### INTRODUCTION

### MATERIAL AND METHODS

*is the cumulative milk yield at 305 days of animal*

_{ij}*j*on herd-year-season of calving

*i*, HYS

*is the fixed effect of herd-year-season*

_{i}*i*, b

_{n}is the linear covariable for 305-day yield as a function of age at calving, x

*is the age of cow at calving, in months; a*

_{ij}*is the additive genetic effect of animal*

_{ij}*j*on herd-year-season of calving

*i*, e

*is the residual effect.*

_{ij}_{ijkl}is the i-th test day record of cow j made on day in milk t within herd-year-month of test (HYM) subclass l; β

_{mk}are k

^{th}fixed regression coefficient specific for the m

^{th}subclass of calving age-season classes; HYM

_{l}= fixed effect herd-year-month of testing; u

_{jk}and pe

_{jk}are the k

^{th}random regression coefficients that describe, respectively, the additive genetic effects and the permanent environmental effects on cow j; ∅︀

_{k}(DIM

_{t}) are the LP for the test day record of cow j made on day in milk t, in which k is the n

^{th}parameter of coefficient of LP of 4

^{th}and 5

^{th}orders; and e

_{ijkl}is the random residual. The RRM referring to the fourth and fifth orders LP were designated as RRM

_{4}and RRM

_{5}, respectively. Many studies in literature have pointed out these orders as recommended as well as they have already been used in Canada, Italy and United Kingdom for genetic evaluations (Muir et al., 2007).

**G**and

**P**are covariance matrices of the random regression coefficients,

**R**=

**I**σ

^{2}

_{e}is a diagonal matrix (residual) and ⊗ is a Kronecker product between matrices.

_{min}and DIM

_{max}are minimum and maximum values for the days in milk (DIM).

*t*-th standardized days in milk (DIM

^{*}

_{t}), the

*k*-th polynomials is given as follows:

*k*/2 = (

*k*−1)/2 if

*k*is odd and

*m*is an index number needed to determine the

*k*-th polynomial.

*i*for test day

*t*was calculated by:

*α̂*

*is a (k*

_{i}_{a}×1) vector of the estimates of additive genetic random regression coefficients specific to the animal

*i*, and

**z**

_{t}is a (k

_{a}×1) vector of LP coefficients evaluated at day

*t*. An example for fifth order polynomial was presented as follow:

**z′**

_{305}for fifth-order LP used under study was as follows:

*i*was obtained by summing the EBVs from day 6 to 305 days in milk, which was illustrated for example for a fifth order LP as follows:

^{2}

_{a}was the additive genetic variance for the trait and r

^{2}is the correlation between the true breeding value and estimated breeding values (Misztal and Wiggans, 1988).

_{0}) implied that restricted likelihood functions of the models did not differ when the number of parameters increased. The calculated value of LRT was compared to the chi-square table (

*x*

^{2}) with ten degrees of freedom with level of significance set at 5%.

### RESULTS

_{4}and lower RV for RRM

_{5}. The difference in the −2LogL between RRM

_{4}and RRM

_{5}tested by LRT was significant (p<0.05) by the chi-square statistic. Thus the null hypothesis of equality of RRM was rejected.

_{4}and RRM

_{5}were equal 0.23 and 0.24 and slightly higher than 0.21, the value obtained for 305MY by LM (Table 2). Additive genetic variances from RRM

_{4}(402,908.3 kg

^{2}) and RRM

_{5}(400,119.7 kg

^{2}) were higher than the value from LM (311,000 kg

^{2}). The estimates of heritability, additive genetic and permanent environmental variances of test day yields from 6 to 305 days in milk ranged from 0.16 to 0.27, from 3.76 to 6.88 kg

^{2}and from 11.12 to 20.21 kg

^{2}, respectively (Figure 1).

_{4}and RRM

_{5}, respectively (Figure 2). High additive genetic correlations were observed between adjacent test day milk yields and were close to 1 mainly during mid-lactation, but decreased with the increasing of distance between test days.

_{4}and RRM

_{5}models increased from 11% to 31%, when progeny sizes decreased from 200 to 399 to 10 to 24 compared to those from LM (Table 3). For cows, differences between standard deviations of 305MY EBVs from RRM

_{4}and RRM

_{5}, and those from LM ranged from 26% to 31%, depending on number of test days.

_{4}and RRM

_{5}models for bulls increased from 0.86 to 0.95 with the increase in bulls’ progeny size (Table 4). Rank correlations were higher than 0.80 for cows and, in general, increased from 0.83 to 0.87 when the number of test days increased from 6 to 10.

_{4}and RRM

_{5}were equal to 0.87 and 0.86 for all cows and 0.89 (for both RRM) for all bulls (Table 5). These correlation estimates decreased from 0.89 to 0.69 when proportion of selection of top bulls was 10% and decreased to 0.85 when proportion of selection was 1%. When the proportion of selection decreased from 60% to 10%, the rank correlation decreased from 0.78 to 0.57.

_{4}(Figure 3). EBVs of test days for bull S1 estimated from RRM were higher than those estimated for bull S3 mainly between 6 and 60 and between 150 and 305 days in milk. Although S5 showed a flatter trajectory of EBVs on days in milk with higher EBVs between 6 and 120 days in milk, S4 presented higher test-day EBVs between 120 and 305 days in milk compared to S5.

