Standard errors of solutions in large scale mixed models, application to linear and curvilinear effects of inbreeding on production traits.

Many approaches for using linear mixed models do not produce standard errors of solutions. However, knowing the standard errors allows for statistical tests. Even if exact estimation of standard errors is not feasible in large mixed models, there are methods to approximate them. We based this on Mixed Model Conjugate Normal Equations associated with a Preconditioned Conjugate Gradient (PCG) solver. The advantage of associating both methods is that the right hand side vector normally accumulated by PCG can be easily changed to a function of solutions vector k allowing direct solution for Φ=C-1k using regular PCG solving programs. The square root of k’Φ=k’C-1k gives the standard error associated with the function of solutions described by k. Often a block of C-1 is needed. Its elements were obtained by computing linear functions of element of
this block and by backsolving to obtain the needed elements. In matrix notation let K be the coefficients of the linear functions and D a matrix containing the values obtained by computing K’C-1K. The elements of the block were then obtained as (KK’)-1KDK’(KK’)-1. This method was applied to study linearity of inbreeding depression on milk, fat and protein test-day yields. Inbreeding effects were estimated using linear, quadratic and cubic regressions on inbreeding coefficients inside breeds in a test-day model similar to the one used in the Walloon Region of Belgium. The pedigree contained 956,516 animals. A total of 5,596,038 first lactations test-day records from 660,407 cows were used. Results had contrasting behaviors, however evaluation of plotted inbreeding effect and the associated confidence interval showed that between 0 an 10% inbreeding differences among evaluations of inbreeding depression were small.