Second-order approximations for selection coefficients at polygenic loci.

I determine the second-order approximation for the phenotypic distribution of a quantitative trait, ignoring the effects of epistasis and linkage disequilibrium, conditioned on the presence of a specified genotype at one underlying locus of small effect. I demonstrate that this approximation has an error that is third order in the allelic or genotypic effects, independent of the form of the phenotypic distribution. I also show that the approximation of analogous form for the phenotypic distribution conditioned on the presence of a specified allele at a single locus is also correct to second order. Both approximations allow for dominance and are consistent in the sense that computing marginal fitnesses from approximations based on genotypic deviations and those based on average allelic effect yield the same answers. Surprisingly, the second-order approximations derived here yield the same approximation for dynamics at a single locus as first-order approximations used earlier thus justifying earlier stability computations based on these first-order approximations.