NOTATIONS
We denote the following to describe our methods:
U
i
,
k
{\displaystyle U_{i,k}}
is the score statistic for the
i
t
h
{\displaystyle i^{th}}
variant from the
k
t
h
{\displaystyle k^{th}}
study
V
i
j
,
k
{\displaystyle V_{ij,k}}
is the covariance of the score statistics between the
i
t
h
{\displaystyle i^{th}}
and the
j
t
h
{\displaystyle j^{th}}
variant from the
k
t
h
{\displaystyle k^{th}}
study
U
k
{\displaystyle \mathbf {U_{k}} }
is the vector of score statistics of rare variants in a gene from the
k
t
h
{\displaystyle k^{th}}
study.
V
k
{\displaystyle \mathbf {V_{k}} }
is the variance-covariance matrix of score statistics of rare variants in a gene from the
k
t
h
{\displaystyle k^{th}}
study, or
V
k
=
c
o
v
(
U
k
)
{\displaystyle \mathbf {V_{k}} =cov(\mathbf {U_{k}} )}
S
{\displaystyle S}
is the number of studies
U
i
,
k
{\displaystyle U_{i,k}}
and
V
i
j
,
k
{\displaystyle V_{ij,k}}
are described in detail in RAREMETALWORKER method .
SINGLE VARIANT META ANALYSIS
Single variant meta-analysis score statistic can be reconstructed from score statistics and their variances generate by each study, assuming that samples are unrelated across studies. Define meta-analysis score statistics as
U
m
e
t
a
,
i
=
∑
k
=
1
S
U
i
,
k
{\displaystyle U_{meta,i}=\sum _{k=1}^{S}{U_{i,k}}}
and its variance
V
m
e
t
a
,
i
=
∑
k
=
1
S
V
i
i
,
k
{\displaystyle V_{meta,i}=\sum _{k=1}^{S}{V_{ii,k}}}
Then the score test statistics for the
i
t
h
{\displaystyle i^{th}}
variant
T
m
e
t
a
,
i
{\displaystyle T_{meta,i}}
asymptotically follows standard normal distribution
T
m
e
t
a
,
i
=
U
m
e
t
a
,
i
/
V
m
e
t
a
,
i
=
∑
k
=
1
S
U
i
,
k
/
∑
k
=
1
S
V
i
i
,
k
∼
N
(
0
,
1
)
{\displaystyle T_{meta,i}=U_{meta,i}{\bigg /}{\sqrt {V_{meta,i}}}=\sum _{k=1}^{S}{U_{i,k}}{\bigg /}{\sqrt {\sum _{k=1}^{S}{V_{ii,k}}}}\sim \mathbf {N} (0,1)}
BURDEN META ANALYSIS
VT META ANALYSIS
SKAT META ANALYSIS
Formulae for RAREMETAL
Test
Statistics
Null Distribution
Notation
Single Variant
T
=
∑
i
=
1
n
U
i
/
∑
i
=
1
n
V
i
{\displaystyle T=\sum _{i=1}^{n}{U_{i}}{\bigg /}{\sqrt {\sum _{i=1}^{n}{V_{i}}}}}
T
∼
N
(
0
,
1
)
{\displaystyle T\sim \mathbf {N} (0,1)}
U
i
is the score statistic from study
i
;
{\displaystyle U_{i}{\text{ is the score statistic from study }}i;}
V
i
is the variance of
U
i
.
{\displaystyle V_{i}{\text{ is the variance of }}U_{i}.}
un-weighted Burden
T
b
=
∑
i
=
1
n
U
i
/
∑
i
=
1
n
V
i
{\displaystyle T_{b}=\sum _{i=1}^{n}{\mathbf {U_{i}} }{\Big /}{\sqrt {\sum _{i=1}^{n}{\mathbf {V_{i}} }}}}
T
b
∼
N
(
0
,
1
)
{\displaystyle T_{b}\sim \mathbf {N} (0,1)}
U
i
is the vector of score statistics from study
i
,
o
r
{\displaystyle \mathbf {U_{i}} {\text{ is the vector of score statistics from study }}i,or}
U
i
=
{
U
i
1
,
.
.
.
,
U
i
m
}
;
{\displaystyle \mathbf {U_{i}} =\{U_{i1},...,U_{im}\};}
V
i
is the covariance of
U
i
.
