### INTRODUCTION

*Bos indicus*breeds capable of withstanding heat stress, diseases, low feed supply and low management level with

*Bos taurus*dairy breeds of high genetic ability for milk yield. This mating scheme has resulted in a variety of crossbred groups with different levels of additive and non-additive genetic abilities.

### MATERIALS AND METHODS

### Description of the study area, animals and data

*Bos indicus*) used as a foundation stock for crossbreeding were bought from farmers in the Southern and Western Ethiopia, respectively. The herd from Debre Zeit research center was B and their crosses with Friesian (F).

### Lactation curve function and parameters

*=*

_{t}*at*

^{b}*e*

*where*

^{−ct}*t*is the length of time since calving. The MIG assumes the parameter

*b*of the IG function to be equal to one, i.e., y

*=*

_{t}*ate*

*. Here, the log-transformed linear form of MIG ((*

^{−ct}*ln(y*

_{t}*/t)*=

*ln(a)*+

*(−ct)*) was used to fit to the monthly test-day milk data using the regression procedure (SAS, 2003). The y

*is milk yield (kg) at time*

_{t}*t*(d) after calving, and

*a*,

*b*and

*c*are parameters of the functions. The parameter

*a*is a scaling factor associated with the average yield,

*b*is related to pre-peak curvature and

*c*is related to post-peak curvature. Parameters of the MIG function were obtained as an output from the regression analysis fitting the MIG function to each lactation of each cow. Cows with lactation lengths shorter than 90 d (three monthly test-day records) were excluded from the analysis and longer lactations were truncated to 305-d lactation length. A total of 59,413 monthly test-day milk records were used to estimate the parameters of the MIG function.

### Data analysis

*Y*= Observation on_{ijkl}*l*th cow that calved in*i*th herd-year-season of calving,*j*th parity,*k*th breed group subclasses,*μ*= Overall mean,*HYS*=_{i}*i*th calving herd-year-season subclasses (*i*= 1 to 325),*P*=_{j}*j*th parity (*j*= 1, 2, 3, 4, 5, 6 and ≥7),*BG*_{k}*= k*th breed group (*k*= 1 to 23; Table 1), and*e*= residual associated with y_{ijkl}._{ijkl}

*b LL*

*, where*

_{ijkl}*b*= regression coefficient of LY on LL, and

*LL*

*= lactation length of the*

_{ijkl}*l*th cow that calved in

*i*th herd-yr-season of calving,

*j*th parity, and from

*k*th breed group. The LSM for breed groups and parities were compared using Bonferroni t-tests (SAS, 2003).

### Estimation of variance components and genetic parameters

*y*= the vector of observations (i.e., LY, IY, PY, YD, DP, LL, ln(a) and c),*b*= vector of fixed effects of herd-year-season and parity subclasses, and LL covariate for LY only,*g*= vector of fixed breed effects,*a*= vector of random animal additive genetic effects,*pe =*vector of random permanent environmental effects,*e*= vector of random residual effects,*X*= incidence matrices relating records to fixed herd-year-season and parity, and LL covariate for LY only,*Q*= matrix relating observations to breed effect (through H, F, J, and S breed fractions) and general heterosis (through heterozygosity fraction),*Z*= incidence matrices relating records to animal additive genetic effects, and*W*= incidence matrices relating records to permanent environmental effects.

*y*to be Xb+ Qg. The vector of direct animal additive genetic effects was assumed to have a normal distribution with mean zero and variance

*A*is the additive numerator relationship matrix among animals in the population, and

*I*is the identity matrix, and

*y*

*is vector of observation for the*

_{i}*ith*trait;

*b*

*is vector of fixed effects (i.e., HYS, parity, and LL covariate for LY only), g*

_{i}_{i}is vector of cow breed and general heterosis effects for the

*ith*trait;

*a*is vector of animal direct genetic effects for the

_{i}*ith*trait;

*X*,

_{i}*Z*, and

_{i}*W*are incidence matrices that relate fixed (herd-year-season and parity subclasses and covariate), animal direct genetic and permanent environmental effects to observations for the

_{i}*i*th trait, respectively and Q

_{i}is matrix relating observations to breed effect (through H, F, J, and S breed fractions) and general heterosis (through heterozygosity fraction). The model assumed the expected value for trait

*i*(

*y*) to be

_{i}*X*+

_{i}b_{i}*Q*. The vectors of direct animal additive genetic effects, permanent environment and residual effects for each trait were assumed to be normally distributed with mean zero. The variance-covariance structure of the random effects for a bivariate animal repeatability model could be described as:

_{i}g_{i}_{a1a2}, σ

_{pe1pe2}and σ

_{e1e2}are additive, permanent environment and residual covariances between traits 1 and 2, A is the numerator relationship matrix, and I is an identity matrix.

### Estimation of breeding values and sire rank correlations

*û*is the EBV of an individual for a particular trait,

*ĝ*is a vector of GLS estimates of differences between breeds H, F, J, and S and breed B,

*p*' is the transpose of the vector of fractions of H, F, J, and S breeds in an individual, and

*â*is the predicted random animal additive genetic value (i.e., from

*p*'

*ĝ*; Elzo and Famula, 1985; Arnold et al., 1992; Koonawootrittriron et al., 2002). The ranks of the sires EBV for each trait were tested using Spearman’s Rank correlations for significant association between the ranks of the sires for pairs of traits using the CORR procedure of the SAS (SAS, 2003). Genetic trends across years were evaluated using a regression analysis of mean yearly EBV for each trait calculated from solutions for each herd-year-season contemporary group and regressed on year. The year data collection started, i.e., 1977, was considered as the base year.