Linkage disequilibrium

Linkage disequilibrium statistics are computed by two methods, stats.pairwise_LD() and stats.matrix_LD().

The available statistics are:

The statistics listed in the table below are available as methods or attributes of this class. The documentation provides more information regarding the usage.

Code

Statistic

Equation

Reference

D

Standard linkage disequilibrium

(1)

1

Dp

Lewontin’s \(D'\)

(2)

2

r

Correlation coefficient

(3)

3

rsq

Correlation coefficient

(3)

3

Reference

  1. Lewontin and Kojima (Evolution 1960 14:458-472).

  2. Lewontin (Genetics 1964 49:49-67).

  3. Hill and Robertson (Theor. Appl. Genet. 1968 38:226-231).

To compute linkage disequilibrium statistics, we assume pair of alleles at two different sites that are respectively at relative frequencies \(p_1\) and \(p_2\) while the genotype constituted by the two alleles is at frequency \(p_{12}\). The standard linkage disequilibrium is:

(1)\[D = p_{12} - p_1 p_2\]

The standardized linkage disequilibrum is computed as:

(2)\[D' = \frac{D}{k}\]

where:

  • \(k = p_1 p_2\) if \(D\) < 0 and \(p_1 p_2 < (1-p_1) (1-p_2)\),

  • \(k = (1-p_1) (1-p_2)\) if \(D\) < 0 and \(p_1 p_2 \ge (1-p_1) (1-p_2)\),

  • \(k = p_1 (1-p_2)\) if \(D\) > 0 and \(p_1 (1-p_2) < (1-p_1) p_2\), and

  • \(k = (1-p_1) p_2\) otherwise.

Finally, the pairwise correlation coefficient \(r^2\) is computed as follows:

(3)\[r^2 = \left( \frac{D}{\sqrt{p_1 p_2 (1-p_1) (1-p_2)}} \right) ^2\]