Multibreed Genetic Parameters and Predicted Genetic Values for First Lactation 305-d Milk Yield, Fat Yield, and Fat Percentage in a Bos taurus ´ Bos indicus Multibreed Dairy Population in Thailand[1]^/

Skorn Koonawootrittriron^*, Mauricio A. Elzo^2/, and Sornthep Tumwasorn^3/

Sakon Nakhon Agricultural Research and Training Center

Pungkone, Sakon Nakhon 47160, Thailand.

* Correspondence: Present address: Sakon Nakhon Agricultural Research and Training Center, P.O. Box 3,

Pungkone, Sakon Nakhon 47160, Thailand. E-mail: skorn@access.rit.ac.th

[1]^/ This research was supported by the Florida Agricultural Experiment Station and a grant from

the Thailand Research Fund under the Royal Golden Jubilee Project,

and approved for publication as Journal Series No. R-08419.

^2/ Department of Animal Sciences, University of Florida, Gainesville, FL 32611-0910, USA

Abstract

Estimates of multibreed covariance components, genetic parameters, and predicted genetic values for first lactation 305-d milk yield (MY), 305-d fat yield (FY), and 305-d fat percentage (FP) were obtained using production records from 481 purebred and crossbred cows in a Thai multibreed dairy population. Multibreed covariance components were estimated by restricted maximum likelihood procedures using a generalized expectation-maximization algorithm suitable for multibreed populations. Two multibreed sire-maternal grandsire models (BTBI and HO) that accounted for intrabreed additive genetic effects, and intralocus intra and interbreed nonadditive genetic effects were used for the analyses. The BTBI model considered Bos taurus (BT) and Bos indicus (BI) fractions, whereas the HO model accounted for Holstein (H) and Other breeds (O) fractions. Additive and nonadditive genetic covariance estimates for base population BI (or O) were larger than those for base population BT (or H) for all traits while base environmental covariance estimates for BI (or O) were smaller than BT (or H). Ranges of multibreed heritability estimates for MY (0.07 to 0.42), FY (0.08 to 0.45), and FP (0.04 to 0.39) across all breed group combinations represented in the data set were within the ranges of unibreed estimates found elsewhere. Estimates of interactibilities (ratios of multibreed nonadditive genetic variances to phenotypic variances) were smaller than those of heritabilities for all traits. Multibreed additive and nonadditive genetic correlations ranged from 0.37 to 0.79 for (MY,FY), and they were close to zero for (MY,FP) and (FY,FP). Ranges of sire additive, nonadditive, and total multibreed predicted genetic values (MPGV) were wider for MY and FY in the HO than in the BTBI model. The opposite occurred for FP. Sire rankings by additive, nonadditive, and total MPGV in the BTBI were highly correlated (0.98) to those in the HO model. High correlations (0.99) also existed between sire rankings by additive, nonadditive, and total MPGV within models. The analyses conducted here showed the feasibility of using multibreed procedures for prediction of genetic values and estimation of covariance components in highly unbalanced small multibreed field dairy datasets. However, the large standard errors of prediction obtained here pointed out the need for substantially larger and more balanced multibreed datasets to obtain more reliable genetic predictions useable for genetic improvement programs in Thailand.

Key words: covariance components, dairy cattle, milk production, multibreed population, sire evaluation

Introduction

Milk production in Thailand is based on crossbred animals whose composition is usually 75% or more Holstein (H), the remainder coming from various Bos indicus (Native, Red Sindhi, Brahman) and, to a lesser extent, from Bos taurus (Jersey, Red Dane) breeds. The composition of this crossbred population has been largely determined by a government policy that suggested in 1971 the use of H semen and crossbreeding with local animals to improve milk production in Thailand. Subsequently, crossbred bulls (5/8 H and higher) were produced in Thailand by the Dairy Farming Promotion Organization (DPO) and the Department of Livestock Development. Currently, semen from H and crossbred H bulls is been used nationwide through artificial insemination (AI) services (Sukhato and Kengvikkum, 2000).

