Editing Quantitative genetics (section)

==== Gene effects redefined====
The gene-model effects ('''a''', '''d''' and '''-a''') are important soon in the derivation of the ''deviations from substitution'', which were first discussed in the previous ''Allele Substitution'' section. However, they need to be redefined themselves before they become useful in that exercise. They firstly need to be re-centralized around the population mean ('''G'''), and secondly they need to be re-arranged as functions of '''β''', the ''average allele substitution effect''.

Consider firstly the re-centralization. The re-centralized effect for '''AA''' is '''a• = a - G''' which, after simplification, becomes '''a• = 2''q''(a-''p''d)'''. The similar effect for '''Aa''' is '''d• = d - G = a(''q''-''p'') + d(1-2''pq'')''', after simplification. Finally, the re-centralized effect for '''aa''' is '''(-a)• = -2''p''(a+''q''d)'''.<ref name="Falconer 1996"/>{{rp|116–119}}

Secondly, consider the re-arrangement of these re-centralized effects as functions of '''β'''. Recalling from the "Allele Substitution" section that β = [a +(q-p)d], rearrangement gives '''a = [β -(q-p)d]'''. After substituting this for '''a''' in '''a•''' and simplifying, the final version becomes '''a•• = 2q(β-qd)'''. Similarly, '''d•''' becomes '''d•• = β(q-p) + 2pqd'''; and  '''(-a)•''' becomes '''(-a)•• = -2p(β+pd)'''.<ref name="Falconer 1996"/>{{rp|118}}