Estimation of additive and dominance genetic variances for - CGIL ...

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386 ( ) M.J.R. Pante et al.rAquaculture 204 2002 383–392 Table 2 Number of sires, dams, offspring and average body weight Ž kg. for the three populations of rainbow trout Population Number Number Number of Average body s.d. Ž kg. of sires of dams offspring weight Ž kg. 1 154 512 69,920 3.71 1.18 2 165 456 62,897 3.70 1.27 3 278 682 67,280 4.16 1.55 of the likelihood function Ž Jensen et al., 1996 . . Six different models were studied by including all or different subsets of the following effects in the animal model: the random additive genetic effect, the random parental dominance genetic effect, the random common environmental effect due to full-sibs, and the inbreeding coefficient as a covariate: ysXhqZ 1aqe Ž 1. ysXhqbFqZ 1aqe Ž 2. ysXhqZ 1aqZ2cqe Ž 3. ysXhqbFqZ 1aqZ2cqe Ž 4. ysXhqbFqZ 1aqZ3dqe Ž 5. ysXhqbFqZ 1aqZ2cqZ3dqe Ž 6. where y is a vector of observations of animals; h is a vector of the fixed effects of generation)location)cage)sex; a, d, c are random effects of direct additive genetic, parental dominance interaction, and common environment due to full-sib groups, respectively; b is the linear regression of y on inbreeding coefficients; F is the coefficient of inbreeding; X, Z 1, Z 2, Z3 are the corresponding incidence matrices relating the effects to y; and e is the vector of random residuals. Eq. Ž. 1 is the simple additive genetic model Ž A . ; Eq. Ž. 2 is the Ž AqF . model with additive genetic effects and the inbreeding coefficients as a linear covariate; Eq. Ž. 3 is model Ž AqCE. with the additive genetic and common environment effects; Eq. Ž 4. is model Ž AqCEqF . with the additive genetic and common environment effects and with inbreeding coefficients as linear covariate; Eq. Ž. 5 is model Ž AqDqF . with additive and dominance genetic effects and inbreeding coefficients as covariate; and Eq. Ž. 6 is the full model Ž AqDqCEqF . , where all effects are fitted into the model. A model where only the additive and dominance genetic effects Ž AqD . are present was not included in the study because the inbreeding coefficient has to be fitted to obtain proper estimates of the dominance genetic effects Ž Hoeschele and Van Raden, 1991 . . The assumptions for the parameter means and variances were: 2 y X b A sa 0 0 0 a 0 2 a d 0 0.25Dsd 0 0 d 0 2 c 0 0 Isc 0 E s ;Var s , c e 0 0 e 0 0 0 2 Ise
( ) M.J.R. Pante et al.rAquaculture 204 2002 383–392 387 where s 2 is the additive genetic variance, s 2 is the dominance genetic variance, s 2 a d c is the common environment variance, s 2 is the error variance, A is the additive genetic e relationship matrix, D is the dominance genetic relationship matrix and I is an identity matrix. Note that parental dominance variance is one-quarter of the offspring dominance variance. Following the algorithm of Hoeschele and Van Raden Ž 1991 . , inbreeding was accounted for when inverting the A matrix but was ignored when inverting the D matrix. However, with low levels of inbreeding, including the individual inbreeding coefficients as a covariate in the model may be sufficient to account for ignoring inbreeding in forming the inverse of the D relationship matrix Ž Hoeschele and Van Raden, 1991 . . Moreover, the covariance between the additive and dominance genetic effects was also ignored due to the large size of the data set. The estimated variance components were expressed as ratios of the total phenotypic Ž 2 . 2 2 2 variance s for each model: the additive genetic variance ratio as h ss rs , i.e. t a t the heritability; the dominance genetic variance ratio as d 2 ss 2rs 2 d t ; and the common environmental variance ratio as c2 ss 2rs 2 c t . Log likelihood ratio tests were carried out among the six models to determine the most suitable of the six models. The likelihood ratio is LRsy2logwŽ rNz . rNz Ž .xs y2wlogŽ rNz . yŽlogŽ Nz .x, where logŽ Nz . is the maximum of the likelihood function for the full model and logŽ rNz . is the maximum of the likelihood function for the reduced model subject to r number of constrained effects or parameters ŽLynch and Walsh, Table 3 Ž 2 . Ž 2 . Ž 2 Estimates of additive h , dominance d , and common environmental c . variances expressed as ratio of Ž . Ž 2 total phenotypic variance with their corresponding standard errors s.e. , and the residual variance s . e for each of the model for the three populations of rainbow trout 2 2 2 2 Population Model h "s.e. d "s.e c "s.e. s e 1 A 0.366"0.018 0.549 Aq F 0.354"0.018 0.554 AqCE 0.156"0.021 0.041"0.005 0.637 AqCEq F 0.158"0.021 0.038"0.004 0.637 Aq Dq F 0.162"0.023 0.144"0.005 0.635 Aq DqCEq F 0.158"0.020 0.000"0.000 0.038"0.004 0.637 2 A 0.365"0.019 0.559 Aq F 0.344"0.018 0.570 AqCE 0.087"0.018 0.062"0.006 0.681 AqCEq F 0.089"0.017 0.055"0.005 0.681 Aq Dq F 0.083"0.017 0.218"0.005 0.683 Aq DqCEq F 0.082"0.017 0.161"0.013 0.014"0.013 0.684 3 A 0.462"0.020 0.558 Aq F 0.441"0.019 0.572 AqCE 0.180"0.022 0.058"0.006 0.707 AqCEq F 0.185"0.022 0.053"0.005 0.705 Aq Dq F 0.167"0.022 0.216"0.005 0.713 Aq DqCEq F 0.167"0.012 0.215"0.004 0.000"0.000 0.714
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Estimation of additive and dominance genetic variances for - CGIL ...

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