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

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