Editing Quantitative genetics (section)

====Cousins crossings====
[[File:Inbreeding- Cousins First.jpg|thumb|175px|right|Pedigree analysis first cousins]]

These are derived with methods similar to those for siblings.<ref name="Crow & Kimura"/>{{rp|132–143}}<ref name="Falconer 1996"/>{{rp|82–92}} As before, the ''co-ancestry'' viewpoint of the ''inbreeding coefficient'' provides a measure of "relatedness" between the parents '''P1''' and '''P2''' in these cousin expressions.

The pedigree for ''First Cousins (FC)'' is given to the right. The prime equation is '''f<sub>Y</sub> = f<sub>t</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>1D</sub> + f<sub>12</sub> + f<sub>CD</sub> + f<sub>C2</sub> ]'''. After substitution with corresponding inbreeding coefficients, gathering of terms and simplifying, this becomes ''' f<sub>t</sub> = (1/4) <nowiki>[</nowiki> 3 f<sub>(t-1)</sub> + (1/4) <nowiki>[</nowiki>2 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 <nowiki>]]</nowiki> ''', which is a version for iteration—useful for observing the general pattern, and for computer programming. A "final" version is ''' f<sub>t</sub> = (1/16) [ 12 f<sub>(t-1)</sub> + 2 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 ] '''. [[File:Inbreeding- Cousins Second.jpg|thumb|175px|left|Pedigree analysis second cousins]]

The ''Second Cousins (SC)'' pedigree is on the left. Parents in the pedigree not related to the ''common Ancestor'' are indicated by numerals instead of letters. Here, the prime equation is ''' f<sub>Y</sub> = f<sub>t</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>3F</sub> + f<sub>34</sub> + f<sub>EF</sub> + f<sub>E4</sub> ]'''. After working through the appropriate algebra, this becomes ''' f<sub>t</sub> = (1/4) <nowiki>[</nowiki> 3 f<sub>(t-1)</sub> + (1/4) <nowiki>[</nowiki>3 f<sub>(t-2)</sub> + (1/4) <nowiki>[</nowiki>2 f<sub>(t-3)</sub> + f<sub>(t-4)</sub> + 1 <nowiki>]]]</nowiki> ''', which is the iteration version. A "final" version is ''' f<sub>t</sub> = (1/64) [ 48 f<sub>(t-1)</sub> + 12 f<sub>(t-2)</sub> + 2 f<sub>(t-3)</sub> + f<sub>(t-4)</sub> + 1 ] '''. [[File:Cousin Inbreeding.jpg|thumb|250px|left|Inbreeding from several levels of cousin crossing.]]

To visualize the ''pattern in full cousin'' equations, start the series with the ''full sib'' equation re-written in iteration form: ''' f<sub>t</sub> = (1/4)[2 f<sub>(t-1)</sub> + f<sub>(t-2)</sub> + 1 ]'''. Notice that this is the "essential plan" of the last term in each of the cousin iterative forms: with the small difference that the generation indices increment by "1" at each cousin "level". Now, define the ''cousin level'' as '''k = 1''' (for First cousins), '''= 2''' (for Second cousins), '''= 3''' (for Third cousins), etc., etc.; and '''= 0''' (for Full Sibs, which are "zero level cousins"). The ''last term'' can be written now as: ''' (1/4) [ 2 f<sub>(t-(1+k))</sub> + f<sub>(t-(2+k))</sub> + 1] '''. Stacked in front of this ''last term'' are one or more ''iteration increments'' in the form '''(1/4) [ 3 f<sub>(t-j)</sub> + ... ''', where '''j''' is the ''iteration index'' and takes values from '''1 ... k''' over the successive iterations as needed. Putting all this together provides a general formula for all levels of ''full cousin'' possible, including ''Full Sibs''. For '''k'''th ''level'' full cousins, '''f{k}<sub>t</sub> = ''Ιter''<sub>j = 1</sub><sup>k</sup> { (1/4) [ 3 f<sub>(t-j)</sub> + }<sub>j</sub> + (1/4) [ 2 f<sub>(t-(1+k))</sub> + f<sub>(t-(2+k))</sub> + 1] '''. At the commencement of iteration, all f<sub>(t-''x'')</sub> are set at "0", and each has its value substituted as it is calculated through the generations. The graphs to the right show the successive inbreeding for several levels of Full Cousins. 
[[File:Inbreeding- Csns Half.jpg|thumb|200px|left|Pedigree analysis half cousins]]

For ''first half-cousins (FHC)'', the pedigree is to the left.  Notice there is just one common ancestor (individual '''A'''). Also, as for ''second cousins'', parents not related to the common ancestor are indicated by numerals. Here, the prime equation is ''' f<sub>Y</sub> = f<sub>t</sub> = f<sub>P1,P2</sub> = (1/4) [ f<sub>3D</sub> + f<sub>34</sub> + f<sub>CD</sub> + f<sub>C4</sub> ]'''. After working through the appropriate algebra, this becomes ''' f<sub>t</sub> = (1/4) <nowiki>[</nowiki> 3 f<sub>(t-1)</sub> + (1/8) <nowiki>[</nowiki>6 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 <nowiki>]]</nowiki> ''', which is the iteration version. A "final" version is ''' f<sub>t</sub> = (1/32) [ 24 f<sub>(t-1)</sub> + 6 f<sub>(t-2)</sub> + f<sub>(t-3)</sub> + 1 ] '''. The iteration algorithm is similar to that for ''full cousins'', except that the last term is '''(1/8) [ 6 f<sub>(t-(1+k))</sub> + f<sub>(t-(2+k))</sub> + 1 ] '''. Notice that this last term is basically similar to the half sib equation, in parallel to the pattern for full cousins and full sibs. In other words, half sibs are "zero level" half cousins.

There is a tendency to regard cousin crossing with a human-oriented point of view, possibly because of a wide interest in Genealogy. The use of pedigrees to derive the inbreeding perhaps reinforces this "Family History" view. However, such kinds of inter-crossing occur also in natural populations—especially those that are sedentary, or have a "breeding area" that they re-visit from season to season. The progeny-group of a harem with a dominant male, for example, may contain elements of sib-crossing, cousin crossing, and backcrossing, as well as genetic drift, especially of the "island" type. In addition to that, the occasional "outcross" adds an element of hybridization to the mix. It is ''not'' panmixia.