Dairy data collected by DPO were used to obtain sire additive genetic predictions starting in 1996, using intrabreed prediction models and procedures (D.P.O., 1999). Separate evaluations were conducted for purebred H and crossbred (H ´ Other (O) breeds). The intrabreed genetic evaluation procedure used by DPO deviated genetic predictions from a base expressed in Bos taurus (BT) and Bos indicus (BI) fractions. Improved versions of intrabreed animal models were used by Koonawootrittriron et al. (2002) to evaluate all animals in a 1991-2000 accumulated DPO data set. Koonawootrittriron et al. (2002) considered two genetic bases, the one used by DPO, based on BT and BI fractions (BTBI model) and another one that deviated genetic predictions from a Holstein (H) and Other Breed (O) base (HO model). The HO base was used to account for the fact that nearly all animals in the DPO population had some H fraction.

Because of the existence of purebred and crossbred animals in the DPO data set, multibreed genetic evaluation procedures (Elzo, 1983; Elzo and Famula, 1985) need to be explored. Multibreed procedures yield more detailed genetic predictions (additive, nonadditive, total) than the previously used intrabreed procedures (additive only). Multibreed additive genetic predictions refer to allelic deviations from a common multibreed genetic base. Multibreed nonadditive genetic effects consider intrabreed and interbreed intralocus interactions between sire and dam alleles. Multibreed total genetic predictions are simply the sum of additive and nonadditive genetic predictions. The same two genetic bases studied in unibreed models by Koonawootrittriron et al. (2002), however, because of the complexity of multibreed models and the small size of the available DPO data set, sire-maternal grandsire multibreed models will be used instead of animal models.

Thus, the objectives of this study were: 1) to estimate additive, nonadditive, and total genetic covariance components and genetic parameters for 305-d milk yield, 305-d fat yield, and 305-d fat percentage, using multibreed sire-maternal grandsire BTBI and HO models, and 2) to compare sire additive, nonadditive, and total multibreed predicted genetic values (MPGV) obtained from the multibreed BTBI and HO models.

Materials and Methods

Animals, Records, and Traits

The base DPO data set used here was the same one used by Koonawootrittriron et al. (2002). This unedited data set consisted of first lactation 12,505 monthly test-day milk yields, and 10,042 monthly test-day fat percentages from 921 cows located in 68 farms in Central Thailand, and collected between 1991 and 2000. Test-day fat yields (10,042) were computed by multiplication of test-day milk yields and test-day fat percentages. Subsequently, 1) 305-d fat percentage (FP) was computed as the average of all test-day fat percentages, and 2) 305-d milk yield (MY) and 305-d fat yield (FY) were computed using formula [1] in Koonawootrittriron et al. (2002). For completeness, this formula is:

[1]

where TPY is the 305-d production yield (MY or FY) of a cow, P₁ is the test-day production yield sample in the first month after calving, D₁ is the interval between five days after calving and the last day of the fist sampling month, P_i is the test-day production yield sample in month i (i = 2, … , k), D_i is the interval between the last day of month i - 1 and i (i = 2, … , k), P_k+1 is the test-day production yield sample in the last month of reaching 305-d in lactation, and D_k+1 is the interval between the 305-d of lactation and the last day of the month before reaching 305-d in lactation.

A program was written in SAS to perform these computations (SAS, 1990).

The resulting data set containing MY, FY, and FP was used as input file for program THAIPED (Elzo, 2000a) to create a pedigree and an edited data file. Only cows with measurements in all three traits (MY, FY, and FP) were included in the edited data file. The edited data file was tested for connectedness between sires and herd-year-season subclasses using program THAICSET (Elzo, 2000b). Cow records from single sire herd-year-seasons were eliminated. Only herd-year-season subclasses with two or more sires, one of them represented in two or more herd-year-season subclasses became part of the largest connected data set. The resulting multibreed connected data set contained MY, FY, and FP from 481 cows. The connected data set and the pedigree file were used as input files for the MREMLEM program (Elzo, 2001) to compute additive and nonadditive genetic, environmental, and phenotypic covariance components, as well as covariance ratios (heritabilities, interactibilities, and genetic, environmental, and phenotypic correlations).

