Regression analysis

Measure the parents. Regress the offspring phenotype on: (1) the phenotype of one parent or (2) the mean of the parental parental phenotypes. In either case, the covariance between the parental phenotype and the offspring genotype is $\left({1 \over 2}\right)V_a$. Now the regression coefficient between one parent and offspring, $b_{P \rightarrow O}$, is

$$\begin{eqnarray*} b_{P \rightarrow O} &=& \frac{\mbox{Cov}_{PO}}{\mbox{Var}(P)... ...t)V_a \over V_p} \\ &=& \left({1 \over 2}\right)h^2_N \quad . \end{eqnarray*}$$

In short, the slope of the regression line is equal to one-half the narrow sense heritability. In the regression of offspring on mid-parent value,

$$\begin{eqnarray*} \mbox{Var}(MP) &=& \mbox{Var}\left(\frac{M+F}{2}\right) \\ ... ...\frac{1}{4} \left(2V_p\right) \\ &=& \frac{1}{2} V_p \quad . \end{eqnarray*}$$

Thus, $b_{MP \rightarrow O} = \frac{1}{2}V_a/\frac{1}{2}V_p = h^2_N$. In short, the slope of the regression line is equal to the narrow sense heritability.