_{4}and RRM

_{5}models was 8% lower than the number of bulls evaluated by LM models in the class 0.30 to 0.39 of reliability of EBVs (Table 7). There was from 9% to 17% more bulls in the classes between 0.40 to 0.49 and 0.70 to 079 when RRM models were compared to LM models. The classes of reliability above 0.80 to 0.89 presented between 58% and 136% more bulls when the number of bulls estimated by RRM models was compared to the number of bulls by LM model.

_{4}and RRM

_{5}models increased in average from 4% to 17% with the decrease in bulls’ progeny size compared to the average of reliabilities estimated by LM, whose values ranged between 0.41 and 0.89 (Table 8). The gain of reliability in parentheses ranged from 8% to 33% to 1% to 13% with the decrease in progeny size. For cows, the average gain in reliability was between 23% and 24% for every class of test day. Moreover, the gain in reliability of cows of each class ranged from 11% to 49% for cows with 6 test day records to 0% to 102% for cows with 10 records.

### DISCUSSION

_{4}and RRM

_{5}models, AIC, BIC, −2LogL and LRT indicated RRM

_{4}as the best fit of lactation curve (Table 1). In literature, models with higher orders of LP were indicated as the best fit according to AIC, BIC, −2LogL and residual values (Biassus et al., 2010; Aliloo et al., 2014). Although RV was lower for RRM

_{5}, increasing the order of polynomials did not affect breeding values and their reliabilities as well as the estimates of genetic parameters in this study.

_{4}and RRM

_{5}and slightly higher than 0.21 obtained by the LM model but the all values showed the same magnitude (Table 2). Additive genetic variances for 305MY estimated by RRM

_{4}and RRM

_{5}were about 29% higher than that in LM model. The residual value decreased about 10% when models were fitted by fourth (RRM

_{4}) and fifth (RRM

_{5}) order LP. Similarly, Biassus et al. (2010) compared models fitted by LP from third to sixth orders, which showed that the differences between RV of models decreased from 14% to 5% when the polynomial orders increased. Çankaya et al. (2014) compared models fitted from second to fourth orders and the differences between RV of models decreased from 24% to 10%. Residual variance values presented by Takma and Akbas (2009) decreased from 30% to 7% when adjusted models from second to sixth orders.

_{4}and RRM

_{5}were higher than that estimated from LM (Table 3). When bulls’ progeny size decreased from higher classes of progeny size (200 to 299) to lower classes (10 to 24), the distribution of EBVs increased from 11% to 30% higher than that in the LM, which indicate that bulls with less information presented larger changes in distribution of EBVs around the mean promoted by RRM. For cows, the change in the standard deviation of EBVs was around 28% higher in RRM models compared to LM, considering 6, 7, 8, 9, or 10 test days by lactation. Melo et al. (2007), using data from Brazilian Holsteins, found that standard deviations of EBVs from RRM was 22% higher compared to the standard deviations from 305-day LM for cows and the differences were up to 3% for bulls with progeny size higher than 49 and up to 22% for bulls with lower progeny size. Lidauer et al. (2003) reported an increase about 9% for young bulls with at least 20 progenies and about 3% for active bulls with 60 progenies of Finnish dairy cattle. Therefore the increase in the standard deviations of mean values of EBVs suggested that estimates from RRM changed the distribution of values of EBVs of bulls and cows and consequently changed the ranking of top bulls and cows.

_{4}and RRM

_{5}models increased from 0.86 to 0.96 according to the increase of the progeny size classes, which indicate that the increase in the amount of information approximate the estimation of EBVs of LM and RRM models. On other hand, as the amount of information (progeny size) decreased, the differences in the ranks of EBVs were higher between models, which suggest, in this case, that the re-ranking of bulls was higher for bulls with less progeny size. For cows, there was a substantial difference in the correlations between RRM models and LM in general, but, according to the number of test days, these differences were similar for cows with 6, 7, 8, 9, or 10 test days. Thus these results confirm the assumption that RRM may change the ranking of top animals. These changes in ranking became more evident when a selection of bulls and cows was applied by RRM

_{4}and RRM

_{5}EBVs of cows and EBVs of bulls with progeny size higher than 49 (Table 5). The correlations decreased from 0.87 to 0.57 when 10% of cows were selected by RRM EBVs and increased as the proportion of selection decreased to 40% and 60% of cows. The selection of top bulls by RRM

_{4}and RRM

_{5}EBVs decreased the rank correlation from 0.89 to 0.70 (10% of bulls) and from 0.89 to 0.87 (1% of bulls). In order to illustrate the changes in the rankings of animals, the top ten bulls from the 10% of best bulls are shown (Table 6). Although the rank correlations between 10% of bulls were strong, the position of some bulls in relation to the other bulls may have large changes as observed for bulls S4, S8, and S9. The trajectory of test-day EBVs on days in milk may show important information to explain why the cumulative EBVs for 305MY from RRM models were higher compared to EBVs from LM, which may be observed in the trajectory of the best five bulls selected for EBVs at 305 days in Figure 3. For example, bull S3 was ranked as the first best bull by LM EBVs but it did not present the best initial EBVs compared to EBVs for S1 (6 to 60 days in milk) or the best final EBVs (150 to 305 days in milk), although the EBVs in mid-lactation (60 to 150 days) were equal. Therefore, the bull S1 was better than bull S3 in regard to the trajectories of test day EBVs because RRM models were able to estimate the test day EBVs (and cumulative EBVs for 305MY) with more precision. Consequently, the differences between the two bulls became more evident, which could not have been identified by LM.