Correlated responses to selection have been estimated using two independent approaches: genetic trends and control population. Genetic trends use all data from all generations of selection but its estimations depend on the model, and therefore response to selection depends on the genetic parameters used in the model. In the control population approach, the number of data is smaller but differences between control and selected population will have only genetic bases if both populations are raised contemporarily.
2.4.1. Genetic Parameters and Genetic Trends
A repeatability model was used to analyze OR in the selection period:
where y
ijklm is the trait OR; YS
i is the effect of year-season of the mating day (31 levels for OR); PO
j is the effect of the parity order (four levels for OR); L
k is the effect of lactation status at mating (two levels: lactating and nonlactating does when mated); a
ijkl is the additive value of the animal; p
ijkl is the permanent environmental effect; and e
ijklm is the residual effect.
For individual weights and IGR, as well as for growth variability traits, the animal model used was:
where y
ijklmn is IW28, IW63 and IGR; YS
i is the fixed effect of year-season in which the animal was growing (29 levels); PO
j is the effect parity in which the animal was born (five levels); NBA
k is the effect of the number of rabbits born alive when the animal was born (17 levels); d
ijkl is the random dam (or female) effect between parities (851 levels); c
ijklm is the common litter effect (2683 levels); a
ijklmn is the additive value of animal; and e
ijklmn is the residual effect.
Univariate analysis for OR was performed to estimate the heritability of the selection trait. To account for the selection process and to estimate the heritability of each growth trait as well as the correlation between OR and growth traits, bivariate analyses including OR were performed.
Data augmentation [
14,
15] was performed to analyze the data in order to have the same design matrices for all traits since different models for OR and growth traits were used. Augmented data are not used for inferences but allow the simplification of computing.
After data augmentation, the model for all traits was:
where
is a vector of augmented data;
are known incidence matrices; and
is the (co)variance residual matrix. Records of different individuals were assumed to be conditionally independent, given the parameters, but a correlation between residuals of different traits of the same individual was allowed.
Hence, sorting the data by individual, the residual (co)variance matrix can be written as
, with
being the
residual (co)variance matrix between OR and the growth trait analyzed and
being an identity matrix of the same order as the number of individuals. Bounded uniform priors were used to represent vague previous knowledge of environmental effects,
. Prior knowledge concerning the other random effects was represented by assuming that they were normally distributed, conditionally on the associated variance components. Thus, for the additive genetic effects:
where
is a vector of zeroes and
is the genetic variance covariance matrix. Sorting the data by individual as before, this matrix can be written as
, where
is the
genetic (co)variance matrix between traits analyzed and
is the known additive genetic relationship matrix between elements of the additive genetic effects vector.
The distribution of permanent environmental effects was assumed to be normal and of the form:
where
is a vector of zeroes and
is the permanent effect matrix. Sorting the data by individual, this matrix can be written as
, with
being the
permanent variance matrix between traits analyzed and
being the identity matrix with the same order of the number of levels of permanent effects.
The distribution of the random dam effects between parities was assumed to be normal and of the form:
where
is a vector of zeroes and
is the random dam effects between parities matrix. Sorting the data by individual, this matrix can be written as
, with
being the
permanent variance matrix between traits analyzed and
being the identity matrix with the same order of the number of levels of the random dam effects between parities.
Finally, the distribution of the common litter effects was assumed to be normal and of the form:
where
is a vector of zeroes and
is the common litter effects matrix. Sorting the data by individual, this matrix can be written as
, with
being the
common litter variance matrix between traits analyzed and
being the identity matrix with the same order of the number of levels of the common litter effects.
For all analyses, bounded flat priors were used for matrices , , , and .
To estimate the correlations between growth traits, trivariate analyses including OR and two growth traits were performed. Hence, sorting the data by individual, the residual (co)variance matrix can be written as , with being the residual (co)variance matrix between OR and the growth traits analyzed and being an identity matrix of the same order as the number of individuals. As previous analyses, bounded uniform priors were used for environmental effects and normally distributed priors, conditionally on the associated variance components that were used for random effects. The additive genetic effects, the permanent environmental effects, the random female effects between parities and the common litter effects were the same as described above. As in the previous model, all effects are independent among them. However, sorting the data by individual as before, the matrix can be written , where is the genetic (co)variance matrix between traits analyzed and is the known additive genetic relationship matrix described previously; the matrix can be written as , with being the permanent variance matrix between traits analyzed and being the identity matrix with the same order of the number of levels of permanent effects; the matrix can be written as , with being the permanent variance matrix between traits analyzed and being the identity matrix with the same order of the number of levels of the random dam effects between parities; and the matrix can be written as , with being the permanent (co)variance matrix between traits analyzed and being the identity matrix with the same order of the number of levels of the common litter effects.
Marginal posterior distributions of all unknowns were estimated using a Gibbs sampling procedure using the program TM [
16]. Different confidence intervals were estimated: k
95% is the guaranteed value of the interval [k, 1] containing the 95% of the probability, P is the probability of the estimation being higher (or lower) than 0.00, P
0.10 and P
0.30 are the probability of the estimation being higher (or lower) than 0.10 and 0.30, respectively, and P
r is the probability of relevance [
13,
14]. We considered a heritability to be irrelevant when it was lower than 0.10 [
4,
5,
6,
17]. In the case of correlation, we considered to be an irrelevant value all correlations in absolute value lower than 0.30, since the percentage of the variance explained by the other trait (r
2) is <10%. After some exploratory analyses, two chains were used, each of 1,000,000 iterations, with a burning period of 200,000 iterations. Only every 100th iteration was saved. Features of marginal posterior distributions of parameters were obtained using the package R code. Convergence was tested using the Z criterion of Geweke and Monte Carlo sampling errors were computed.
2.4.2. Selected versus Control Population
The model assumed for analyzing OR using the offspring of the control and selected population was:
where y
ijklm is the trait OR; Line
i is the effect of the line (two levels: control and selected); YS
j is the effect of year-season (three levels); PO
k is the effect of parity (two levels: at second and fourth gestation); L
l is the effect of lactation state of the doe (two levels: lactating and nonlactating does when mated); p
ijklm is the effect of the doe (105 levels); and e
ijklmn is the residual of the model.
The model assumed for analyzing individual growth traits and their variabilities was:
where y
ijklm is IW28, IW63 and IGR; YS
i is the fixed effect of year-season in which the animal was growing (four levels); PO
j is the effect parity in which the animal was born (four levels); NBA
k is the effect of the number of rabbits born alive when the animal was born (16 levels); d
ijkl is the random dam effect between parities (96 levels); c
ijklm is the common litter effect (302 levels); and e
ijklmn is the residual effect.
Bounded uniform priors were used for all unknowns with the exception of the dam and common litter effects, which were considered normally distributed. Dam effect was with mean and variance , where I is a unity matrix and is the dam effect variance of the trait. Common effect was with mean and variance , where I is a unity matrix and is the common effect variance of the trait. Residuals were normally distributed with mean and variance . The priors for the variances were also bounded uniform positive. Features of the marginal posterior distribution of differences between line means were estimated by using the Gibbs sampling algorithm. Similarly to previously described genetic estimations, chains of 1,000,000 samples each were used, with a burning period of 200,000. One sample out of each 100 was saved to avoid high correlations between consecutive samples. Features of marginal posterior distributions of differences between line means were obtained using the package R code. Convergence was tested using the Z criterion of Geweke and Monte Carlo sampling errors were computed.