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= Kelly's ZnS = Kellys $$Z_{nS} $$ 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

$$D_{ij} = p_{ij}-p_ip_j $$

where $$p_{ij} $$ is the frequency of the alternative allele at i and j co-ocurring and 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 the squared correlation of allelic identity between loci $$i $$ and $$j $$

$$\delta_{ij} = \frac{D_{ij}}{p_i(1 - p_i)p_j (1-p_j)}$$

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