### INTRODUCTION

### MATERIAL AND METHODS

*y*

*is the n*

_{ijkln}^{th}observation, recorded on the lactation day (t) of the animal (l) from genetic group (k), corresponding to the calving season (j), and control class of HYM (i);

*HYM*

*the fixed effect of the n*

_{i}^{th}control class of HYM (i = 1,…, 3070);

*EC*

*the fixed effect of the j*

_{j}^{th}calving season (j = 1 and 2);

*GG*

*the fixed effect of the k*

_{k}^{th}cow genetic group (k = 1,…,6);

*b*

*the f*

_{f}^{th}regression coefficient for linear (

*f*= 1) and quadratic (

*f*= 2) effects of the cows’ age

*x*

*at calving, in months;*

_{l}*β*

*are the r*

_{r}^{th}fixed regression coefficients, specific for modeling the average lactation curve of the population,

*α*

*and*

_{lr}*γ*

*are the r*

_{lr}^{th}random regression coefficients that describe the trajectory of the additive genetic effects and permanent environmental effects of the l

^{th}cow, respectively;

*ψ*

*(*

_{r}*t*) is the covariate of regression function according to lactation day (t);

*K*

*is the adjustment order for the fixed regression coefficients corresponding to the average lactation curve of the population (*

_{o}*K*

*= 3);*

_{o}*K*

*and*

_{a}*K*

*are the adjustment orders of the Legendre’s polynomial for the genetic additive random effects and permanent environmental effects, respectively, which ranged from 3 to 5, and*

_{pe}*ɛ*

*is the residual random effect associated with*

_{ijkln}*y*

*.*

_{ijkln}*K*

*,*

_{a}*K*

*, where*

_{pe}*K*

*and*

_{a}*K*

*represent the orders of Legendre’s polynomials adjusted for additive genetic effects and permanent environmental effects, respectively. For example, the model Leg3,4 denotes an analysis that adjusts Legendre’s polynomials of third and fourth order to the additive genetic effects and permanent environmental effects, respectively.*

_{pe}*C*

_{1}(∑) (Bozdogan, 2000); relative percentage of complexity reduction (RPCR) = [

*C*

_{1}(∑

*)−*

_{k}*C*

_{1}(∑

*)/*

_{kR}*C*

_{1}(∑

*)]×100 (Bozdogan, 2000), where p is the number of model parameters; n is the total number of observations; TDIM is the total number of days in milk;*

_{k}*f*o total number of residue classes; NDIM number of days in milk existing in the l

^{th}class; and ∑

*and ∑*

_{k}*are the matrices of (co)variance and correlation of the model parameters;*

_{kR}*C*

_{1}(∑) = rank(∑) Log[trace(∑)/rank(∑)]−Log(|∑|). Lower values associated to these criteria indicate better model fit, except for RPCR where higher percentages indicate better fit. The (co)variance components were estimated by restricted maximum likelihood method using the Wombat program (Meyer, 2007).

_{i}) (Table 2) and milk yield up to 305 DIM, as well as the genetic and permanent environmental correlations between these persistency measures and between these measures and 305MY. Lower values of persistency measures indicate greater lactation persistency, except for PS

_{5}, where higher values indicate greater lactation persistency.

*i)*on any day (

*t)*in milk, for example, to the RRM Leg3,3, Leg3,4 and Leg3,5 (

*K*

*= 3) can be obtained as follows:*

_{a}*i*), and

*t*), then, the estimated breeding value (EBV) of the animal (

*i*) on day (

*t*) is:

_{305}) of the animal (

*i*) is achieved by:

##### (3)

*Z*

_{c305MYg}is the vector sum of the coefficients of the Legendre’s polynomials, specific to milk yield over the full lactation.

*y*

*is the mean EBV of the d*

_{d}^{th}year of birth;

*X*

*is the d*

_{d}^{th}year of birth; and

*b*

_{0}and

*b*

_{1}, are respectively, the intercept and the linear regression coefficient (genetic tendency).

### RESULTS AND DISCUSSION

*.*, 2009). In general, the comparison criteria used in this study indicated that Leg3,5 and Leg5,5 models showed the best fit to the data. On analysis, it was observed that the estimates of the residual variance and genetic variance of both models were similar (Figure 3a and 3b), indicating that increasing the order of the Legendre’s polynomial for modeling additive genetic effect causes little distinction in heritability estimates between the models (Figure 3c). There was also a greater variation in these estimates between the DIM throughout the lactation period when used, in the Leg5,5 model, the Legendre’s polynomial of order 5 to modeling this additive genetic effect, suggesting that it is a less robust model. Moreover, the complexity of the Leg5,5 model was higher than that of Leg3,5 model (Table 3). Therefore, it is recommended the Leg3,5 model to describe the changes in the (co)variance components in milk yield during lactation of Girolando cows.

*.*(2000) and Bignardi et al. (2009) also reported that the additive genetic effect modeling with lower order Legendre’s polynomials for the permanent environmental effect are sufficient to capture most of the genetic and permanent environmental variability observed in the shape of lactation curve. However, other authors have reported better fits with Legendre’s polynomials of the same order for both random effects (Cobuci et al., 2006).

