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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?

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    $\begingroup$ none of your references seem to have links? $\endgroup$
    – shigeta
    Apr 24, 2014 at 17:12
  • $\begingroup$ Yeah, where are the links? $\endgroup$
    – Did
    Aug 24, 2014 at 8:45
  • $\begingroup$ Thanks for noticing. don't know what happened to them. I added the links. $\endgroup$
    – Remi.b
    Aug 24, 2014 at 18:17

1 Answer 1

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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.

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