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Quantitative genetics
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===Siblings covariances=== Covariance between half-sibs ('''HS''') is defined easily using allele-substitution methods; but, once again, the dominance contribution has historically been omitted. However, as with the mid-parent/offspring covariance, the covariance between full-sibs ('''FS''') requires a "parent-combination" approach, thereby necessitating the use of the gene-model corrected-cross-product method; and the dominance contribution has not historically been overlooked. The superiority of the gene-model derivations is as evident here as it was for the Genotypic variances. ====Half-sibs of the same common-parent (HS)==== The sum of the cross-products '''{ common-parent frequency * half-breeding-value of one half-sib * half-breeding-value of any other half-sib in that same common-parent-group }''' immediately provides one of the required covariances, because the effects used [''breeding values''—representing the allele-substitution expectations] are already defined as deviates from the genotypic mean [see section on "Allele substitution – Expectations and deviations"]. After simplification. this becomes: ''' cov(HS)<sub>A</sub> = {{sfrac|1|2}} pq a<sup>2</sup> = {{sfrac|1|4}} s<sup>2</sup><sub>A</sub> '''.<ref name="Crow & Kimura"/> {{rp|132–141}}<ref name="Falconer 1996"/> {{rp|134–147}} However, the ''substitution deviations'' also exist, defining the sum of the cross-products '''{ common-parent frequency * half-substitution-deviation of one half-sib * half-substitution-deviation of any other half-sib in that same common-parent-group }''', which ultimately leads to: ''' cov(HS)<sub>D</sub> = p<sup>2</sup> q<sup>2</sup> d<sup>2</sup> = {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> '''. Adding the two components gives: '''cov(HS) = cov(HS)<sub>A</sub> + cov(HS)<sub>D</sub> = {{sfrac|1|4}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> '''. ====Full-sibs (FS)==== As explained in the introduction, a method similar to that used for mid-parent/progeny covariance is used. Therefore, an ''unadjusted sum of cross-products'' (USCP) using all products—{''' parent-pair-frequency * the square of the offspring-genotype-mean '''}—is adjusted by subtracting the '''{overall genotypic mean}<sup>2</sup> ''' as ''correction factor (CF)''. In this case, multiplying out all combinations, carefully gathering terms, simplifying, factoring, and cancelling-out is very protracted. It eventually becomes: '''cov(FS) = pq a<sup>2</sup> + p<sup>2</sup> q<sup>2</sup> d<sup>2</sup> = {{sfrac|1|2}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> ''', with no dominance having been overlooked.<ref name="Crow & Kimura"/> {{rp|132–141}}<ref name="Falconer 1996"/> {{rp|134–147}} ====Applications (siblings)==== The most useful application here for genetical statistics is the ''correlation between half-sibs''. Recall that the correlation coefficient (''r'') is the ratio of the covariance to the variance [see section on "Associated attributes" for example]. Therefore, ''' r<sub>HS</sub> = cov(HS) / s<sup>2</sup><sub>all HS together</sub> ''' = ''' [{{sfrac|1|4}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> ] / s<sup>2</sup><sub>P</sub> = {{sfrac|1|4}} H<sup>2</sup> '''.<ref>Note that texts that ignore the dominance component of cov(HS) erroneously suggest that r<sub>HS</sub> "approximates" ( {{sfrac|1|4}} h<sup>2</sup> ).</ref> The correlation between full-sibs is of little utility, being ''' r<sub>FS</sub> = cov(FS) / s<sup>2</sup><sub>all FS together</sub> ''' = ''' [{{sfrac|1|2}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|4}} s<sup>2</sup><sub>D</sub> ] / s<sup>2</sup><sub>P</sub> '''. The suggestion that it "approximates" (''{{sfrac|1|2}} h<sup>2</sup>'') is poor advice. Of course, the correlations between siblings are of intrinsic interest in their own right, quite apart from any utility they may have for estimating heritabilities or genotypic variances. It may be worth noting that '''[ cov(FS) − cov(HS)] = {{sfrac|1|4}} s<sup>2</sup><sub>A</sub> '''. Experiments consisting of FS and HS families could utilize this by using intra-class correlation to equate experiment variance components to these covariances [see section on "Coefficient of relationship as an intra-class correlation" for the rationale behind this]. The earlier comments regarding epistasis apply again here [see section on "Applications (Parent-offspring"].
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