^{2}= 0.23). From then, gradually decreased until the end of lactation with values of 0.18 on day 305 (Figure 4). Lower values observed in early lactation can be attributed to the higher permanent environmental variance and to the lower additive genetic variance in this period. Similar trend was observed by Dorneles et al. (2009a) for Holsteins in Brazil, with heritability estimates ranging from 0.14 to 0.20. In general, genetic correlations were higher between adjacent DIM and lower among the more distant ones (Table 5). DIM in mid lactation were highly correlated, with correlations greater or equal to 0.90 between DIM on day 155 with DIM from day 105 to 255 (Table 5 and Figure 5a). The permanent environmental correlations were also high between adjacent DIM, decreasing between the farthest ones (Figure 5b). It can also be observe low and negative genetic correlation between the extreme controls (5 and 305). Negative genetic correlations between extreme controls during lactation period have been reported in the literature (Lopez-Romero and Carabaño, 2003; Costa et al., 2005; Bignardi et al., 2009; Pereira et al., 2010).

*Bos indicus*animals.

_{3}and PS

_{7}measures, and lower in the PS

_{4}, PS

_{5}, PS

_{6}, and PS

_{8}measures (Table 6), suggesting that there is an important contribution from the genetic component to this trait, which could be included among the criteria for selection in Girolando breed. Differences between heritability estimates of persistency measures may be associated to the period (phase) of lactation used in the calculation for persistency (Madsen, 1975). Cobuci et al. (2012), when working on data from Holstein using fourth-order polynomials, reported heritability estimate of 0.33 for PS

_{3}, equal to the obtained in the present study. In general, the values obtained in this study were higher than those reported by Pereira et al. (2012) for Gir, and Biassus et al. (2010) and Dorneles et al. (2009b) for the Holstein breed in Brazil, which ranged from 0.04 to 0.32 and were obtained with RRM in third and fourth-order Legendre’s polynomials.

_{5}with PS

_{2}and PS

_{4,}and between PS

_{2}and PS

_{6}(Table 6). It can also be observed negative estimates between PS

_{5}and the other measures, probably due to the definition of PS

_{5}, since higher values of this measure indicate greater persistency, unlike the other measures. In general, permanent environmental correlations between these measures were also high, except the correlations among PS

_{2}and PS

_{1}, PS

_{6}, PS

_{7}or PS

_{8}; PS

_{6}and PS

_{4}or PS

_{9}; and PS

_{5}and PS

_{9}, which were low. In general, values from −0.92 to 1.00 among persistency measurements obtained in this study were similar to those reported by Pereira et al. (2012) for Gir (0.86 to 0.99), by Dorneles et al. (2009b) for Holstein (−0.87 to 0.94) and by Freitas et al. (2010) for Guzerat cattle (−0.99 to 0.97) in Brazil.

_{1}, PS

_{7}, or PS

_{8}and 305MY were close to zero, and close to those reported by Kistemaker (2003), who estimated genetic correlation of 0.06 between PS

_{7}and 305MY. However, Dorneles et al. (2009b), Cobuci et al. (2012) and Khorshidie et al. (2012) studying the Holstein breed, and Jakobsen et al. (2002), studying the Danish breed, obtained values close to zero between PS

_{5}and 305MY. These values confirm that there is a little genetic association between milk yield and persistency, indicating that animals having the same level of milk yield may present different levels of lactation persistency. Therefore, animals with greater EBV for persistency are not exactly those with greater EBV for 305MY (Cobuci et al., 2007). This fact can be confirmed when comparing the EBV for measures of persistency and 305MY from the top five sires with more than 25 daughters (Table 7), in which the fourth best sire (S4) classified according to its EBV to 305MY is the first one to all measures of persistency.

_{7}presented such characteristics therefore can be recommended for evaluation of lactation persistency in Girolando cattle.

_{7}and 305MY is much lower at high intensities (<10%), however, as expected, this percentage increased as the proportion of selected animals was higher (Figure 6). It was also noted that this value is higher in cows than in sires, but when the sire selection is restricted by the number of daughters, this percentage becomes much higher. Similar trends were observed by Cobuci et al. (2007) for Holsteins and by Pereira et al. (2012) for Gir.

_{7}). The EBV from sires S1 to S4 were higher than those from S5 and showed an increasing trend throughout lactation. Nevertheless, it is important to note that these five sires had different ratings for both traits (305MY and PS

_{7}).

_{7}(Table 8). In general, the results revealed a significant genetic progress for 305MY in the Girolando population during the study period, a trend which was more pronounced in sires than in cows, with 31.73% increase over the study period. These results indicate efficiency of the breeding program for Girolando cattle.

_{7}was close to zero (0.02, Table 6), confirming that the selection for 305MY does not lead to improvement in lactation persistency. Similar trends were observed by Cobuci et al. (2007) for Holsteins and by Pereira et al. (2012) for Gir. Based on these results, an alternative for the improvement of both traits implies in using selection indexes that combine lactation persistency and 305MY and enabling the simultaneous selection of the most productive and persistent animals for milk yield (Chaves, 2009). The study revealed that selection for increased to milk yield up to 305 days did not cause genetic progress for lactation persistency of the animals belonging to the Genetic Improvement Program for Girolando cattle. The use of the measure persistency PS

_{7}proposed by Kistemaker (2003), under the RRM using 3 and 5-order Legendre’s polynomials for additive genetic effects and permanent environmental effects, respectively, would be the most suitable option for use in genetic evaluation for milk yield and lactation persistency of the breed. The use of selection indexes for simultaneous evaluation of both genetic traits can be a viable alternative. However these are not the only trait to be considered in genetic evaluation, mainly in production systems of tropical regions.