1. Pedigree-based Evaluations

Inbreeding Coefficients

The inbreeding coefficient of an individual is the probability that two alleles chosen at random from the maternal and paternal haplotypes are identically by descent (IBD). This parameter estimates the extent to which the individual may suffer from inbreeding depression and predicts the homogeneity of its offspring. It can be calculated with

Animal <- pedInbreeding(Pedig)
mean(Animal$Inbr[Animal$Indiv %in% Phen$Indiv])

## [1] 0.01943394

Whether mating to a sound inbred individual should be favored or avoided depends on whether the breeder wishes offspring with uniform or heterogeneous breeding values. More important than the inbreeding coefficient of an animal itself, however, is the expected inbreeding coefficient of the offspring, which should be low. The expected inbreeding coefficient of the offspring is equal to the kinship of the parents.

Kinship

The kinship between two individuals is the probability that two alleles randomly chosen from both individuals are IBD. A matrix containing the kinship between all pairs of individuals can be computed with function pedIBD. It is half the numerator relationship matrix. The R code below computes the kinship between the female with ID 276000812750188 and all male selection candidates that have a breeding value larger than 1.0 and a pedigree with at least 5 equivalent complete generations.

pKin   <- pedIBD(Pedig, keep.only=Phen$Indiv)
isMale <- Pedig$Sex=="male" & (Pedig$Indiv %in% Phen$Indiv[Phen$BV>1.0])
males  <- Pedig$Indiv[isMale & summary(Pedig)$equiGen>5]
pKin[males, "276000812750188", drop=FALSE]

##                 276000812750188
## 276000811902819      0.28955731
## 276000812749676      0.06626139
## 276000812689003      0.03807358
## 276000812750190      0.18099209
## 276000813155662      0.17488755
## 276000812688988      0.16645344
## 276000812771544      0.01719556

The males with lowest kinship should be favoured for mating.

Breed Composition

Mating decisions should not only depend on the breeding value of the male and the kinship between male and female, but also on the native contribution of the male. Many endangered breeds have been graded up with commercial high-yielding breeds. These increasing contributions from other breeds displace the original genetic background of the endangered breed, decrease the genetic contribution from native ancestors, and reduce the conservation value of the breed.

For computing the breed composition of individuals it should be taken into account that the breed name of founders born recently is in fact unkown even if they have been classified as purebred. Hence, for these individuals, the breed name should be changed to "unknown". Below, the breed name of founders born after 1970 is changed to "unknown" if they had been classified as Hinterwald cattle. Thereafter, function pedBreedComp was used to estimate the contribution of each individual from each foreign breed and from native founders. The contribution an individual has from native founders is called the native contribution NC of the individual. Finally, column NC containing the native contributions was added to the pedigree.

Pedig <- prePed(PedigWithErrors, thisBreed="Hinterwaelder", lastNative=1970)

cont  <- pedBreedComp(Pedig, thisBreed="Hinterwaelder")
Pedig$NC <- cont$native
head(cont[rev(Phen$Indiv), 2:6])

##                    native    unknown     unbek0  Fleckvieh Vorderwaelder
## 276000812749835 0.6096039 0.09570312 0.03677368 0.11021423    0.14770508
## 276000812749569 0.5945129 0.14160156 0.04071045 0.11599731    0.10717773
## 276000891724277 0.0000000 0.50000000 0.50000000 0.00000000    0.00000000
## 276000891920412 0.5501404 0.12109375 0.03527832 0.10354614    0.18994141
## 276000812496874 0.3734741 0.41601562 0.03430176 0.08294678    0.09326172
## 276000811287745 0.5804138 0.09765625 0.03259277 0.08963013    0.19970703

The columns are ordered so that the most influential foreign breeds come first. It can be seen that the contribution from native founders varies considerably between individuals. Individuals with high genetic contribution from native founders should be favored for mating provided that the kinship between male and female is sufficiently low.

Kinship at Native Alleles

Since animals with high native contributions tend to be related, the inbreeding level could increase considerably when introgressed genetic material is removed from the population. This could be avoided by restricting the incease in kinship at native alleles in the population. The kinship at native alleles is also called the native kinship. An R-Object containing the information needed to estimate native kinship from pedigree can be obtained as:

pKinatN <- pedIBDatN(Pedig, thisBreed="Hinterwaelder", keep.only=Phen$Indiv)

For pairs of individuals the native kinship can be estimated as:

pKinatN$of <- pKinatN$Q1/pKinatN$Q2
pKinatN$of["276000891862786","276000812497659"]

## [1] 0.182625

However, the kinship at native alleles between two individuals says nothing about the amount of genetic material the individuals have from native ancesors. It only quantifies how different these native alleles are. Hence, in any case, the native contributions of the individuals should also be considered:

Pedig[c("276000891862786", "276000812497659"),c("Born","NC")]

##                 Born        NC
## 276000891862786 2004 0.6164551
## 276000812497659 2004 0.6274414

The native contributions of both indivividuals are larger than the mean NC of the phenotyped individuals, which is 0.44. However, their native kinship is larger than the average, which is

pKinatN$mean

## [1] 0.0776925

The correlation between the kinship and the native kinship is high (as expected):

subKin    <- pKin[Phen$Indiv, Phen$Indiv]
subNatKin <- pKinatN$of[Phen$Indiv, Phen$Indiv]
diag(subKin)    <- NA
diag(subNatKin) <- NA
cor(c(subKin), c(subNatKin), use="complete.obs")

## [1] 0.8916129

The correlation is even higher if only individuals with high native contributions are considered.

