Editing Quantitative genetics (section)

==== Applications (parent-offspring)====
The most obvious application is an experiment that contains all parents and their offspring, with or without reciprocal crosses, preferably replicated without bias, enabling estimation of all appropriate means, variances and covariances,  together with their standard errors. These estimated statistics can then be used to estimate the genetic variances. Twice ''the difference between the estimates of the two forms of (corrected) parent-offspring covariance'' provides an estimate of '''s<sup>2</sup><sub>D</sub>'''; and twice the ''cov(MPO)'' estimates '''s<sup>2</sup><sub>A</sub>'''. With appropriate experimental design and analysis,<ref name="S & T"/><ref name="Snedecor & Cochran"/><ref name="Kendall & Stuart"/> standard errors can be obtained for these genetical statistics as well. This is the basic core of an experiment known as ''Diallel analysis'', the Mather, Jinks and Hayman version of which is discussed in another section.

A second application involves using ''regression analysis'', which estimates from statistics the ordinate (Y-estimate), derivative (regression coefficient) and constant (Y-intercept) of calculus.<ref name="S & T"/><ref name="Snedecor & Cochran"/><ref name="Draper & Smith">{{cite book|last1=Draper|first1=Norman R.|last2=Smith|first2=Harry|title=Applied regression analysis.|date=1981|publisher=John Wiley & Sons|location=New York|isbn=0-471-02995-5|edition=Second|url-access=registration|url=https://archive.org/details/appliedregressio0000drap}}</ref><ref name="Balaam 1972"/> The ''regression coefficient'' estimates the ''rate of change'' of the function predicting '''Y''' from '''X''', based on minimizing the residuals between the fitted curve and the observed data (MINRES). No alternative method of estimating such a function satisfies this basic requirement of MINRES.  In general, the regression coefficient is estimated as ''the ratio of the covariance(XY) to the variance of the determinator (X)''. In practice, the sample size is usually the same for both X and Y, so this can be written as '''SCP(XY) / SS(X)''', where all terms have been defined previously.<ref name="S & T"/><ref name="Draper & Smith"/><ref name="Balaam 1972"/> In the present context, the parents are viewed as the "determinative variable" (X), and the offspring as the "determined variable" (Y), and the regression coefficient as the "functional relationship" (ß<sub>PO</sub>) between the two. Taking '''cov(MPO) = {{sfrac|1|2}} s<sup>2</sup><sub>A</sub> ''' as '''cov(XY)''', and ''' s<sup>2</sup><sub>P</sub> / 2 ''' (the variance of the mean of two parents—the mid-parent) as '''s<sup>2</sup><sub>X</sub>''', it can be seen that '''ß<sub>MPO</sub> = [{{sfrac|1|2}} s<sup>2</sup><sub>A</sub>] / [{{sfrac|1|2}} s<sup>2</sup><sub>P</sub>] = h<sup>2</sup> '''.<ref>In the past, both forms of parent-offspring covariance have been applied to this task of estimating '''h<sup>2</sup>''', but, as noted in the sub-section above, only one of them ('''cov(MPO)''') is actually appropriate. The '''cov(PO)''' is useful, however, for estimating '''H<sup>2</sup>''' as seen in the main text following.</ref> Next, utilizing '''cov(PO) = [ {{sfrac|1|2}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|2}} s<sup>2</sup><sub>D</sub> ]''' as '''cov(XY)''', and ''' s<sup>2</sup><sub>P</sub>''' as '''s<sup>2</sup><sub>X</sub>''', it is seen that ''' 2 ß<sub>PO</sub> = [ 2 ({{sfrac|1|2}} s<sup>2</sup><sub>A</sub> + {{sfrac|1|2}} s<sup>2</sup><sub>D</sub> )] / s<sup>2</sup><sub>P</sub> = H<sup>2</sup> '''.

Analysis of ''epistasis'' has previously been attempted via an ''interaction variance'' approach of the type '' s<sup>2</sup><sub>AA</sub> '', and '' s<sup>2</sup><sub>AD</sub>'' and also '' s<sup>2</sup><sub>DD</sub>''. This has been integrated with these present covariances in an effort to provide estimators for the epistasis variances. However, the findings of epigenetics suggest that this may not be an appropriate way to define epistasis.