I am trying to understand the concept of heritability and from what I can gather, the heritability of a factor (say birth weight) must be closely related to the correlation coefficient of that factor when you do a linear regression of the factor between parent (say mother) and child.
So my question is this - is it possible to calculate the heritability from the correlation coefficient, and if so, what is the formula.
No. But a strong correlation may nevertheless be an indicator for heritability (it makes it more likely).
The reason is, that the correlation in the phenotype (birth weight) is not only due a correlation in the genotype but also a correlation in the environment of the child and its mother (culture, economical situation, genes, etc of their respective mothers).
If you were able to remove the association of the environments and of the genotypes and the environments by an intervention (eg. shuffling the zygotes (ideally of parents who were shuffled as zygotes as well)) then the correlation coefficient of the phenotype between one of two parents and the child would be half the heritability. $corr=\frac{1}{2} H^2$
Phenotype ($P$), Genotype ($G$), Environment ($E$)
$P_{parent}=G_{parent}+E_{parent}$
$P_{child}=G_{child}+E_{child}$
$ \begin{eqnarray*} corr(P_{parent},P_{child}) &=& \frac{COV(P_{parent},P_{child})}{\sqrt{VAR(P_{parent})}\cdot \sqrt{VAR(P_{child})}} \\ &=& \frac{COV(P_{parent},P_{child})}{ VAR(P)}\\ &=& \frac{COV(G_{par}+E_{par},G_{chi}+E_{chi})}{ VAR(P)}\\ &=& \frac{COV(G_{p},G_{c})+ COV(G_{p},E_{c})+ COV(E_{p},G_{c})+ COV(E_{p},E_{c})}{VAR(P)}\\ &=& \frac{\frac{1}{2}VAR(G)+ COV(G_{p},E_{c})+ COV(E_{p},G_{c})+ COV(E_{p},E_{c})}{VAR(P)}\\ &\overset{(1)}{=}& \frac{\frac{1}{2}VAR(G)}{VAR(P)}\\ &\overset{(2)}{=}&\frac{1}{2}H^2 \end{eqnarray*} $
$(1): \,\,\,$if you managed to get $COV(G_{p},E_{c}), COV(E_{p},G_{c}),COV(E_{p},E_{c}) = 0 $
$(2): \,\,\,H^2=\frac{VAR(G)}{VAR(P)}$ - the definition of heritability