Genetic Diversity

The genetic diversity of a population is the probability that two alleles chosen at random from the population are not IBD. It is one minus the average kinship of the individuals. A simple estimate can be obtained as

1 - mean(pKin[Phen$Indiv, Phen$Indiv])

## [1] 0.9755853

The diversity of this population is high. A high genetic diversity enables to avoid inbreeding and ensures that polygenic traits have a high additive variance, which is required to achieve genetic gain. However, the diversity of this population seems to be extremely high. This could have two reasons:

• The diversity is indeed very high because the individuals are in fact crossbred individuals with genetic contributions from many unrelated breeds, or

• The completeness of the pedigrees is insufficient for such an evaluation.

For this breed it is unclear whether pedigree completeness is low because pedigrees of indivduals from other breeds were cut, or because previously unregistered animals have been registered. Hence, pedigrees of introduced individuals from other breeds should not be cut to reduce this uncertainty.

A parameter which depends not so much on the completeness of the pedigrees is the diversity at native alleles.

Kinship and Diversity at Native Alleles

The kinship at native alleles in the population is the probability that two alleles chosen at random from the population are not IBD, given that both alleles originate from native founders. The mean kinship at native alleles is

pKinatN$mean

## [1] 0.0776925

The genetic diversity at native alleles is one minus the kinship at native alleles (named conditional gene diversity in Wellmann, Hartwig, and Bennewitz 2012). Thus, it can be calculated as

1 - pKinatN$mean

## [1] 0.9223075

Note that incomplete pedigrees result in an overestimation of the genetic diversity. This is not the case for the genetic diversity at native alleles because alleles originating from founders born after 1970 were classified to be non-native. Hence, the diversity at alleles originating from these individuals does not affect the estimate.

A high diversity at native alleles is important if a goal of the breeding program is to remove introgressed genetic material from the population. Without maintenance of a high diversity at native alleles, inbreeding coefficients will soon rise to an unreasonable level.

Native Effective Size

The native effective size of a population is defined as the size of an idealized random mating population for which the genetic diversity decreases as fast as the diversity at native alleles decreases in the population under study (Wellmann, Hartwig, and Bennewitz 2012). Thus, the native effective size quantifies how fast the diversity at native alleles decreases. In contrast, the effective size quantifies how fast the genetic diversity decreases.

In a population without historic introgression the native effective size (native Ne) is equal to the effective size (Ne) as estimated by Wellmann and Bennewitz (2011). But in a population with historic introgression it can be argued that the effective size is not useful to describe the history of a population because even a small amount of introgression with unrelated individuals prevents a drop of genetic diversity, so the effecive size would be infinite.

For estimating the native effective size, the diversity at native alleles needs to be estimated at various times, and then the native effective size is estimated from the slope of a regression function.

An estimate of the native effective size is automatically provided when the kinship at native alleles is computed and parameter is specified:

pKinatN  <- pedIBDatN(Pedig, thisBreed="Hinterwaelder", keep.only=Phen$Indiv, nGen=6)

## Number of Migrant Founders: 237
## Number of Native  Founders: 150
## Individuals in Pedigree   : 1658
## Native effective size     : 49.5

The native Ne is stored as an attribute of the result:

attributes(pKinatN)$nativeNe

## [1] 49.5

Thus, the diversity at native alleles decreases as fast as in an idealized population consisting of 49.5 individuals.

Effective Size

The effective size Ne of a population is the size of an idealized random mating population for which the genetic diversity decreases as fast as in the population under study. It is commonly estimated from the mean rate of increase in coancestry (Cervantes et al. 2011), whereby the increase in coancestry between any pair of individuals i and j is computed as $\Delta c_{ij}=1-\sqrt[\frac{g_i+g_j}{2}]{1-c_{ij}}$, where c_ij is the kinship between i and j, and g_i, g_j are the numbers of equivalent complete generations of individuals i and j. The effective size is then estimated as $N_e=\frac{1}{2\overline{\Delta c}}$. Thus, the effective size can be estimated as:

id     <- Summary$Indiv[Summary$equiGen>=4 & Summary$Indiv %in% Phen$Indiv]
g      <- Summary[id, "equiGen"]
N      <- length(g)
n      <- (matrix(g, N, N, byrow=TRUE) + matrix(g, N, N, byrow=FALSE))/2
deltaC <- 1 - (1-pKin[id,id])^(1/n)
Ne     <- 1/(2*mean(deltaC))
Ne

## [1] 97.95922

However, the above formula assumes that ancestors are missing at random. In populations with historic introgression, pedigrees of individuals from foreign breeds are often cut, so their parents are not missing at random. In fact, even a small amount of introgression suffices to prevent a decrease of genetic diversity, so that the effective size of such a population is ∞.

