3.3. Estimation of Genetic Parameters

3.3.1. Statistical Methods

Estimation of genetic and environmental parameters was performed by two-trait analysis using calving difficulty trait along with birth weight. The different sex (grandcows and grandsires) were analyzed separately. Both univariate and bivariate linear animal models were used. The basic bivariate model for estimating variance components for first parity traits was:





where yi and yj are observed birth weight, calving difficulty score. The vector [b’i b’j] contains for all traits the fixed effects of herd-year, season, and combination of sex of the calf and age group of the cow. For birth weight, the fixed effect of age at weighing in days was included in the model. The X and Z matrices are incidence matrices relating the observations to the fixed and random effects, respectively; m is a vector of maternal genetic effects; a is a vector of additive genetic effects of the animals; and e is a vector of random residuals. The models for later parity traits also included a random permanent environmental effect of the cow, pe. For random effects, the means were zero and the variances were as follows:













Where A is the relationship matrix. For bivariate analyses with first-and later-parity traits, the permanent environmental effect of cow was only included for later parities, and residual covariances were assumed to be zero since the traits were measured on different animals. Covariances between genetic and environmental effects were assumed to be zero, and no variances due to dominance or epistatic effects were assumed to exist. Covariances were estimated using the average information algorithm (Jensen et al., 1997) for restricted maximum likelihood included in the DMU package (Jensen and Madsen, 1994). The convergence criterion was chosen so that the norm of update vector for the covariance components was less than 10−4. Asymptotic standard errors of covariance components were computed from the inverse average information matrix. Standard errors of genetic correlations were obtained by Taylor series expansions (Madsen and Jensen, 2000).

Heritabilities. Direct and maternal heritabilities on the observable scale were calculated as σ2a/σ2P and σ2m/σ2P, respectively, where σ2P= σ2m+σa,m + σ2a + σ2e for all traits in the first parity and σ2P= σ2pe + σ2m+ σa,m + σ2a+ σ2e for later parities. Heritabilities on the underlying continuous scale were approximated from the heritabilities on the observable scale to enable comparisons with other studies, using a transformation described by Gianola (1982):









where ηk is the score for response, πk is the probability of response in the kth category (k = 1, 2, . . ., m), and zk is the ordinate of a standard normal density function corresponding to thresholds between categories k and k + 1. For two response categories as for birth weight, it reduces to: h2underlying = h2observed [π(1 -π)]/z2, as shown in Dempster and Lerner (1949).

Implementation and software. Variance-covariance parameters for all models were estimated using manual computed or the software package, ASREML (Gilmour et al., 2001). Heritabilities (h2) and all correlations (genetic-rg, residual-re, and phenotypic-rp) were computed using estimated variance-covariance matrices. In ASREML, it was possible to get standard errors for all estimated parameters or its ratios. In total, there were 61 blocks of genetic analyses (31 for univariate and 30 for bivariate analyses). It was not our aim to compute correlations (covariances) among type traits or between calving difficulty and birth weight at first and later parities.