Table 1. Numbers of sires, dams, and calves by breed-group-of-sire ´ breed-group-of-dam combination

	Breed-group-of-sire
Breed-group-of-dam	Holstein	(0.63-0.99)H (0.37-0.01)O ^1^/	Jersey
(0.8-1.0)H (0.2-0.0)O	78 ^2^/	4	1
	115 ^3^/	9	3
	127 ^4^/	9	3
(0.6-0.8)H (0.4-0.2)O	103	6	2
	168	12	2
	178	12	2
(0.4-0.6)H (0.6-0.4)O	76	5	1
	92	7	1
	106	7	1
(0.2-0.4)H (0.8-0.6)O	17	2	-
	15	2	-
	17	2	-
(0.0-0.2)H (1.0-0.8)O	14	1	-
	15	2	-
	15	2	-

^1/ H = Holstein, O = Other breeds: Native, Brahman, Red Sindhi, Sahiwal, Jersey, Red Dane; ^2/ Number of sires; ^3/ Number of dams; ^4/ Number of females with records.

Table 1 contains numbers of sires, dams, and females with records in the multibreed connected data set by breed-group-of-sire ´ breed-group-of-dam combination. On the sire side, these numbers clearly reflect the government suggestion of utilizing H semen on the existing cow population. On the dam side, the number suggest that the preferred cows were 60% to 80% H, with smaller numbers above 80% H and below 60% H. Use of straightbred sires of breeds other than H was almost nonexistent. The crossbred H bulls used amounted to less than 6% of the H sires. This may have been due in part to lack of connectedness (e.g., single-sire herd-year-seasons) of crossbred sires. Numbers of sires, dams, and cows with records classified according to their BT and BI fractions were similar to numbers in the first two columns of Table 1.

Table 2 shows the phenotypic means and standard deviations for each trait by breed-group-of-sire ´ breed-group-of-dam combination as defined in Table 1. Table 2 also contains means and standard deviations by breed group of sire (last 3 rows), and by breed group of dam (last column), and for the complete multibreed data set (last 3 cells of the Overall column). Daughters of H sires produced more milk and fat than daughters of other breed groups. Sixty to eighty percent H crossbred cows had higher milk and fat production levels than any other breed group, although their means were close to 80% to 100% H and 40% to 60%H dams.

Table 2. Means and standard deviations for milk yield, fat yield, and fat percentage by breed-group-of-sire ´ breed-group-of-dam combination

	Breed-group-of-sire
Breed-group-of-dam	Holstein	(0.63-0.99)H (0.37-0.01)O ^1^/	Jersey	Overall
(0.8-1.0)H (0.2-0.0)O	4,116.11±1,167.91^2^/	3,541.22±1,134.84	3,670.33±389.69	4,069.27±1,160.11
	150.39±36.55 ^3^/	133.33±45.10	148.00±9.54	149.22±44.67
	3.66±0.38 ^4^/	3.79±0.40	3.93±0.25	3.67±0.38
(0.6-0.8)H (0.4-0.2)O	4,202.53±1,153.57	3,381.75±1461.46	3,073.50±983.59	4,139.47±1,188.67
	155.28±44.06	129.50±54.60	111.00±42.43	153.21±45.14
	3.73±0.43	3.86±0.33	3.55±0.21	3.73±0.42
(0.4-0.6)H (0.6-0.4)O	4,058.58±1,125.15	4,113.14±1,004.98	.	4,063.04±1,109.17
	150.18±37.23	157.86±35.14	.	150.94±41.59
	3.72±0.41	3.83±0.34	.	3.74±0.41
(0.2-0.4)H (0.8-0.6)O	3,774.29±1,619.58	2,531.50±617.30	.	3,643.47±1,583.13
	138.94±58.11	90.50±36.06	.	133.84±57.51
	3.71±0.35	3.50±0.57	.	3.69±0.37
(0.0-0.2)H (1.0-0.8)O	3,597.93±1,270.55	2,270.50±161.93	.	3,500.59±1,220.52
	133.40±49.91	114.00±8.49	.	131.12±47.18
	3.71±0.37	4.15±0.34	.	3.76±0.41
Overall	4,106.41±1,177.22	3,495.25±1,218.39	3,557.17±657.62	4,058.90±1,184.45
	151.29±44.77	133.38±46.37	141.50±33.74	149.97±44.92
	3.70±0.40	3.83±0.37	3.88±0.37	3.72±0.40

