Livestock Research for Rural Development 12 (2) 2000  Citation of this paper 
In order to compare different animal models, the methodology of mixed models under animal models was used to predict (co)variance components of 23 traits related to reproduction traits of 1,456 litters and growth and slaughter traits of 3,845 Californian and New Zealand White rabbits raised in southeastern Brazil. The (co)variance components, obtained by four different models in single trait analysis, were used to estimate genetic parameters. The four animal models considered as random effects only the animal direct genetic effects (model 1), the animal direct and permanent effects of litters or common effects of does as permanent environment effects (model 2), the animal direct and maternal genetic effects, uncorrelated to each other, and permanent environmental effects (model 3), and as in model 3, but with correlated animal direct and maternal effects (model 4). All the models considered the fixed effects of contemporary groups, parity, sex and the covariates  level of inbreeding of litters or rabbits, and litter sizes, where these were applied. The models were compared based on likelihood test and the "best" model is proposed for each trait. Permanent environmental effects were important for all traits and should be considered in animal models that analyze reproductive, growth and slaughter traits of rabbits of these two breeds. The magnitude of the c² term varied from 0 to 0.39. Maternal genetic effects were not important for reproductive traits, but significantly affected growth and slaughter traits, and were more important for New Zealand White than for Californian rabbits. Maternal heritabilities varied from 0.03 to 0.14. As models 1, 2, 3 and 4 gave different estimates for genetic parameters, total heritabilities were calculated for all traits. Total heritabilities were low for reproductive traits (from 0 to 0.14), and moderate for growth (from 0.03 to 0.36) and slaughter traits (0.02 to 0.23). Although New Zealand White and Californian rabbits perform similarly, the results showed that the genetic parameters for these two breeds are different and should not be analyzed together. The models chosen for each trait serve as guides for proposition of animal models in single or multi trait analysis of rabbit data.
As rabbits can be fed with forages and do not compete directly with humans for grains, they can be a very important source of high quality animal protein in developing countries. However the knowledge of genetics of rabbit breeds , including two of the most used ones, New Zealand White and Californian, under tropical conditions is not well established and more studies are needed (Lukefahr and Cheeke 1991; Lukefahr et al 1992; Ferraz 1993; Ferraz et al1994).
In selection programs applied to rabbit production, reproductive, growth and slaughter traits can be considered. However, the larger the number of traits, the smaller the genetic gain obtained in each trait. The decision about which trait should be considered depends not only on the economics involved, but also on genetic parameters of the traits and the "practical" importance of the trait. The definition is taken by the geneticist and the breeder, always considering market reasons.
Studies on genetic parameters of several traits of rabbits have been made by some authors. Khalil et al (1986) made an important review article on this subject. However, the studies of parameters estimated for populations raised under tropical or subtropical conditions are not very many. Among such studies can be placed those of Nunes and Polastre (1988), Baselga and Camacho (1990), Camacho and Baselga (1990), Ferraz et al (1991a,b), Moura et al (1991a,b), Polastre et al (1991), Santacreu and Blasco (1991), Baselga et al (1992), Ferraz et al (1992), Lukefahr et al (1992), Ferraz et al (1994), Lukefahr et al (1994), Rochambeau et al(1994) and Ayyat et al. (1995), using different models, including animal models.
Genetic parameters are ratios of variances and covariances, so these estimates are needed for the complete genetic knowledge of populations. The estimation of variance and covariance components evolved from Henderson’s Methods I and III (Henderson 1953), through maximum likelihood (Hartley and Rao 1967), iterative maximum likelihood (Cunningham and Henderson 1968; Thompson 1969), MINQUE (Rao 1970; LaMotte 1970), Restricted Maximum Likelihood  REML (Patterson and Thompson 1971) and MIVQUE (Rao 1971). Best Linear Unbiased Predictors  BLUP (Henderson 1949; Henderson et al 1959), using methodology of mixed models is becoming the preferred method of estimation for animal breeders (Henderson 1988). REML co(variance) component estimation is also becoming the most commonly used algorithm in such estimations.
The objectives of this study were to compare different animal models used to estimate genetic parameters of eight reproductive, nine growth and six slaughter traits of Californian and New Zealand White rabbits raised under subtropical conditions in southeastern Brazil, using single trait analysis with mixed models methodology. The aim was not only to have a better knowledge of genetics of those breeds in subtropics but also to try to choose the "best" models for each one of the traits that can be used as selection criteria.
Material and Methods
Data came from records of 1,456 litters (for reproductive traits), 3,845 rabbits (for growth traits) and 2,195 carcasses (for slaughter traits) of Californian and New Zealand White rabbits, born between 1988 and 1992. Table 1 shows the distribution of data between the breeds.
The animals had been raised at the Rabbit farm on the Campus of the University of Sao Paulo, located in Pirassununga (22ºS and 47ºW, 750 m above sea level). Average temperatures were around 22.5ºC from January to March, 19ºC from April to June (decreasing each month) and from July to September (increasing each month) and 22ºC from October to December. Sunlight hours had similar trends to temperature. The rabbits were housed in metal cages and fed a commercial pelleted feed (18% crude protein and 17% fibre). Reproduction started after reaching 130 days of age.
Table 1: Distribution of data used for estimation of genetic parameters, between Californian and New Zealand White breeds. 

