Reynold's genetic distance
Keywords: formules

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## Reynold's genetic distance for short-term evolution (1983) [Reynolds, J. 83]

DReynold = − ln ( 1 − θ ) DReynold = -ln (1 - %theta)
the following notation is used, for all 𝚹 estimators:
n ˉ = ∑ i = 1 r n i r , n c = ( r n ˉ − ∑ i = 1 r n i 2 r n ˉ ) ( r − 1 ) , p lu ˜ = ∑ i = 1 r n i p ilu ˜ r n ˉ , α il ˜ = 1 − ∑ u = 1 v 1 p ilu 2 ˜ bar n = sum _{i=1} ^r {{n_i} over r } , n_c = (r bar n - sum _{i=1} ^r {n^2 _i over {r bar n}}) over (r-1), tilde p _lu = sum _{i=1} ^r {n_i tilde p _ilu} over {r bar n}, tilde %alpha _il = 1 - sum ^{v_1} _{u=1} tilde p _ilu ^2

## unweighted average of single-locus ratio estimators, ̃𝛉 U:

θ U ˜ = 1 m ∑ l = 1 m θ l ˜ tilde %theta _U = 1 over m sum from l=1 to m tilde %theta _{l}
, with θ l = a l a l + b l %theta _{l} = {a_l} over {a_l + b_l} . The estimates of the components of variance of interest for the lth locus are within populations, r is the number of populations examined:
b l = 2 ∑ i = 1 r n i α il ˜ r ( 2 n ˉ − 1 ) b_l = 2 sum from {i=1} to r {{ n_i tilde %alpha _{il}} over {r(2 bar n - 1)}}
and between populations:
a l = [ 2 ∑ i = 1 r n i ∑ u = 1 v l ( p ilu ˜ − p lu ˜ ) 2 − ( r − 1 ) b l ] 2 ( r − 1 ) n c a_{l}= {[ 2 sum ^{r}_{i=1} n_{i} sum ^{v_{l}}_{u=1} ( tilde{p}_{ilu} - tilde{p}_{lu}) ^{2} - ( r - 1) b_{l}] } over {2 (r - 1) n_c}
When there are just two populations, r = 2, the usual genetic distance situation obtains, and the most convenient computing formulas for the variance components are (used in Populations)
a l = 1 2 ∑ u ( p 1lu ˜ − p 2lu ˜ ) 2 − ( n 1 + n 2 ) ( n 1 α 1l ˜ + n 2 α 2l ˜ ) 4n 1 n 2 ( n 1 + n 2 − 1 ) a_{l}={1} over {2} sum _{u} ( tilde{p}_{1lu}-tilde{p}_{2lu}) ^{2} - {( n_{1}+n_{2}) ( n_{1} tilde{%alpha }_{1l}+n_{2} tilde{%alpha }_{2l}) } over {4n_{1}n_{2} ( n_{1}+n_{2}-1 ) }
a l + b l = 1 2 ∑ u ( p 1lu ˜ − p 2lu ˜ ) 2 + ( 4n 1 n 2 − n 1 − n 2 ) ( n 1 α 1l ˜ + n 2 α 2l ˜ ) 4n 1 n 2 ( n 1 + n 2 − 1 ) a_{l}+b_{l} = {1} over {2} sum _{u} ( tilde{p}_{1lu}- tilde{p}_{2lu}) ^{2} + {( 4n_{1}n_{2}-n_{1}-n_{2}) ( n_{1} tilde{%alpha }_{1l}+n_{2} tilde{%alpha }_{2l}) } over {4n_{1}n_{2} ( n_{1}+n_{2}-1) }