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