Item 
Californian 
New Zealand White 
Reproductive traits (litters) 
662 
794 
Growth traits (rabbits) 
2,010 
1,835 
Slaughter traits (rabbits) 
1,149 
1,046 
Number of sires 
42 
56 
Number of dams 
161 
180 
Number of animal with pedigree for reproductive traits 
744 
746 
Number of animals with pedigree for growth and slaughter traits 
2,138 
1,968 
Traits that were analyzed were:
Reproductive traits
Reproduction data for each doe and parturition were recorded: litter size at birth total (LSB) and alive (LSBA), litter size at 21 days of age (LS21), litter size at weaning (LSW), litter weights at birth (LWB), 21 days (LW21) and weaning (LWW) and mortality from birth to weaning (MORT).
Growth traits
The rabbits were weekly weighed from weaning to 11 weeks, and the traits were: weaning weight (WW), weights at 5, 6, 7, 8, 9, 10 and 11 weeks (W5, W6, W7, W8, W9, W10, W11), besides average daily gain (ADG).
Slaughter traits
At slaughter, each animal was individually weighed before and after slaughter, giving the traits: weight at slaughter (WS), carcass weight (CW), viscera weight (VW), head weight (HEAD), skin weight (SKIN) and carcass yield (CY).
Inbreeding coefficients for litters and does were calculated using a modification of K. Meyer’s program DFNRM, made by Van Vleck (1991, personal communication). Pedigree information was used as far it existed. Data were analyzed by mixed model methodology for animal models, in single trait analysis. using the software DFREML (Meyer 1988, 1989), modified by Boldman and Van Vleck (1991) for use of Sparspak, a sparse matrix solver, according to suggested procedures described by Ferraz (1992). Parameters estimated were heritability for direct and maternal effects, total heritability (as proposed by Dickerson 1947, 1970), besides the c² term (ratio between the variance due to permanent environment and phenotypic variance). Phenotypic variance was also estimated.
Analyseswere done according to the general model:
y = X$ + Za + Zm +Zc + e, where:
y =vector of dependent variable, the observations;
X = incidence matrix of fixed effects;
$ = vector of fixed effects, including sex (for growth and slaughter traits), parity, year and trimester (for reproductive traits), contemporary groups (for growth and slaughter traits), linear and quadratic effects of covariable level of inbreeding of does and rabbits, linear effects of covariable age at weaning (for weaning traits), linear and quadratic effects of litter size at weaning and linear effects of age at slaughter (for slaughter traits);
Z = incidence matrix for random effects;
a = vector of random animal direct genetic effects;
m = vector of random animal maternal genetic effects;
c = vector of random permanent environmental effects of does (for reproductive traits) or common effects of litters (for growth and slaughter traits);
e = vector of random errors, NID (0, F²).
