Kelly's ZnS

Kelly's $Z_{nS}$ is a test statistic that can be used to test a genetic region for deviations from the neutral model, based on the squared correlation of allelic identity between loci.

Details

Given loci $i$ and $j$, $D_{ij}$ the Linkage Disequilibrium between these loci, is denoted as

$D_{ij}=p_{ij}-p_{i}p_{j}$

where $p_{ij}$ is the frequency of the alternative allele at i and j co-occurring and $p_{i}$ and $p_{j}$ the frequency of the alternative allele at $i$ and $j$ respectively.

a standardised measure of this is $\delta _{ij}$ the squared correlation of allelic identity between loci $i$ and $j$

$\delta _{ij}={\frac {D_{ij}^{2}}{p_{i}(1-p_{i})p_{j}(1-p_{j})}}$

Where $Z_{nS}$ averages $\delta _{ij}$ over all pairwise combinations between S loci.

$Z_{nS}={\frac {2}{S(S-1)}}\sum _{i=1}^{S-1}\sum _{j=i+1}^{S}\delta _{ij}$

Usage

Inflated $Z_{nS}$ scores indicate a deviation from the neutral model and can be used as a potential signature of previous selection