f_{BC})

With no previous inbreeding or genetic relationship between A and B or between B and C this reduces to f_{PQ} = 1/8.

Worked Examples

With coancestries, unlike the previous path analyses, it is usual to start with the early generations and systematically work down through the pedigree until the parents of the chosen individual are reached. To be able to calculate the coancestries between actual parents it is also necessary to calculate the hypothetical coancestries between other members of each generation. Thus, a running tally is kept as you progress down through the generations. This avoids having to double back later to find missing values.

Two of the following four examples were used earlier to calculate coefficients of inbreeding by path analysis:

Example | Figure | |

1 | 21 | First cousin marriages |

2 | 22 | Uncle - niece marriage |

3 | 23 | Double first cousin marriage |

4 | 24 | Close inbreeding (full sibs) |

__Example 1__

Figure 21 First Cousin Marriages

__Find F _{Q}__

The coancestries of the members of the early generations are as follows:

f_{AB} = f_{CD} = f_{CE} = f_{CF} = f_{DF} = f_{EF} = 0. Also F_{A} = F_{B} = F_{C} = F_{D} = F_{E} = F_{F} = 0.

Therefore, f_{DE } (full sibs) = 1/4(2f_{AB} + f_{AA} + f_{BB}) and since

f_{AA }= 1/2(1 + F_{A}) = 0.5 and f_{BB} = 1/2(1 + F_{B}) = 0.5

Then f_{DE} = 1/4(0 + 0.5 + 0.5) = 0.25

f_{GH} = F_{I} = 1/4(f_{CE} + f_{CF} + f_{DE} + f_{DF})

= 1/4(0 + 0 + 0.25 + 0)

= 0.0625

f_{IJ} = f_{KL} = f_{KM} = f_{KN} = f_{LN} = f_{MN} = 0

and F_{J} = 0

Therefore, f_{LM }(full sibs) = 1/4(2f_{IJ} + f_{II} + f_{JJ})

f_{II } = 1/2(1 + F_{I}) = 0.53125 and f_{JJ} = 1/2(1 + F_{J}) = 0.5

Therefore f_{LM} = 1/4(0 + 0.53125 + 0.5) = 0.2578125

Finally, F_{Q} = f_{OP} = 1/4(f_{KM} + f_{KN} + f_{LM} + f_{LN})

= 1/4(0 + 0 + 0.2578125 + 0)

= 0.064453125 = __6.45%__

Example 2

Figure 22 Uncle - niece Marriage

__Find F _{G}__

f_{AB} = f_{CE} = f_{DE} = 0

Also F_{A} = F_{B} = F_{C} = F_{D} = F_{E} = F_{F} = 0

f_{CD} (full sibs) = 1/4(2f_{AB} + f_{AA} + f_{BB})

Since f_{AA} = 1/2(1 + F_{A}) = 0.5 and f_{BB} = 1/2(1 + F_{B}) = 0.5

Then f_{CD} = 1/4(0 + 0.5 + 0.5) = 0.25

Therefore F_{G }= f_{CF }(overlapping generations) = 1/2(f_{CD} + f_{CE})

= 1/2(0.25 + 0)

= 0.125 = __12.5%__

Example 3

** **Sometimes it is necessary, as shown earlier, to present a pedigree in alternative forms to facilitate the calculation of coancestries. The following are two different ways of depicting the same pedigree:

Figure 23 Double First Cousin Marriage

Conventional Diagram |
Modified Diagram |

__Find F _{K}__

__Using first diagram__:

f_{AB} = f_{AC} = f_{AD} = f_{BC} = f_{BD} = f_{CD} = f_{EG} = f_{EH} = f_{FG} = f_{FH} = 0

Also F_{A} = F_{B} = F_{C} = F_{D} = F_{E} = F_{F} = F_{G} = F_{H} = 0

and f_{AA} = 1/2(1 + F_{A}) = f_{BB} = 1/2(1 + F_{B}) = f_{CC} = 1/2(1 + F_{C}) = f_{DD} = 1/2(1 + F_{D}) = 0.5

Then f_{EF }(full sibs) = 1/4(2f_{AB} + f_{AA }+ f_{BB})

= 1/4(0 + 0.5 + 0.5) = 0.25

and f_{GH }(full sibs) = 1/4(2f_{CD} + f_{CC} + f_{DD})

= 1/4(0 + 0.5 + 0.5) = 0.25

__Using second diagram__

Therefore, F_{K }= f_{IJ} = 1/4(f_{EF} + f_{EH} + f_{GF} + f_{GH})

= 1/4(0.25 + 0 + 0 + 0.25)

= 0.125 = __12.5%__

Example 4

Figure 24 Close Inbreeding (full sibs)

__Find F _{I}__

f_{AB} = 0

F_{A} = F_{B} = F_{C} = F_{D} = 0

f_{CD} (full sibs) = 1/4(2f_{AB} + f_{AA} + f_{BB})

f_{AA} = 1/2(1 + F_{A}) = 0.5 and f_{BB} = 1/2(1 + F_{B}) = 0.5

Therefore f_{CD} = 1/4(0 + 0.5 + 0.5) = 0.25 = F_{E} = F_{F}

f_{EF} (full sibs) = 1/4(2f_{CD} + f_{CC} + f_{DD})

f_{CC} = 1/2(1 + F_{C}) = 0.5 and f_{DD} = 1/2(1 + F_{D}) = 0.5

Therefore f_{EF} = 1/4(0.5 + 0.5 + 0.5)

= 0.375 = F_{G} = F_{H}

f_{GH }(full sibs) = 1/4(2f_{EF} + f_{EE} + f_{FF})

f_{EE} = 1/2(1 + F_{E}) = 0.625 and f_{FF} = 1/2(1 + F_{F}) = 0.625

Therefore, F_{I} = f_{GH} = 1/4(0.75 + 0.625 + 0.625)

= 0.5 = __50%__

[1]Alternative names for Coancestry are the Coefficient of Kinship, the Coefficient of Consanguinity and the Coefficient of Parentage. Lasley (1978) and Warwick and Legates (1979) employ a similar method to calculate F using covariance tables. The covariance between two prospective parents is equal to twice their coancestries and therefore twice the inbreeding coefficient of their progeny (F). For further information on coancestries see Falconer (1989).

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