Animal Molecular Breeding, 2013, Vol.3, No.2, 4
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random effect. The same model was fitted on all the
traits and the traits were analyzed separately. The
general formulation of the mixed model fitted is as
follows.
Yijkl = ų + si + cj + pk + eijkl
where, all the abbreviations are same as described in
first model. The formulation of model in matrix
notation is as follows.
Y = 1 ų + xb + za + e
Where, 1 is the column vector of the means; ų is the
over all mean; B is the column vector of fixed effects;
a is the column vector of random effects; z is an
incidence matrix of 0’ s and 1’ s; x is an incidence
matrix of o’ s, 1’ s & -1’ s and x-x values for the
discrete effects, and e is a column vector of the
random errors. This model is same as first model,
except the random effect may be correlated.
2.5
Univariate
In univariat analysis, birth weight, weaning weight, 6
month weight and first six monthly greasy fleece yield
were analyzed separately. The same model was fitted
on all four traits. The general formulation of the mixed
model fitted is as followes.
Yijkl = µ+ Ai + cj + pk + eijkl
Where, A
i
is random effect of it animal and all the
other abbreviations are same as described in earlier
models. The formulation of general single trait animal
model is matrix notation is as follows:
y = xf + z
a
+ e.
Where, y is a vector of Nx 1 records; f is a vector of
fixed environmental effects of sex and year; and
covariable was taken here; a is vector of breeding
values for additive direct genetic effects fitted shich is
random; s is a N* n F design matrix for fixed effects
with column ranks N*F*; z is a N* NR design matrix
for random animal effects, where z=1, and e is a
vector of N random residual errors.
2.6
Multivariate Model
In multivariate analysis, all the four mentioned traits
were taken simultaneously for analysis. The multi trait
animal model used to estimate parameters is as
follows.
Y
ijkl
= µ+ A
i
+ c
j
+ p
k
+ e
ijkl
.
Where all the abbreviations are same as described in
univariate model. The above multi trait animal model,
in matrix notation, for 4 traits used is as follows:
y = x
b
+ z
u
+ e
where, y is a vector of Nt* 1 of records; b is a vector
of fixed environmental effects of sex and years. No
covariable was taken here; µ is a vector of breeding
values for additive direct genetic effects fitted, which
is random; e is a vector of N random residual errors
and x and z are incidence matrices relating the records
to the effects of the model.
Three solutions for sire evaluation were used as Best
Linear Unbiased Prediction (BLUP
1
)
values of model
8
of Least squares analysis (Harvey, 1990), univariate
(
BLUP
2
)
and multivariate (BLUP
3
)
solutions of
REML using animal model (Mayer, 1998), which
utilized information from all the known relationship.
Under univariate and multivariate and multivariate
animal model no sire effect was fitted. The sire
solutions was sorted out from the solution of all the
animals used for comparison with BLUP values
obtained under model 8. On the basis of these BLUP
values sires were ranked.
2.7
Rank Correlation
The spearman’s rank correlation between BLUP
values obtained by above methods was worked out
(
Steel and Torrie, 1980) as follows.
Where, r is the rank correlation; n is the number of
sires; d
i
is the difference between rank of the sire
ranked by two methods.
The significance of the rank correlation was tested by
students t-test as follows:
Note: with (n-2) degree of freedom
)1 )(1 (
1
2
6
 
n
nn
d
r
i
i
2
r
n r t
1
2