Kelly's ZnS

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Kelly's ${\displaystyle Z_{nS}}$ 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.^[1]

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

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

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

a standardised measure of this is ${\displaystyle \delta _{ij}}$ the squared correlation of allelic identity between loci ${\displaystyle i}$ and ${\displaystyle j}$^[2]

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

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

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

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

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