^1/ H = Holstein, O = Other breeds: Native, Brahman, Red Sindhi, Sahiwal, Jersey, Red Dane; ^2/ milk yield (kg); ^3/ fat yield (kg); ^4/ fat percentage (%).

Estimation of Multibreed Covariance Components and Genetic Predictions

Covariance components were estimated using multibreed restricted maximum likelihood procedures (MREMLEM; Elzo, 1994, 1996). The MREMLEM procedure uses a generalized expectation-maximization (GEM) algorithm (Dempster et al., 1977) to compute MREMLEM covariance components. The MREMLEM program (Elzo, 2001) computes additive and nonadditive genetic, and environmental covariance matrices simultaneously. To ensure that all variances and covariances were within their permissible ranges, the Cholesky matrices of covariance matrices were computed first, and then the covariance matrices themselves by multiplying the Cholesky matrices by their transposes (Elzo, 1996).

A disadvantage of expectation-maximization algorithms is that they do not compute the information matrix. However, although the MREMLEM did not provide standard errors, large standard errors of estimation of covariance components should be expected because of the unbalancedness and the small size of the data set.

Multibreed Model. The model used was a multibreed sire-maternal grandsire model that accounted for intrabreed additive genetic effects, and intralocus intra and interbreed nonadditive genetic effects. Although desirable, the structure and size of the data set prevented the use of a multibreed animal model. Preliminary analyses showed that the extreme unbalancedness of the data caused confounding and multicollinearity among additive and nonadditive sire genetic group effects, thus unbiased estimates of them were impossible to obtain. Consequently, the final form of the multibreed sire-maternal grandsire model contained random additive and nonadditive sire genetic effects, and accounted for additive and nonadditive genetic as well as environmental heterogeneity of variances and covariances across breed group combinations (BT and BI in the BTBI model, and H and O, in the HO model). Because sire genetic group effects were not included in the model, sire additive, nonadditive, and total multibreed genetic predictions for the BTBI and HO models were deviated from common pooled additive-nonadditive genetic bases within each model (BTBI multibreed base, and HO multibreed base).

The data set permitted the analysis of at most two traits at a time. Thus, there were three computer runs for the BTBI and the HO models: 1) MY and FY, 2) MY and FP, and 3) FY and FP. These analyses yielded two estimates of variances for each trait and a single covariance. Each pair of variance estimates (for MY, FY, and FP) was averaged to produce a single variance estimate.

The multibreed sire-maternal grandsire model for a two-trait analysis was as follows.

y = Xb + Z_mg g_mg +Z_a u_a + Z_n u_n + e [2]

E[y] = Xb+ Z_mg g_mg

var(y) = Z_a G_a Z_a’ + Z_n G_n Z_n’ + R

where

y = vector of cow records, ordered by traits within cows,

b = vector of fixed environmental effects: herd-year-seasons, and a covariate for age of dam effects modeled as a function of the BT fraction of the dam (BTBI model), and the H fraction of the dam (HO model). Seasons were classified as winter (November to February), summer (March to June), and rainy (July to October).

g_mg = vector of fixed maternal granddam and unknown maternal grandsire regression genetic group effects. In the DPO data set, the g_mg vector contained dam regression genetic group effects because all dams of cows had unknown sires.

