### INTRODUCTION

### MATERIALS AND METHODS

### Data

### Models

*is the fixed test-year–month effect*

_{i}*i*; b

*is the*

_{jm}*m*th fixed regression coefficient specific to parity

*j*(one level for the first lactation, and four levels each for the second through fifth lactations); u

*and p*

_{km}*are mth RR coefficients specific to cow*

_{km}*k*for AG and PE effects, respectively; w(t

*)*

_{kl}*is a covariate associated with DIM*

_{m}*t*

*for TD record*

_{kl}*l*of cow

*k*; and e

*is a random residual effect associated with Y1. The covariates of the fixed regression coefficient for parity effect are fifth-order Legendre polynomials [14], with the exponential term of the Wilmink function (e*

_{ijkl}^{−0.05t}) as a sixth-order covariate [15,16]. The covariates of the RR coefficients for AG and PE effects are second-order Legendre polynomials [17,18] in accordance with the official genetic evaluation model for production traits in Japan [19]. Generally, herd effect for TD records was included in the RR-TD model. Because only the records of cows with extremely long lactations (i.e., more than 451 days) were used for analysis, it was difficult to make a contemporary group of each herd and to reliably estimate herd effects and AG effects simultaneously in our preliminary study. Therefore, we did not account for herd effect in the model. The mean square errors that we obtained (Figure 2) were similar to the residual variances reported by Bohmanova et al [5] which account for herd effect. Therefore, we consider that the estimation accuracies of the models in our current study are similar to those in another study [5] that accounted for herd effect.

*p*) and random (

*q*) regressions in Model_M2; the covariates of fixed and RR are second- and first-order Legendre polynomials (F2R1) and third- and second-order Legendre polynomials (F3R2), respectively.

**G**and

**P**are AG and PE (co)variance square matrices, respectively, of RR coefficients; ⊗ is the Kronecker product;

**A**is the AG relationship for animals;

**R**is the identity matrix for cows; and

**R**is a diagonal matrix of residual variance for each record. The DMU program [20] was used for REML to estimate the variance components and obtain the solutions of the regression coefficients for AG and PE effect. Mean square errors (

### Predicting RR coefficients and daily milk yields in M2 from those in M1

*+p*

_{m}*of Model_M2). We set two regression equations: the equation of combined RR coefficients of M2 on the RR coefficients for AG effect of M1 (REG_1) and that on the coefficients for AG and PE effects of M1 (REG_2). Multiple regression analysis was performed by using the REG procedure of the SAS software package [21]. The predictive values for daily milk yields of several DIM in M2 were calculated by using the combined RR coefficients predicted from REG_1 and REG_2, and the correlation coefficients between these values and those obtained by using the combined RR coefficients of Model_M2 were calculated. In addition, the predicted daily milk yields in M2 were calculated by using these combined RR coefficients and the solutions of fixed test-year–month effects and fixed regression coefficients. The correlation coefficients between these values and TD milk yields were calculated for every 15 successive days of M2.*

_{m}### RESULTS AND DISCUSSION

### Comparison of errors among different models

### Relationships between RR coefficients in M1 and M2

### Predicting RR coefficients and daily milk yields in M2 from those in M1

*R*

^{2}) for REG_1 and REG_2 of the combined intercepts of M2 were much higher than those for the combined first-order coefficients of M2 (Table 3). The differences of

*R*

^{2}between REG_1 and REG_2 for the same objective variables were small; for the intercept and first-order coefficient, these values were 0.022 and 0.011, respectively, in the first lactation and 0.046 and 0.024 in later lactations. The small differences in

*R*

^{2}arose from the very weak correlations between all of the RR coefficients for the PE effect of M1 and those for the AG and PE effects of M2 (Table 2). Therefore, we considered that the RR coefficients for the PE effect of M1 had little effect as explanatory variables in the multiple-regression equation for predicting the RR coefficients of M2. The standardized partial regression coefficients for the intercept for the AG effect of M1 were much larger than those for the other explanatory variables in the equation of the combined intercepts of M2 (Table 4). This result indicates that the intercepts for the AG effect of M1 have a large effect in the equations that predict the RR coefficients of M2.

*R*

^{2}for multiple regression of the RR coefficients after 305 DIM on those within the first 305 DIM were moderate to high when the intercepts after 305 DIM were the objective variables. The predictive values for daily milk yield after 305 DIM that were obtained by using the random-regression coefficients for AG effect within the first 305 DIM were highly correlated with those calculated by using the coefficients after 305 DIM. These results suggest that milk production after 305 DIM can be predicted by using the random-regression coefficient estimates for AG effect within the first 305 DIM. Combining these random-regression coefficients with the fixed regression coefficients of the lactation curve related to an individual cow’s environment (e.g., herd or calving season) will provide an estimate of the total milk yield during that cow’s lactation. Predicting the milk yield after 305 DIM or the total lactation yield for individual cows will facilitate the timing of insemination or drying off to optimize individual lactation periods.