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) }