u_a = vector of random sire and maternal grandsire additive direct genetic effects,

u_n = [u_n1 u_n2 u_n3 ]’ = vector of random sire nonadditive intralocus direct genetic effects: two intrabreed (BT´BT, BI´BI, BTBI model; H´H, O´O, HO model) and one interbreed (BT´BI, BTBI model; H´O, HO model). Sire nonadditive intralocus direct genetic effects are defined to be due to intralocus intrabreed and interbreed interactions between alleles from a sire and alleles from dams of all breed groups mated to a particular sire. These interactions measure the average interactive ability of a sire across all breed groups of dams, i.e., the interactive ability of a sire within a sire subclass. Intralocus interactions within a sire subclass yield an incomplete assessment of dominance effects. To have a complete assessment of dominance effects a statistical model would also have to have terms for the interactive ability of a dam within a dam subclass, and for the interactive ability of a specific sire ´ dam combination. Thus, the sire maternal grandsire model used here predicts only an average dominance effect in the progeny of an individual sire and dams of all breed groups in a multibreed population. It is an approximation to a complete multibreed model capable of measuring all intrabreed and interbreed dominance effects. The complete nonadditive model was not computationally feasible with the DPO data set.

e = vector of residual effects,

X = incidence matrix relating cow records to elements of b. The elements of X are ones, zeroes, and BT fraction of dams (BTBI model) or H fraction of dams (HO model),

Z_mg = incidence matrix relating cow records to elements of g_mg. The elements of Z_mg for the BTBI model are the BT fractions of maternal granddams, BT fractions of unknown maternal grandsires, and zeroes. For the HO model, the elements of Z_mg are H fractions of maternal granddams, H fractions of unknown maternal grandsires, and zeroes. For example, a 7/8 BT 1/8 BI daughter of a BT sire and a 3/4 BT 1/4 BI dam would contribute with a .5 to her maternal granddam group, and a 1 to her unknown maternal grandsire group.

Z_a = incidence matrix relating cow records to elements of u_a . The elements of Z_a are ones if sires of cows are known, .5’s if maternal grandsires of cows are known, and zeroes.

Z_n = incidence matrix relating cow records to elements of u_n through probabilities of intrabreed and interbreed intralocus combinations in females with records. The elements of Z_n are probabilities of alleles of base populations BT and(or) BI in a single locus. Matrix Z_n = block-diagonal {Z_n1 Z_n2 Z_n3}. For the BTBI model: 1) the elements of Z_n1 are the intralocus probabilities of (BT_{sire of cow}, BT_{dam of cow}), 2) the elements of Z_n2 are the intralocus probabilities of (BI_{sire of cow}, BI_{dam of cow}), and 3) the elements of Z_n3 are the intralocus probabilities of [(BT_{sire of cow}, BI_{dam of cow}) + (BI_{sire of cow}, BT_{dam of cow})]. Substitute H for BT and O for BI in the previous probabilities. For example, a 3/4 BT 1/4 BI daughter of a BT sire and a 1/2 BT 1/2 BI dam would contribute with .5 to Z_n1, 0 to Z_n2, and .5 to Z_n3.

