# Definition of Linkage Desiquilibrium (LD)

According to wiki, linkage disequilibrium $D$ equals

$$D = x_{11} - p_1\cdot q_1$$

where:

$$\begin{matrix} \text{Haplotype} & \text{Frequency}\\ A_1B_1 & x_{11}\\ A_{1}B_{2} & x_{12} \\ A_{2}B_{1} & x_{21} \\ A_{2}B_{2} & x_{22} \\ \end{matrix}$$

and

$$\begin{matrix} \text{Allele} & \text{Frequency}\\ A_{1} & p_{1}=x_{11}+x_{12} \\ A_{2} & p_{2}=x_{21}+x_{22} \\ B_{1} & q_{1}=x_{11}+x_{21} \\ B_{2} & q_{2}=x_{12}+x_{22} \\ \end{matrix}$$

According to Hartl and Clark, linkage disequilibrium $D$ equals:

$$D = x_{11}x_{22}-x_{12}x_{21}$$

Question

Can you please prove that these two formulations of linkage disequilibrium are equivalent (assuming that they are equivalent)? If they're not… Do we use different definitions? What are their respective meanings?

• none of your references seem to have links? Apr 24 '14 at 17:12
• Yeah, where are the links?
– Did
Aug 24 '14 at 8:45
• Thanks for noticing. don't know what happened to them. I added the links. Aug 24 '14 at 18:17

Just need to solve the equation. p1 = X11 + X12; q1 = X11 + X21; 1 = X11 + X12 + X21 + X22.

D = X11 - (X11 + X12) * (X11 + X21)

D = X11 - (X11X11 + X11X21 + X11X12 + X12X21)

D = X11 - X11X11 - X11X21 - X11X12 - X12X21

D = X11 * (1 - X11) - X11X21 - X11X12 - X12X21

D = X11 * (X11 + X12 + X21 + X22 - X11) - X11X21 - X11X12 - X12X21

D = X11 * (X12 + X21 + X22) - X11X21 - X11X12 - X12X21

D = X11X12 + X11X21 + X11X22 - X11X21 - X11X12 - X12X21

D = X11X22 - X12X21

Sorry for the very long answer, but I thought it would be more clear if I wrote everything out.