Multivariate estimation of variance and covariance components using restricted maximum likelihood, in dairy cattle : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Animal Science at Massey University
The multivariate estimation of sire additive and residual variances and covariances by Restricted Maximum Likelihood (REML) is addressed. Particular emphasis is given to its application to dairy cattle data when all traits are explained by the same model and no observations are missing. Special attention is given to the analysis of new traits being included in a sire evaluation programme, for which a model has to be developed and no previous estimates of the population parameters exist. Results obtained by using either the multivariate Method 3 of Henderson, multivariate REML excluding the Numerator Relationship Matrix (NRM) or by multivariate REML including the NRM were compared. When a large number of traits were fitted simultaneously the variance-covariance matrix estimated by Method 3 was negative-definite (outside the allowable parameter space). REML estimates obtained while ignoring the NRM were biased. The number and sequence of traits fitted in the analysis affected the estimates at convergence. A canonical transformation of the variance-covariance matrix was undertaken to simplify the computation by means of an Expectation Maximisation (EM) algorithm. Approaches to choosing initial values for their use in iterative methods were compared via their values at convergence and the number of iterations required to converge. To further simplify the use of multivariate REML, three transformations of the Mixed Model Equations (MME) were integrated: the absorption of proven sire effects taken as fixed, a triangular factorisation of the NRM, and the singular value decomposition of the coefficient matrix in the MME. One statistical algorithm (EM) and one mathematical algorithm (Scoring type) were developed to iteratively solve the REML equations on the transformed scale, such that the transformed coefficient matrix of the MME did not need to be inverted at each iteration and the required quantities to build the REML equations were obtained through vector operations. Traits other than Production (TOP) from New Zealand Holstein-Friesian dairy cows were analysed (4 management and 13 conformation characteristics), each trait scored using a linear scale from 1 to 9, with extreme values corresponding to extreme phenotypes. Mixed model methodology was used for the analysis of TOP as no significant departure from normality was observed. To model the TOP, the fixed effects of herd, inspector, age, stage of lactation (linear and quadratic) and breed of dam were tested for significance. Only the effects of inspector and herd were significant for all traits, with breed of dam significantly affecting adaptability to milking, shed temperament and stature. Estimates of phenotypic means and standard deviations, and heritabilities for TOP were: adaptability to milking 5.4 ± 1.7, 0.20; shed temperament 5.5 ± 1.6, 0.12; milking speed 5.7 ± 1.5, 0.11; farmer's overall opinion 5.7 ± 1.7,.14; stature 5.1 ± 1.0, 0.14; weight 4.4 ± 1.0, 0.37; capacity 5.3 ± 1.0, 0.40; rump angle 5.4 ± 0.7, 0.16; rump width 5.2 ± 0.7, 0.08; legs 5.2 ± 0.6, 0.34; udder support 5.3 ± 1.0, 0.63; fore udder 4.9 ± 1.1, 0.48; rear udder 4.9 ± 1.0, 0.33; front teat placement 4.2 ± 0.7, 0.22; rear teat placement 5.2 ± 0.8, 0.22; udder overall 4.8 ± 1.1, 0.42; and dairy conformation 5.3 ± 1.1, 0.32. Large positive phenotypic correlations among management traits were obtained, while the correlations of these traits with type were small and positive when significant. Large and positive correlations among udder traits were found. All traits related to size were positively correlated amongst themselves. Most of the traits were positively correlated with dairy conformation. Estimated genetic correlations for stature and weight with other conformation traits were generally negative. With the exception of udder support, all udder traits were positively correlated amongst themselves. Dairy conformation was positively correlated with most traits, except with stature, rump angle, legs, rear udder and udder overall. The estimates obtained in this study shold be used in the evaluation of Holstein-Friesian sires and cows lor TOP in New Zealand.