Clarifying the Relationship between Average Excesses and Average Effects of Allele Substitutions

Abstract

Fisher's concepts of average effects and average excesses are at the core of the quantitative genetics theory. Their meaning and relationship have regularly been discussed and clarified. Here we develop a generalized set of one locus two-allele orthogonal contrasts for average excesses and average effects, based on the concept of the effective gene content of alleles. Our developments help understand the average excesses of alleles for the biallelic case. We dissect how average excesses relate to the average effects and to the decomposition of the genetic variance.

Keywords: average effects; average excesses; effective gene content; models of genetic effects; non-equilibrium populations.

Figure 1

Graphical interpretation of the decomposition of the genotypic values (5) through the statistical excess (in black) and the statistical (in gray) formulations of NOIA for one locus with two alleles. For simplicity, a case with equal allele frequencies (p₁ = p₂ = 1/2) is shown. The specific genotypic values (circles; G₁₁ = 1, G₁₂ = 3, G₂₂ = 2) displaying overdominance and a fixation index (F = − 2/5) have been chosen for facilitating the visualization of the parameters of interest. The size of the circles represents the frequency of the genotypes. Horizontal dashed lines emphasize coincident arrow edges, the upper one corresponding to the population mean phenotype, μ = 2.55. The regression independent variable of the statistical formulation is the gene content, whereas the one of the statistical excess formulation is scaled by c = 1/(1 + F) = 5/3 and it works as an effective gene content. For both cases, the independent variable, w, is rescaled by its expectation as shown in Table 1.