Van Vleck L D
U.S. Department of Agriculture, Lincoln, NE 68583-0908.
J Anim Sci. 1990 Dec;68(12):4014-25. doi: 10.2527/1990.68124014x.
Rules for forming the mixed-model equations for the reduced animal model with all relationships and including maternal effects have been set out by Quaas and Pollak. They also have shown how to simplify the mixed-model equations when genetic group effects are included in the model with what has become known as the Q-P transformation. Westell has given rules for calculating the coefficients for the Q-P transformed equations that are associated with the inverse of the numerator relationship matrix and genetic group effects. Those rules can be extended to include maternal effects and genetic groups for maternal as well as direct effects. As with the rules of Quaas and Pollak for the equations for the reduced animal model, a similar set of rules can be obtained for the genetic groups model after the Q-P transformation. The rules are derived easily by examining the algebraic results of absorbing the direct and maternal breeding value equations for non-parents into the parent breeding value, group and fixed effects equations. These rules involve Westell's rules and the inverse elements of the genetic (co)variance matrix for direct and maternal additive genetic effects. The rules make calculation of breeding values for parents for models including direct and maternal genetic group effects nearly as easy as for models without genetic group effects. Back solution for direct and maternal breeding values of non-parents similarly is as simple as when genetic group effects are not in the model.
夸斯和波拉克已经阐述了构建包含所有亲缘关系且纳入母体效应的简化动物模型的混合模型方程的规则。他们还展示了在模型中纳入遗传群体效应时，如何通过所谓的Q - P变换来简化混合模型方程。韦斯特尔给出了计算与分子亲缘关系矩阵的逆以及遗传群体效应相关的Q - P变换方程系数的规则。这些规则可以扩展到包括母体效应以及母体和直接效应的遗传群体。与夸斯和波拉克针对简化动物模型方程的规则一样，在进行Q - P变换后，对于遗传群体模型也可以得到类似的一组规则。通过研究将非亲本的直接和母体育种值方程吸收到亲本育种值、群体和固定效应方程中的代数结果，很容易推导出这些规则。这些规则涉及韦斯特尔的规则以及直接和母体加性遗传效应的遗传（协）方差矩阵的逆元素。这些规则使得计算包含直接和母体遗传群体效应模型中亲本的育种值几乎与计算不包含遗传群体效应模型中的育种值一样容易。同样，非亲本直接和母体育种值的反向求解与模型中不存在遗传群体效应时一样简单。