The differences among the models refers to the number of random effects considered. Model 1 considered only the animal direct genetic effect, Model 2 the animal direct genetic effect and the permanent environmental effect of doe or common effect of litters, Model 3 also considered the effects included in Model 2 plus the animal maternal genetic effect, uncorrelated with the direct effect and Model 4 considered all the effects included in Model 3, but in this case the animal direct and maternal genetic effects were considered correlated.
To compare animal models, it was assumed that the higher the likelihood function, the more the model explained the data. Likelihood function is higher when new parameters are included in the model. So, to go from model 1 to model 2, model 2 to model 3 and model 3 to model 4, one parameter was added each time. To compare the difference between the values of the likelihood function of two models, the methodology used was that described by Rao (1973) and Mood et al (1974). This method is based on the fact that that the difference 2[log7_{i}  log7_{i’}] has Qui squared distribution, where 7_{i} and 7_{i’ }are the values of likelihood function, after the convergence criteria of the iterative process has been reached (in this case the variance of the function in the last 5 evaluations should not be larger than 1 x 10^{9}) in the different models. The number of degrees of freedom of this comparison is equal to the number of parameters that were added to the model (one in the case of the comparisons made here). Significance was tested not only at level of P<0.05, but also a "practical" significance, based on variation of values of genetic parameters was considered in the choice of the "best" model.
Results and Discussion
The definition of the correct model is important, because the more complex the model, the larger the time needed for solution. This is even more important in multitrait analysis, because CPU time is a function of n_{3}, where n is the number of variance and covariance components to be estimated. If you go from a Model 1 to Model 4 in a single trait analysis, the number of parameters to be estimated goes from one ( the variance component for the animal direct genetic effect) to three (animal direct and maternal genetic effect and the covariance between them), and CPU time will be close to 9 times larger. In a twotrait analysis, the number of variance components goes from three (both direct effects and their covariance) to nine and CPU time will be increased in the order of 9^{3}/3^{3} or 27 times larger. This "rule" is only a guide to estimate the processing time, but it depends of course on several factors, like the genetic relationship among animals, established in A matrix (the relationship matrix) and it’s inverse, A^{1}, The case will be even worse for three or fourtrait analysis. With the right models defined in singletrait analysis, a great waste of time can be avoid.
The Quisquared value and it’s significance for the likelihood test for each trait is given in Table 2. The analysis shows that the models affect the results differently, depending on whether the traits are reproductive or growth and slaughter.
Table 2: Quisquared value (1 degree of freedom) for likelihood test used to compare different animal models used for (co)variance components estimation in 23 traits of rabbits 

