The $D$ Statistic
For two biallelic loci with alleles $A/a$ (frequencies $p_A, p_a$) and $B/b$ (frequencies $p_B, p_b$), the disequilibrium coefficient is:
\[D = p_{AB} - p_A p_B\]where $p_{AB}$ is the observed haplotype frequency. Under linkage equilibrium $D = 0$. $D$ ranges between $D_{\min}$ and $D_{\max}$, which depend on allele frequencies.
Normalised Measures
Two standardised measures dominate the literature:
\[D' = \frac{D}{D_{\max}} \quad \text{if } D > 0, \qquad D' = \frac{D}{D_{\min}} \quad \text{if } D < 0\] \[r^2 = \frac{D^2}{p_A\, p_a\, p_B\, p_b}\]$|D’| = 1$ indicates no recombination has been observed between the haplotypes; $r^2 = 1$ indicates complete allelic correlation (perfect proxies). $r^2$ is the more useful metric for GWAS power because it directly determines the loss of power when using a proxy SNP: effective sample size scales as $N \cdot r^2$.
Decay with Recombination
LD decays geometrically each generation under random mating:
\[D_t = D_0\,(1 - c)^t\]where $c$ is the recombination fraction between loci and $t$ is the number of generations. At $c = 0.5$ (unlinked loci), LD halves each generation. In humans, substantial LD typically extends 10–100 kb.
Haplotype Blocks and LD Pruning
| Operation | Definition | ||
|---|---|---|---|
| Haplotype block | Genomic region with limited historical recombination and high internal $ | D’ | $ |
| LD pruning | Remove SNPs within a window until no pair exceeds an $r^2$ threshold (e.g., $r^2 > 0.1$) | ||
| LD clumping | Keep the most-significant SNP per locus; assign nearby SNPs at $r^2 > $ threshold to it |
LD-independent SNP sets from pruning are used for PCA, heritability estimation, and relatedness calculation.