Kelly's ZnS explained

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.^[1]

Details

Given loci

i and j , D _{ij} the Linkage Disequilibrium between these loci, is denoted asD _{ij} = p _{ij} - p _{ip} _{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 ^[2] \delta _{ij} = (D ^{2} _{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} = (2)/(S(S -1 )) \sum ^{S -1} _{i =1} \sum ^{S} _{j = i +1} \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

Notes and References

1. Kelly. J. K.. July 1997. A test of neutrality based on interlocus associations. Genetics. 146. 3. 1197–1206. 10.1093/genetics/146.3.1197 . 0016-6731. 1208047. 9215920.
2. Book: Hartl, Daniel L.. Principles of population genetics. 1989. Sinauer Associates. Andrew G. Clark. 0-87893-302-6. 2nd. Sunderland, Mass.. 18558351.