Californian Comparison between Models 
New Zealand White Comparison between Models 

Trait 
21 
32 
43 
21 
32 
43 
LSB 
3.42 
65.7* 
1.50 
4.8* 
0.72 
0.13 
LSBA 
5.6* 
69.9* 
0.68 
2.07 
1.79 
0.10 
LS21 
3.76 
0.00 
0.40 
0.68 
0.00 
0.18 
LSW 
3.46 
0.00 
0.62 
1.59 
0.00 
0.11 
LWB 
1.04 
0.16 
0.04 
5.1* 
0.17 
0.27 
LW21 
0.98 
0.36 
0.00 
1.41 
0.00 
0.00 
LWW 
0.10 
0.04 
1.48 
1.72 
0.00 
0.13 
MORT 
4.66* 
0.40 
0.16 
4.7* 
0.00 
0.84 
WW 
317.8* 
5.2* 
0.28 
391.5* 
4.7* 
0.15 
W5 
257.5* 
3.58 
0.22 
285.5* 
6.5* 
0.17 
W6 
49.3* 
1.82 
1.04 
183.4* 
6.7* 
0.22 
W7 
179.6* 
0.00 
0.22 
163.0* 
4.7* 
0.22 
W8 
161.6* 
0.04 
0.62 
83.3* 
8.1* 
2.12 
W9 
112.1* 
0.00 
0.54 
128.0* 
7.9* 
0.14 
W10 
57.0* 
0.06 
0.70 
61.3* 
8.4* 
0.38 
W11 
26.0* 
0.00 
0.46 
84.2* 
4.9* 
1.44 
ADG 
3.8* 
0.00 
0.08 
30.8* 
1.64 
0.06 
WS 
50.0* 
0.10 
0.58 
49.5* 
4.7* 
2.58 
CW 
54.32* 
0.02 
0.66 
48.6* 
8.1* 
0.42 
VW 
1471.2* 
0.00 
0.52 
26.5* 
4.6* 
0.86 
HEAD 
15.1* 
0.00 
0.18 
15.4* 
0.00 
0.16 
SKIN 
40.1* 
0.00 
0.70 
53.9* 
2.44 
0.56 
CY 
13.9* 
4.1* 
0.10 
14.0* 
1.48 
0.72 
* Stat5istically significant at P<0.05 
For both breeds, the common environmental effects of litters strongly affected the estimation of (co)variance components for growth and slaughter traits, but very few reproductive traits were affected by permanent environmental effects of does. That can be explained by the age of animals when measurements were made. Rabbits are animals that grow very fast and there is an interval of only around 50 days between weaning and slaughter, and in this interval the effects of litters still exist. These results show that is very important to include permanent environmental effects when (co)variances are estimated with rabbit data.
When comparing Model 3 with Model 2 (the latter includes maternal genetic effects), only LSB and LSBA for reproductive traits, and WW and CY were influenced by this source of variation in Californian rabbits. However, no reproductive traits and all growth traits (except ADG) and WS, CW, VW were affected by genetic maternal effects in New Zealand White rabbits. This is very important, because the inclusion of genetic maternal effects sometimes is confounded with permanent environmental effects of does and hard to separate or explain. In Californian rabbits, maternal effects can be excluded from the animal models in the majority of traits. The differences between Models 4 and 3 (includes the correlation between direct and maternal genetic effects) were not detected in any one of the 23 traits analyzed in both Californian and New Zealand White rabbits, which is also an important finding so as to avoid wasting CPU time.
Table 3 presents the values of total heritability, which considers heritabilities for direct and maternal effects and also their correlation. These values are very important in the choice of "best" models, because if no statistical difference has been detected by the likelihood test, but total heritability changes in what can be considered a "practical" difference, a more complex model can be considered "better" than the one detected by the statistical test.
Table 3: Total heritabilities for 23 reproductive, growth and slaughter traits of Californian and New Zealand White rabbits, estimated by four different animal models. 

