### INTRODUCTION

*Bos indicus*) and Holstein Friesian as exotic (

*Bos taurus*) cattle in the year 1980. The level of inheritance of Friesian cattle in Karan Fries is 50% to 75% (Gurnani et al., 1986). For maintaining high level of milk production/productivity of Karan Fries cattle and their further improvement, it is necessary to execute proper program of genetic evaluation of males and females for selection of animals of high genetic merit. So, there is a need to estimate the genetic parameters of test-day milk yields using RRM for genetic evaluation of dairy cattle. Genetic parameters like heritability and genetic correlations among test-day milk yields have been estimated in different

*Bos taurus*breeds (Swalve, 1995; Rekaya et al., 1999; Kettunen et al., 2000; Faro et al., 2008; Elahi Torshizi et al., 2012). However, no literature is available regarding estimation of genetic parameters for monthly test-day milk yields (MTDMY) using RRM in Karan Fries cattle. The objective of this investigation was to estimate the genetic parameters for first lactation monthly test-day milk yields using RRM in Karan Fries cattle.

### MATERIALS AND METHODS

### Data structure

### Model

^{*}is Kronecker product function; I is identity matrix and R is diagonal matrix of homogenous residual variances.

### Function used in random regression model

*P*

_{0(x)}*= 1*, and

*P*

_{1(x)}*= x.*Then, in general, the

*n+1*polynomial is described by the following recursive equation:

_{i}days were standardized to the interval −1 to +1 with the following formula:

*t*

*and*

_{min}*t*

*were the earliest and latest age represented in data (Schaeffer, 2004). Legendre Polynomial was defined within the range of values from −1 to +1.*

_{max}### Estimation of genetic parameters and (co) variances

*G*is additive genetic variance covariance matrix. The permanent environmental variance of test-day milk yields was estimated as

*P*is permanent environmental variance covariance matrix. A mixed model analysis was carried out to obtain restricted maximum likelihood estimate of covariance components using WOMBAT software (University Of New England, Armidale, NSW, Australia) (Meyer, 2010).

^{th}TD milk yield;

^{th}TD milk yield. The genetic correlations among different MTDMY were calculated from the analysis of variance and covariance among test-day milk yields. The Variance components for different test day milk yields were also estimated using Harvey’s LSMLMW software (Ohio State University, Columbus, OH, USA) (Harvey, 1990) designed for complete mixed model analysis. In analysis the fixed effect of year and season of calving, age at calving and random effect of sires on phenotypes of test-day milk yield in cows in first lactation were considered. Heritability was estimated using paternal half sib method.

_{g}):

*r*

*is the genetic correlation between milk yields on TD i and j;*

_{gij}*σ*

*is the genetic covariance between milk yields on TD i and j;*

_{aij}^{th}TD yield and

*σ*

^{2}

*aj*is the additive genetic variance in j

^{th}TD yield.

### RESULTS AND DISCUSSION

### Random regression coefficients

_{0}, A

_{1}, A

_{2}, and A

_{3}) and five permanent environmental random regression coefficients (P

_{o}, P

_{1}, P

_{2}, P

_{3}, and P

_{4}). The estimated variances (a

_{i}, a

_{i}) and covariances (a

_{i}, a

_{j}) among the additive genetic and permanent environment random regression coefficients using LP in first lactation of Karan Fries cattle have depicted in Table 2 and 3. Eigen values represent the amount of variation explained by the corresponding eigen function (Kirkpatrick et al

*.*, 1990). First three eigen values 3.26 (86.89%), 0.43 (11.57%). and 0.06 (1.54%) of the additive genetic CF accounted for at least 99% of the sum of all eigen values but the first four eigenvalues 8.77 (64.61%), 2.99 (22.04%), 1.31 (9.64%), and 0.42 (3.08%) for permanent environment effect accounted 99% of total variation. It showed that the three main eigen values and associated eigen function explained the most of the additive genetic and permanent environmental effect variance and little variation was associated to other eigen values for additive genetic effects and permanent environment effects. The size of the first eigen values indicated that selection based on this would result in quick change in average milk yield. Similar result was reported by Elahi Torshizi et al. (2012) who reported the first three eigen values 14.11 (93.44%), 0.62 (4.10%), and 0.37 (2.45%) for additive genetic effect and first three eigen values 16.17 (93.43%), 0.65 (3.74%), and 0.36 (2.05%) for permanent environment effects using 3rd order of LP.

### Variances of test-day milk yields

_{A}), permanent environment (VE

_{p}) phenotypic variance and residual variances estimated for first lactation of Karan Fries cattle using LP are presented in Table 4. Perusal of the table showed that the highest V

_{A}was observed for the TD-6 (2.24 kg

^{2}) and the lowest was observed for TD-2 (1.30 kg

^{2}). In general, V

_{A}increased up to TD-6; thereafter a gradual decline was noticed till the end of lactation and became higher for TD-11. The residual variance (2.34 kg

^{2}) was observed higher than additive genetic variance (2.23 kg

^{2}). The VE

_{p}was observed higher for TD-1 (8.32 kg

^{2}) and lower (5.6 kg

^{2}) for TD-7. It was revealed from the results that the magnitude of VE

_{p}was higher in initial test-days (TD-1 to TD-3) and decreased thereafter with a slight increase at the end of lactation. The magnitude of VE

_{p}and V

_{A}seemed to have inverse relationship across the trajectory of lactation curve estimated by RR-TDM. The magnitude of VE

_{p}was higher in the beginning and end of lactation and relatively lower in the mid-lactation. Contrarily, the magnitude of V

_{A}was lower in the beginning and end of lactation and relatively higher in the mid-lactation. The residual variance was observed to be lower than the permanent environment variance for all the test-day milk yields. Geetha et al. (2007) reported that V

_{A}of test-day milk yields increased at the end of lactation for first lactation milk yield in Murrah buffaloes. Jamrozik et al. (1997) reported higher magnitude of V

_{A}and VE

_{p}over residual variances for first lactation test-day milk yield records of Holstein cows. It was revealed from Figure 1 that the magnitude of additive genetic and permanent environment variances were higher for test-day milk yields at both ends of lactation compared to test-day milk yields in mid lactation. Roose et al. (2004) reported that the additive genetic and permanent environmental variances for test-day milk yields were higher at both ends of lactation using LP function in second and third parity of dairy cattle in the Netherlands.

### Heritability estimates of test-day milk yields

### Genetic and permanent environment correlations between test-day milk yields

_{P}and V

_{A}across the trajectory of lactation curve estimated by RR-TDM. The VE

_{P}has a higher magnitude in the beginning and at the end of lactation and relatively lower value in the mid-lactation while the V

_{A}shows the reverse trend. Thus, RRM allows an animal to be evaluated on the basis of only single test-day or on any number of test-day records. However, because of technical complexity of RR-TDM, these are not much in use in India and proposed for the first time for the analysis of test-day milk yield records in Karan Fries cattle.