The next step is to find out if the completeness of the pedigree is
sufficient for the intended evaluation, and to identify individuals with
sufficient pedigree information. This will be demonstrated at the
example of a pedigree of Hinterwald cattle. Data frame Phen
contains selection candidates with breeding values in column
BV, and data frame PedigWithErrors contains
their pedigree. The column with individual IDs is called
Indiv in both data sets.
The completeness of the pedigree of each individual can be summarized
with function summary:
The best possibility to characterize completeness of pedigree
information by a single value is the number of equivalent complete
generations, averaged over all individuals of the actual breeding
population which were included in the evaluations.
The relevant parameter to identify individuals with insufficient
pedigree information to estimate inbreeding, however, is the
PCI. This is because inbreeding can be detected only if
both maternal and paternal ancestries are known. The harmonic mean
ensures that the less complete paternal pedigree is weighted more
heavily, so the PCI equals zero when either parent is
unkown. Inbreeding coefficients can be valid despite small
PCIs if the most recent founders were indeed unrelated,
e.g. because they were from other breeds.
Individual Specific Parameters
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
## [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$Indiv]
diag(subKin) <- NA
diag(subNatKin) <- NA
cor(c(subKin), c(subNatKin), use="complete.obs")
## [1] 0.8916129
The correlation is even higher if only individuals with high native
contributions are considered.
Population Specific Parameters
Genetic Diversity
The genetic diversity of a population is the probability that two
alleles chosen at random from the population are not IBD. It is one
minus the average kinship of the individuals. A simple estimate can be
obtained as
1 - mean(pKin[Phen$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
## [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] 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 \(\infty\).
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")
References
Cervantes, I., F. Goyache, A. Molina, M. Valera, and J. P. Guitérrez.
2011. “Estimation of Effective Population Size from the Rate of
Coancestry in Pedigreed Populations.” Animal Breeding and
Genetics 128.
Wellmann, R., and J. Bennewitz. 2011. “Identification and
Characterization of Hierarchical Structures in Dog Breeding Schemes, a
Novel Method Applied to the Norfolk Terrier.” Journal of
Animal Science 89: 3846–58.
Wellmann, R., S. Hartwig, and J. Bennewitz. 2012. “Optimum
Contribution Selection for Conserved Populations with Historic
Migration; with Application to Rare Cattle Breeds.” Genetics
Selection Evolution 44 (34).