Californian Models 
New Zealand White Models 

Trait 
1 
2 
3 
4 
1 
2 
3 
4 

LSB 
.222 
.077 
.098 
.072 
.240 
.083 
.027 
.033 

LSBA 
.303 
.110 
.120 
.094 
.207 
.098 
.036 
.052 

LS21 
.148 
.000 
.000 
.013 
.197 
.119 
.118 
.115 

LSW 
.146 
.000 
.002 
.023 
.226 
.100 
.100 
.095 

LWB 
.124 
.057 
.058 
.059 
.129 
.000 
.010 
.002 

LW21 
.311 
.130 
.082 
.095 
.167 
.080 
.079 
.082 

LWW 
.103 
.083 
.079 
.048 
.293 
.126 
.126 
.145 

MORT 
.181 
.000 
.014 
.000 
.190 
.001 
.000 
.039 

WW 
.608 
.030 
.093 
.136 
.554 
.117 
.109 
.116 

W5 
.583 
.090 
.110 
.155 
.532 
.076 
.068 
.090 

W6 
.324 
.000 
.033 
.045 
.516 
.170 
.134 
.158 

W7 
.558 
.209 
.209 
.249 
.533 
.170 
.137 
.153 

W8 
.554 
.194 
.187 
.212 
.434 
.133 
.116 
.170 

W9 
.467 
.187 
.186 
.203 
.516 
.203 
.164 
.184 

W10 
.398 
.241 
.236 
.244 
.516 
.309 
.220 
.243 

W11 
.412 
.264 
.264 
.275 
.541 
.261 
.212 
.245 

ADG 
.439 
.358 
.357 
.365 
.573 
.346 
.312 
.316 

WS 
.723 
.177 
.181 
.129 
.817 
.215 
.092 
.141 

CW 
.712 
.178 
.177 
.129 
.761 
.152 
.049 
.096 

VW 
.240 
.033 
.035 
.029 
.459 
.111 
.044 
.096 

HEAD 
.473 
.159 
.159 
.148 
.568 
.225 
.225 
.270 

SKIN 
.770 
.309 
.311 
.321 
.747 
.088 
.036 
.075 

CY 
.344 
.152 
.055 
.050 
.118 
.000 
.014 
.000 

The analysis of this table explains why a model other than the one chosen by the likelihood test was considered the "best" model. This is shown in Table 4, which presents the models of choice to be used in (co)variance components in the population studied. From the joint analysis of Tables 2 and 3, it can clearly be seen that maternal effect is more important in New Zealand White rabbits than in Californian rabbits, at least in the sample analyzed. This can be a useful indicator for further studies, carried out with other populations, and other breeds in a different environment.
Table 4. Animal models of choice for estimation of (co)variance components estimation in 23 traits of Californian and New Zealand White rabbits raised in southeastern Brazil. 

Trait 
Californian 
New Zealand White 
LSB 
2 
3 
LSBA 
2 
4 
LS21 
2 
2 
LSW 
2 
2 
LWB 
2 
2 
LW21 
2 
2 
LWW 
2 
2 
MORT 
2 
2 
WW 
4 
3 
W5 
3 
4 
W6 
3 
4 
W7 
2 
4 
W8 
2 
4 
W9 
3 
3 
W10 
3 
3 
W11 
3 
3 
ADG 
2 
2 
WS 
2 
4 
CW 
2 
4 
VW 
2 
4 
HEAD 
2 
2 
SKIN 
2 
2 
CY 
3 
3 
Model 1, that considers only the direct effect of the animal should not be used for any trait, because permanent effects of does were very important for all of them. Model 2, that takes into account the direct genetic effect of the animal plus the common effects of litter, is the model of choice for the majority of traits for Californian rabbits. It shows that maternal effects are not very important for animals of that breed, except for growth traits. However, for New Zealand White rabbits, Models 3 and 4 seem to be the most adequate for almost all traits, except for the majority of reproductive traits.
When comparing the models of choice in relation to the type of trait, it can be seem that the majority of reproductive traits are not affected by maternal effects, but were affected by permanent environmental effects, therefore Model 2 can be used. For all growth and slaughter traits, permanent environmental effects are very important for both breeds and, depending on the breed and the importance of maternal effects, Models 2, 3 or even 4 should be used. The complete model, Model 4 was not statistically different from Model 3 and should be used only if the correlation between direct and maternal effects is supposed to be important, as CPU time increases with this choice. That was true for several traits in New Zealand White rabbits. There is a variation in total heritability values from Model 3 to Model 4, but they are not very large. If CPU time is not a problem, Model 4 is preferable to Model 3, as the results are slightly closer to "real" parameters, and it’s likelihood function value is higher, even though not statistically different.
The data presented in this paper identify not only the models of choice to be used in analysis of reproductive, growth and slaughter traits of Californian and New Zealand White rabbits, but also showed the influence of permanent environmental and maternal effects on the estimation of (co)variance components of rabbits. Values of total heritabilities for the traits are shown and their variation among models was important to define the most adequate animal models. The results shown here have to be verified in other populations of rabbits of the same breeds, raised in other environments.
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