Editing Quantitative genetics (section)

====Cross-multiplication rules====
In the following sections on sib-crossing and similar topics, a number of "averaging rules" are useful. These derive from [[Path analysis (statistics)|path analysis]].<ref name="Li 1977"/>  The rules show that any co-ancestry coefficient can be obtained as the average of ''cross-over co-ancestries'' between appropriate grand-parental and parental combinations. Thus, referring to the adjacent diagram, ''Cross-multiplier 1'' is that '''f<sub>PQ</sub>''' = average of ( '''f<sub>AC</sub>''', '''f<sub>AD</sub>''', '''f<sub>BC</sub>''', '''f<sub>BD</sub>''' ) = ''' (1/4) [f<sub>AC</sub> + f<sub>AD</sub> + f<sub>BC</sub> + f<sub>BD</sub> ] ''' = '''f<sub>Y</sub> '''.  In a similar fashion, ''cross-multiplier 2'' states that '''f<sub>PC</sub> = (1/2) [ f<sub>AC</sub> + f<sub>BC</sub> ]'''—while ''cross-multiplier 3'' states that '''f<sub>PD</sub> = (1/2) [ f<sub>AD</sub> + f<sub>BD</sub> ]''' . Returning to the first multiplier, it can now be seen also to be '''f<sub>PQ</sub> = (1/2) [ f<sub>PC</sub> + f<sub>PD</sub> ]''', which, after substituting multipliers 2 and 3, resumes its original form.

In much of the following, the grand-parental generation is referred to as '''(t-2)''', the parent generation as '''(t-1)''', and the "target" generation as '''t'''.