ISSN 1022-7954, Russian Journal of Genetics, 2007, Vol. 43, No. 8, pp. 947–953. © Pleiades Publishing, Inc., 2007.
Original Russian Text © G.R. Svischeva, 2007, published in Genetika, 2007, Vol. 43, No. 8, pp. 1139–1145.
Earlier [1–3], we developed a method for mapping
quantitative trait loci (QT loci) in the offspring of two
panmictic populations differing in the distribution of
the trait studied. The method is based on the decompo-
sition of variances and covariances into the same compo-
nents accounted for by additive, dominance, polygenic,
and residual effects of the trait. It is suitable for analysis
of pedigrees with any structure without inbreeding.
However, large pedigrees of domestic and laboratory
animals usually include crosses between consanguine-
ous individuals, and genetic analysis should take into
account the resultant inbreeding.
A method for analyzing crosses between outbred
lines taking inbreeding into account was developed
recently . The essence of the method is that each pair
of genetically related individuals can be assigned to one
of several types of allelic identity, which is determined
on the basis of the animal origin and the number of
identical alleles in QT loci. The genetic covariance for
each related pair is expressed by a nonlinear function of
the probabilities that the animals belong to these types
of identity. If the covariances are determined by this
method, inbreeding is not formalized as a parameter
and has no numerical characteristic.
This study is a continuation of a series of works on
the development of methods for analyzing QT loci. We
consider the construction of a variance–covariance
model for a hybrid pedigree with any types of crosses.
Introduction of a special parameter characterizing the
degree of inbreeding makes the model versatile.
THE VARIANCE–COVARIANCE MODEL
A quantitative trait
is typically described by a
is the mean of the trait,
is the effect of the QT
locus chosen for the analysis and referred to as the
is the effect of all other QT loci, i.e., the
is the environmental effect.
The values of the trait
in members of a hybrid ped-
igree follow a multivariate normal distribution with the
following likelihood function:
is the vector of expected values of the trait
in the members of the hybrid pedigree and
matrix of the covariance values of the trait for all pairs
The pivotal assumption for hybrid pedigrees is that
the contributions of different loci into the determination
of the trait do not depend on the population origin of the
individuals, and the difference between populations in
the trait value is entirely determined by different allele
frequencies in these loci [1, 2].
Earlier , we neglected the effect of dominance for
QT loci combined into a polygene, which considerably
simpliﬁed the analysis. Now let us complicate the prob-
lem: let all QT loci comprised in the polygene have a
dominance effect. Then, the QT locus taken for analysis
(the major gene) has no formal priorities or advantages
over other loci.
Let us consider an arbitrary single QT locus; it does
not mater whether it is the major gene or part of the
polygene. Let the contributions
of its different geno-
) into the determination of the
, respectively. If
are the fre-
quencies of allele
in individuals from the original
, respectively, then the frequency
LH const 1/2 Vln–=
– 1/2 X E X–()
X E X–(),
A Variance–Covariance Model for Analysis
of Hybrid Pedigrees with Inbreeding
G. R. Svischeva
Institute of Cytology and Genetics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090 Russia;
fax: (383) 333-12-78; e-mail: firstname.lastname@example.org
Received June 19, 2006
—A variance–covariance model is suggested for plotting the distribution of a quantitative trait ana-
lyzed in animal pedigrees resulting from crosses of outbred lines. The model takes inbreeding into account.
A special parameter characterizing the degree of inbreeding has been introduced, which makes the model ver-
satile. Pedigrees with the same structure that contain or not contain inbred individuals have been compared to
analyze the effect of inbreeding on the parameters of the trait distribution, such as the genotypic mean and vari-
ance of the trait.
MODELS AND METHODS