Change of Native Genome Equivalent and Ne

The native genome equivalent NGE of a population is the minimum number of founders that would be needed to create a population consisting of unrelated individuals that has the same diversity at native alleles as the population under study (Wellmann, Hartwig, and Bennewitz 2012). It is assumed that the individuals were unrelated in base year base, which could be well before pedigree recording had started. Between the base year and the year lastNative in which the last native founder was born, the population had a historical effective size of histNe. This assumption implies that the founders of the pedigree were related due to ancestors whose pedigrees had not been recorded.

The decrease of native genome equivalents is estimated below for the time since lastNative=1970 by assuming that individuals were unrelated in year base=1800 and that the historical effective size was histNe=150 between 1800 and 1970. Since the computation for the full pedigree could take some time, the parameters are estimated below from a subset of the pedigree, which are the individuals included in vector keep. Vector keep contains from each birth cohort 50 randomly sampled Hinterwald cattle.

data("PedigWithErrors")
Pedig <- prePed(PedigWithErrors, thisBreed="Hinterwaelder", lastNative=1970)
set.seed(0)
keep <- sampleIndiv(Pedig[Pedig$Breed=="Hinterwaelder",], from="Born", each=50)
cand <- candes(phen   = Pedig[keep,],
               pKin   = pedIBD(Pedig, keep.only=keep), 
               pKinatN= pedIBDatN(Pedig, thisBreed="Hinterwaelder", keep.only=keep), 
               quiet=TRUE, reduce.data=FALSE)

sy <- summary(cand, tlim=c(1970, 2000), histNe=150, base=1800, df=4)

## 
##         t    I  pKin pKinatN    Ne  NGE
## 1970 1970 5.46 0.004   0.005  55.4 5.06
## 1971 1971 5.44 0.004   0.006  55.4 5.01
## 1972 1972 5.42 0.005   0.007  55.4 4.95
## 1973 1973 5.39 0.006   0.009  55.3 4.88
## 1974 1974 5.37 0.006   0.011  55.1 4.79
## 1975 1975 5.35 0.007   0.014  55.0 4.66
## 1976 1976 5.33 0.008   0.015  54.7 4.61
## 1977 1977 5.31 0.009   0.017  54.0 4.54
## 1978 1978 5.29 0.010   0.018  52.7 4.52
## 1979 1979 5.27 0.010   0.018  50.7 4.49
## 1980 1980 5.25 0.011   0.020  48.2 4.44
## 1981 1981 5.23 0.011   0.021  45.5 4.40
## 1982 1982 5.22 0.012   0.023  42.9 4.33
## 1983 1983 5.20 0.012   0.024  40.7 4.29
## 1984 1984 5.18 0.013   0.028  38.9 4.18
## 1985 1985 5.17 0.014   0.030  37.8 4.09
## 1986 1986 5.16 0.015   0.033  37.5 4.01
## 1987 1987 5.15 0.017   0.037  38.0 3.90
## 1988 1988 5.14 0.018   0.042  39.5 3.78
## 1989 1989 5.14 0.017   0.044  42.1 3.71
## 1990 1990 5.14 0.018   0.048  46.2 3.62
## 1991 1991 5.14 0.018   0.049  51.8 3.59
## 1992 1992 5.14 0.017   0.049  59.2 3.58
## 1993 1993 5.15 0.018   0.052  68.7 3.54
## 1994 1994 5.16 0.019   0.052  80.3 3.53
## 1995 1995 5.17 0.017   0.051  93.4 3.56
## 1996 1996 5.18 0.017   0.051 106.6 3.56
## 1997 1997 5.20 0.016   0.050 117.7 3.57
## 1998 1998 5.22 0.014   0.051 124.8 3.55
## 1999 1999 5.24 0.013   0.052 127.8 3.53
## 2000 2000 5.26 0.012   0.054 127.3 3.49

ggplot(sy, aes(x=t, y=Ne)) + geom_line() + ylim(c(0,100))

## Warning: Removed 5 rows containing missing values or values outside the scale range
## (`geom_line()`).

ggplot(sy, aes(x=t, y=NGE)) + geom_line() + ylim(c(0,7))

Not only the native genome equivalents (column NGE), but also the native effective size (column Ne), the mean kinship (column pKin), the mean kinship at native alleles pKinatN, and the generation interval (column I) have been estimated for all birth cohorts t.

It is possible that the diversity at native alleles increases for a short period of time, in which case the estimate of the native effective size would be NA. Choose df<4 to get a smooth estimate for the native effective size.

Change of Breed Composition

For monitoring the introgression from other breeds, the contributions of foreign breeds to all birth cohorts can be estimated with function conttac and then plotted, e.g. with function ggplot.

use  <- Pedig$Breed=="Hinterwaelder" & Pedig$Born %in% (1950:1995)
cont <- pedBreedComp(Pedig, thisBreed="Hinterwaelder")
contByYear <- conttac(cont, cohort=Pedig$Born, use=use, mincont = 0.01)
ggplot(contByYear, aes(x=Year, y=Contribution, fill=Breed)) + geom_area(color="black")