Kelly's Z n 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.
Details Given loci i {\displaystyle i} and j {\displaystyle j} , D i j {\displaystyle D_{ij}} the Linkage Disequilibrium between these loci, is denoted as
D i j = p i j − p i p j {\displaystyle D_{ij}=p_{ij}-p_{i}p_{j}}
where p i j {\displaystyle p_{ij}} is the frequency of the alternative allele at i and j co-occurring and p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} the frequency of the alternative allele at i {\displaystyle i} and j {\displaystyle j} respectively. a standardised measure of this is δ i j {\displaystyle \delta _{ij}} the squared correlation of allelic identity between loci i {\displaystyle i} and j {\displaystyle j}
δ i j = D i j 2 p i ( 1 − p i ) p j ( 1 − p j ) {\displaystyle \delta _{ij}={\frac {D_{ij}^{2}}{p_{i}(1-p_{i})p_{j}(1-p_{j})}}}
Where Z n S {\displaystyle Z_{nS}} averages δ i j {\displaystyle \delta _{ij}} over all pairwise combinations between S loci.
Z n S = 2 S ( S − 1 ) ∑ i = 1 S − 1 ∑ j = i + 1 S δ i j {\displaystyle Z_{nS}={\frac {2}{S(S-1)}}\sum _{i=1}^{S-1}\sum _{j=i+1}^{S}\delta _{ij}}
Usage Inflated Z n S {\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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