On parsimonious and equivalent animal models with (grand) maternal effects and missing (grand) dams

In breeds where a large fraction of animals with records on a maternally affected trait are from dams that have no records and unknown parents, the genetic evaluation of such trait may be hindered by misspecification of the genetic covariance matrix. The specified covariance structure for the additive direct and maternal effects in the regular maternal animal model (MAM) when dams have no records differs from the covariance between relatives with maternal effects. Two solutions are possible. One is to include in the vectors of breeding values for direct and maternal effects the dam or a “phantom” dam if the latter is unknown. As a consequence, the number of equations to be solved in the MAM may increase considerably. Alternatively, one may replace the maternal breeding value of the dam with 2/3 of the maternal breeding of the individual, and −1/3 of the maternal breeding value of the sire of the individual. As this “regression” of breeding values has been largely ignored, the goal of this paper is to present a parsimonious equivalent MAM using such regression. The approach is extended to a similar situation for models with grand maternal effects. Two small numerical examples are used to illustrate the proposed methods.