G_a = var(u_a) = multibreed sire and maternal grandsire additive genetic covariance matrix. (Elzo, 1990a, 1994). Matrix G_a = T^TB_aT, where 1) T is a lower triangular matrix that has ones on the diagonal,.5’s in the off-diagonals between an animal in u_a and his sire if known, .25’s in the off-diagonals between an animal in u_a and his maternal grandsire if known, and zeroes elsewhere, 2) T^T is the transpose of T, and 3) B_a is a block-diagonal matrix (block size = nt ´ nt, nt = number of traits) of sire and maternal grandsire multibreed residual genetic variances and covariances. The nt ´ nt covariance matrix for the i^th sire or maternal grandsire in vector u_a is equal to var(u_ai) – {var(1/2 u_asi), if si is known} – {var(1/4 u_ami), if mi is known}, where the subscripts i = individual (sire or maternal grandsire) in u_a, si = sire of individual in u_a, and mi = maternal grandsire of individual in u_a. If there is no inbreeding, the var(u_ai) = p_BTi var(a_BT) + p_BIi var(a_BI) + (p_BTsi p_BIsi + 1/4 p_BTmi p_BImi) var(a_BTBI) in a sire-maternal grandsire model, where: 1) p_Xy = fraction of base population X (X = BT, BI) alleles in animal y, y = i (sire or maternal grandsire in u_a), si (sire of i), and mi (maternal grandsire of i), 2) var(a_X) = additive genetic variance in base population X, X = BT, BI, and 3) var(a_BTBI) = additive genetic variance due the presence of alleles of base populations BT and BI in the same individual. The var(a_BTBI) is an additional additive variance present in crossbred animals because the mean effects of alleles differs across base populations. This variance was called segregation variance by Wright (1968), and it is proportional to the fraction of heterozygous loci in the parents of an individual. The var(1/2 u_asi) and var(1/4 u_ami) are computed in similar fashion. For the HO model substitute H for BT and O for BI in the preceding formulas. For example, the multibreed additive genetic variance of a 3/4 BT 1/4 BI sire, whose sire was BT, his dam was 1/2 BT 1/2 BI, his maternal grandsire was BT, and his maternal granddam was BI, is equal to 3/4 var(a_BT) + 1/4 var(a_BI) + [1 ´ 0 + 1/4 (1 ´ 0)] var(a_BTBI). Because matrix T relates animals to its ancestors, recurrent formulas can be used to construct G_a and its inverse G_a^-1 (for specific details on these computational procedures with and without inbreeding, see Elzo, 1990a). These recurrent procedures require knowledge of only base population covariances. Multibreed covariances are computed as linear combinations of base covariances as described above for the BTBI and HO cases.

G_n = var(u_n) = var([u_n1 u_n2 u_n3 ]’) = block-diagonal multibreed sire regression nonadditive genetic covariance matrix (Elzo, 1990b, 1994). The term regression refers to the fact sire ´ breed-group-of-dam intralocus interaction effects were modeled as a function of intra and interbreed intralocus interactions. Matrix G_n = block-diagonal{G_n1 G_n2 G_n3}. Each submatrix G_nj, j = 1,2,3, can be written in the same form as the matrix of additive genetic effects. Thus, matrix G_nj = T^TB_niT, where 1) T and T^T are the same matrices described above for G_a, and 2) B_nj is a block-diagonal matrix (nt ´ nt blocks) with elements equal to var(u_nji) – {var(1/2 u_njsi), if si is known} – {var(1/4 u_njmi), if mi is known}, where the subscripts i = individual in u_nj, si = sire of individual in u_nj, and mi = maternal grandsire of individual in u_nj. If there is no inbreeding, the var(u_nji) = var(u_njsi) = var(u_njmi) = var(u_nj). Thus, var(u_nji) = (1 – {1/2 , if si is known} – {1/4 , if mi is known}). Recurrent computational procedures that can be used to compute each submatrix G_nj, j = 1,2,3, and their inverses are described in detail in Elzo (1990b).

and,

R = block-diagonal (nt ´ nt blocks) multibreed residual covariance matrix. The nt ´ nt matrix for the i^th cow with nt records is equal to the sum of the nt ´ nt multibreed residual genetic covariance matrix for cow i + the nt ´ nt multibreed residual environmental covariance matrix for cow i. Residual environmental effects here are assumed to contain environmental effects and nonadditive genetic effects not explained by Z_n u_n in the model. Thus, residual environmental covariance matrices are a function of environmental covariances, and nonadditive genetic covariances due to nonadditive genetic effects not accounted for in the sire-maternal grandsire model. The nt ´ nt multibreed residual genetic covariance matrix for cow i is computed using the same formulas described to compute the diagonal submatrices of G_a above, except that here subscripts i = cow i, si = sire of cow i, and mi = maternal grandsire of cow i. The nt ´ nt multibreed residual environmental covariance matrix for cow i is also computed using the formulas used to compute the diagonal submatrices of G_a above, except that: 1) subscripts i = cow i, si = sire of cow i, and mi = maternal grandsire of cow i, and 2) multibreed residual environmental covariances replace multibreed additive genetic covariances. For additional details on the construction of R see Elzo (1994) and Elzo and Wakeman (1998).

