The use of matrix specifications in defining gene action in genotypic value models and generation mean analysis

Gene action and interaction have been defined in the literature by the use of a variety of notations (Mather 1949; Hayman 1954, 1955, 1957; Jinks 1954; Kempthorne 1954, 1955). This leads to unnecessary complications in understanding the subject. This paper provides a simple convenient way of translating one parameterization into another and illustrates the simple linear relationship between them. The various notations are written in matrix form by the use of a specification matrix. This provides a convenient compact method of presentation of the relevant Equations. The linear relationship between the genetic parameters enables these to be estimated in the most convenient way and then converted to other parameters for the purposes of comparison. The generation means Equations of Hayman (1958) are derived using the matrix formulation as an illustration of the use of the specification matrix.