The additive effect of an allele

In constructing Table 1 I used the quantities $\alpha_1$ and $\alpha_2$, but I didn't tell you where they came from. Obviously, the idea should be to pick values of $\alpha_1$ and $\alpha_2$ that give additive genotypic values that are reasonably close to the genotypic values. A good way to do that is to minimize the squared deviation between the two, weighted by the frequency of the genotypes. So our first big assumption is that genotypes are in Hardy-Weinberg proportions.¹⁰

The objective is to find values for $\alpha_1$ and $\alpha_2$ that minimize:

$$a = p^2[x_{11}-2\alpha_1]^2 + 2pq[x_{12}-(\alpha_1+\alpha_2)]^2 + q^2[x_{22}-2\alpha_2]^2 \quad .$$

To do this we take the partial derivative of $a$ with respect to both $\alpha_1$ and $\alpha_2$, set the resulting pair of equations equal to zero, and solve for $\alpha_1$ and $\alpha_2$.¹¹

$$\begin{eqnarray*} \frac{\partial a}{\partial{\alpha_1}} &=& p^2\{2[x_{11} - 2\a... ...-4q^2[x_{22} - 2\alpha_2] -4pq[x_{12} - (\alpha_1+\alpha_2)] \end{eqnarray*}$$

Thus, $\frac{\partial a}{\partial{\alpha_1}} = \frac{\partial a}{\partial{\alpha_2}} = 0$ if and only if

$$p^2(x_{11} - 2\alpha_1) + pq(x_{12} - \alpha_1 - \alpha_2) = 0$$

$$q^2(x_{22} - 2\alpha_2) + pq(x_{12} - \alpha_1 - \alpha_2) = 0$$ (1)

Adding the equations in (1) we obtain (after a little bit of rearrangement)

$$[p^2x_{11} + 2pqx_{12} + q^2x_{22}]- [p^2(2\alpha_1) + 2pq(\alpha_1 + \alpha_2) + q^2(2\alpha_2)] = 0 \quad .$$ (2)

Now the first term in square brackets is just the mean phenotype in the population, $\bar x$. Thus, we can rewrite equation (2) as:

$${\bar x} = 2p^2\alpha_1 + 2pq(\alpha_1 + \alpha_2) +2q^2\alpha_2$$

$$= 2p\alpha_1(p+q) + 2q\alpha_2(p+q)$$

$$= 2(p\alpha_1 + q\alpha_2) \quad .$$ (3)

Now divide the first equation in (1) by $p$ and the second by $q$.

$$p(x_{11} - 2\alpha_1) + q(x_{12} - \alpha_1 - \alpha_2) = 0$$ (4)

$$q(x_{22} - 2\alpha_2) + p(x_{12} - \alpha_1 - \alpha_2) = 0 \quad .$$ (5)

Thus,

$$\begin{eqnarray*} px_{11} + qx_{12} &=& 2p\alpha_1 + q\alpha_1 + q\alpha_2 \\ ... ...r x}/2 \\ \alpha_1 &=& px_{11} + qx_{12} - {\bar x}/2 \quad . \end{eqnarray*}$$

Similarly,

$$\begin{eqnarray*} px_{12} + qx_{22} &=& 2q\alpha_2 + p\alpha_1 + p\alpha_2 \\ ... ...r x}/2 \\ \alpha_2 &=& px_{12} + qx_{22} - {\bar x}/2 \quad . \end{eqnarray*}$$

$\alpha_1$ is the additive effect of allele $A_1$, and $\alpha_2$ is the additive effect of allele $A_2$. If we use these expressions, the additive genotypic values are as close to the genotypic values as possible, given the particular allele freequencies in the population.¹²