Page 20 - Linear Models for the Prediction of Animal Breeding Values

mean of these records. With repeated measurements it is assumed that there is addi-
tional resemblance between records of an individual due to environmental factors
or circumstances that affect the records of the individual permanently. In other
words, there is an additional covariance between records of an individual due to
non-genetic permanent environmental effects. Thus the between-individual vari-
ance is partly genetic and partly environmental (permanent environmental effect).
The within-individual variance is attributed to differences between successive meas-
urements of the individual arising from temporary environmental variations from
one parity to the other. The variance of observations (var(y)) could therefore be
partitioned as:
var(y) = var(g) + var(pe) + var(te)
where var(g) = genetic variance including additive and non-additive, var(pe) = variance
due to permanent environmental effect, and var(te) = variance due to random tempo-
rary environmental effect.
The intra-class correlation (t), which is the ratio of the between-individual vari-
ance to the phenotypic:
t = (var(g) + var(pe))/var(y) (1.5)
is usually called the repeatability and measures the correlation between the records of
an individual. From Eqn (1.5):
var(te)/var(y) = 1 − t (1.6)
With this model, it is always usually assumed that the repeated records on
the individual measure the same trait, that is, there is a genetic correlation of
1 between all pairs of records. Also, it is assumed that all records have equal vari-
ance and that the environmental correlations between all pairs of records are
˜
equal. Let y represent the mean of n records on animal i. The breeding value may
be predicted as:
a = b( y˜ − m) (1.7)
ˆ
i i
where:
b = cov(a, y)/var(y)
˜
˜
Now:
˜ 2
cov(a, y) = cov(a, g + pe + Σte/n) = s a
and:

˜
var(y) = var(g) + var(pe) + var(te)/n
Expressing the items in terms of Eqns 1.5 and 1.6:
2
var(t) = [t + (1 − t)/n]s y
Therefore:
2 2
y
b = s a /[t + (1 − t)/n]s
= nh /[1 + (n − 1)t]
2
Note that b now depends on heritability, repeatability and the number of records.

4 Chapter 1