Scope for a random regression model in genetic evaluation of beef cattle for growth

Continuous evaluation of dairy cattle with a random regression test-day model requires a fast solving method and algorithm. A new computing technique feasible in Jacobi and conjugate gradient based iterative methods using iteration on data is presented. In the new computing technique, the calculations in multiplication of a vector by a matrix were recorded to three steps instead of the commonly used two steps. The three-step method was implemented in a general mixed linear model program that used preconditioned conjugate gradient iteration. Performance of this program in comparison to other general solving programs was assessed via estimation of breeding values using univariate, multivariate, and random regression test-day models. Central processing unit time per iteration with the new three-step technique was, at best, one-third that needed with the old technique. Performance was best with the test-day model, which was the largest and most complex model used. The new program did well in comparison to other general software. Programs keeping the mixed model equations in random access memory required at least 20 and 435% more time to solve the univariate and multivariate animal models, respectively. Computations of the second best iteration on data took approximately three and five times longer for the animal and test-day models, respectively, than did the new program. Good performance was due to fast computing time per iteration and quick convergence to the final solutions. Use of preconditioned conjugate gradient based methods in solving large breeding value problems is supported by our findings. Show more

Utility of the preconditioned conjugate gradient algorithm with a diagonal preconditioner for solving mixed-model equations in animal breeding applications was evaluated with 16 test problems. The problems included single- and multiple-trait analyses, with data on beef, dairy, and swine ranging from small examples to national data sets. Multiple-trait models considered low and high genetic correlations. Convergence was based on relative differences between left- and right-hand sides. The ordering of equations was fixed effects followed by random effects, with no special ordering within random effects. The preconditioned conjugate gradient program implemented with double precision converged for all models. However, when implemented in single precision, the preconditioned conjugate gradient algorithm did not converge for seven large models. The preconditioned conjugate gradient and successive overrelaxation algorithms were subsequently compared for 13 of the test problems. The preconditioned conjugate gradient algorithm was easy to implement with the iteration on data for general models. However, successive overrelaxation requires specific programming for each set of models. On average, the preconditioned conjugate gradient algorithm converged in three times fewer rounds of iteration than successive overrelaxation. With straightforward implementations, programs using the preconditioned conjugate gradient algorithm may be two or more times faster than those using successive overrelaxation. However, programs using the preconditioned conjugate gradient algorithm would use more memory than would comparable implementations using successive overrelaxation. Extensive optimization of either algorithm can influence rankings. The preconditioned conjugate gradient implemented with iteration on data, a diagonal preconditioner, and in double precision may be the algorithm of choice for solving mixed-model equations when sufficient memory is available and ease of implementation is essential. Show more

Currently, most analyses of parameters in test-day models involve two types of models: random regression, where various functions describe variability of (co)variances with regard to days in milk, and multiple traits, where observations in adjacent days in milk are treated as one trait. The methodologies used for estimation of parameters included Bayesian via Gibbs sampling, and REML in the form of derivative-free, expectation-maximization, or average-information algorithms. The first method is simpler and uses less memory but may need many rounds to produce posterior samples. In REML, however, the stopping point is well established. Because of computing limitations, the largest estimations of parameters were on fewer than 20,000 animals. The magnitude and pattern of heritabilities varied widely, which could be caused by simplifications in the model, overparameterization, small sample size, and unrepresentative samples. Patterns of heritability differ among random regression and multiple-trait models. Accurate parameters for large multi-trait random regression models may be difficult to obtain at the present time. Parameters that are sufficiently accurate in practice may be obtained outside the complete prediction model by a constructive approach, where parameters averaged over the lactation would be combined with several typical curves for (co)variances for days in milk. Obtained parameters could be used for any model, and could also aid in comparison of models. Show more