MREMLEM Algorithm. The starting values used for the two-trait MREMLEM analyses were variance estimates (additive and nonadditive genetic, and environmental) from preliminary single-trait MREMLEM analyses, and zeroes for all covariances between traits. In the estimation step, the multibreed mixed model equations were set up by storing only nonzero elements of the left and right hand sides. Multibreed computational algorithms were used to obtain the inverse of the multibreed additive covariance matrix (Elzo, 1990a), and the inverse of the regression nonadditive genetic covariance matrix (Elzo, 1990b). In the maximization step, covariances were estimated using the Cholesky maximization strategy (Elzo, 1996). The convergence criterion was that the square root of the ratio of the sum of squares of the differences between covariance estimates in two successive GEM iterations, divided by the sum of squares of the covariances in the first of them, was less than 10^-4.

Base Genetic, Environmental, and Phenotypic Covariances. Separate sets of three pairwise runs (MY-FY, MY-FP, and FY-FP) were conducted to estimate base covariance components for the BTBI and HO models. Seven 3´3 matrices were computed for each model. For the BTBI model, these matrices were: 1) two additive genetic intrabreed (BT and BI), 2) three nonadditive genetic intralocus (intrabreed BT/BT and BI/BI, and interbreed BT/BI), and 3) two environmental (BT and BI). The corresponding matrices for the HO model were: 1) two additive genetic intrabreed (H and O), 2) three nonadditive genetic intralocus (intrabreed H/H and O/O, and interbreed H/O), and 3) two environmental (H and O). The elements of each covariance matrix were: var(MY), cov(MY,FY), cov(MY,FP), var(FY), cov(FY,FP), and var(FP).

Multibreed Genetic Covariances and Genetic Parameters. Base covariance estimates were used to compute multibreed covariances and genetic parameters (heritabilities, interactibilities, genetic, environmental, and phenotypic correlations) for specific breed group combinations. Here, interactibility refers to intrabreed and interbreed nonadditive interactions between alleles from individual sires and alleles from all dams mated to them.

Multibreed additive and nonadditive genetic, environmental, and phenotypic covariances were obtained as weighted averages of appropriate base covariances (Elzo, 1994; Elzo and Wakeman, 1998). As an example, consider MY and FY, and breed group combination BT ´ 3/4BT 1/4BI:

1) the (MY,FY) multibreed additive genetic covariance is equal to (probability of BT alleles in breed group combination BT ´ 3/4BT 1/4BI) ´ additive cov(BT_MY, BT_FY) + (probability of BI alleles in breed group combination BT ´ 3/4BT 1/4BI) ´ additive cov(BI_MY, BI_FY) + (probability of BT and BI alleles in 3/4BT 1/4BI and in BT, assumed to be zero in this research) ´ additive cov(BTBI_MY, BTBI_FY),

2) the (MY,FY) multibreed nonadditive genetic covariance is equal to (probability of BI/BT intralocus interactions in breed group combination BT ´ 3/4BT 1/4BI) ´ nonadditive cov(BT/BI_MY, BT/BI_FY) + (probability of BT/BT intralocus interactions in breed group combination BT ´ 3/4BT 1/4BI) ´ nonadditive cov(BT/BT_MY, BT/BT_FY),

3) the (MY,FY) multibreed environmental covariance is equal to (probability of BT alleles in breed group combination BT ´ 3/4BT 1/4BI) ´ environmental cov(BT_MY, BT_FY) + (probability of BI alleles in breed group combination BT ´ 3/4BT 1/4BI)