{\displaystyle \mathbf {V_{i}} {\text{ is the covariance of }}\mathbf {U_{i}} .}
Weighted Burden
T
w
b
=
w
T
∑
i
=
1
n
U
i
/
w
T
(
∑
i
=
1
n
V
i
)
w
{\displaystyle T_{wb}=\mathbf {w^{T}} \sum _{i=1}^{n}{\mathbf {U_{i}} }{\bigg /}{\sqrt {\mathbf {w^{T}} \left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right)\mathbf {w} }}}
T
w
b
∼
N
(
0
,
1
)
{\displaystyle T_{wb}\sim \mathbf {N} (0,1)}
w
T
=
{
w
1
,
w
2
,
.
.
.
,
w
m
}
T
is the weight vector.
{\displaystyle \mathbf {w^{T}} =\{w_{1},w_{2},...,w_{m}\}^{T}{\text{ is the weight vector.}}}
VT
T
V
T
=
max
(
T
b
(
f
1
)
,
T
b
(
f
2
)
,
…
,
T
b
(
f
m
)
)
,
where
{\displaystyle T_{VT}=\max(T_{b\left(f_{1}\right)},T_{b\left(f_{2}\right)},\dots ,T_{b\left(f_{m}\right)}),{\text{ where}}}
T
b
(
f
j
)
=
ϕ
f
j
T
∑
i
=
1
n
U
i
/
ϕ
f
j
T
(
∑
i
=
1
n
V
i
)
ϕ
f
j
{\displaystyle T_{b\left(f_{j}\right)}={\boldsymbol {\phi }}_{f_{j}}^{\mathbf {T} }\sum _{i=1}^{n}{\mathbf {U_{i}} }{\bigg /}{\sqrt {{\boldsymbol {\phi }}_{f_{j}}^{\mathbf {T} }\left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right){\boldsymbol {\phi }}_{f_{j}}}}}
(
T
b
(
f
1
)
,
T
b
(
f
2
)
,
…
,
T
b
(
f
m
)
)
{\displaystyle \left(T_{b\left(f_{1}\right)},T_{b\left(f_{2}\right)},\dots ,T_{b\left(f_{m}\right)}\right)}
∼
M
V
N
(
0
,
Ω
)
,
{\displaystyle \sim \mathbf {MVN} \left(\mathbf {0} ,{\boldsymbol {\Omega }}\right){\text{,}}}
where
Ω
i
j
=
ϕ
f
i
T
(
∑
i
=
1
n
V
i
)
ϕ
f
j
ϕ
f
i
T
(
∑
i
=
1
n
V
i
)
ϕ
f
i
ϕ
f
j
T
(
∑
i
=
1
n
V
i
)
ϕ
f
j
{\displaystyle {\text{where }}{\boldsymbol {\Omega _{ij}}}={\frac {{\boldsymbol {\phi }}_{f_{i}}^{T}\left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right){\boldsymbol {\phi }}_{f_{j}}}{{\sqrt {{\boldsymbol {\phi }}_{f_{i}}^{T}\left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right){\boldsymbol {\phi }}_{f_{i}}}}{\sqrt {{\boldsymbol {\phi }}_{f_{j}}^{T}\left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right){\boldsymbol {\phi }}_{f_{j}}}}}}}
ϕ
f
j
is a vector of
0
s and
1
s,
{\displaystyle {\boldsymbol {\phi }}_{f_{j}}{\text{ is a vector of }}0{\text{s and }}1{\text{s,}}}
indicating the inclusion of a variant using threshold
f
j
;
{\displaystyle {\text{indicating the inclusion of a variant using threshold }}f_{j};}
SKAT
Q
=
(
∑
i
=
1
n
U
i
T
)
W
(
∑
i
=
1
n
U
i
)
{\displaystyle \mathbf {Q} =\left(\sum _{i=1}^{n}{\mathbf {U_{i}^{T}} }\right)\mathbf {W} \left(\sum _{i=1}^{n}{\mathbf {U_{i}} }\right)}
Q
∼
∑
i
=
1
m
λ
i
χ
1
,
i
2
,
where
{\displaystyle \mathbf {Q} \sim \sum _{i=1}^{m}{\lambda _{i}\chi _{1,i}^{2}},{\text{ where}}}
(
λ
1
,
λ
2
,
…
,
λ
m
)
are eigen values of
{\displaystyle \left(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\right){\text{ are eigen values of}}}
(
∑
i
=
1
n
V
i
)
1
2
W
(
∑
i
=
1
n
V
i
)
1
2
{\displaystyle \left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right)^{\frac {1}{2}}\mathbf {W} \left(\sum _{i=1}^{n}{\mathbf {V_{i}} }\right)^{\frac {1}{2}}}
W
is a diagonal matrix of weights.
{\displaystyle \mathbf {W} {\text{ is a diagonal matrix of weights.}}}