Biomedical Engineering Reference
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to control the family-wise error rate
α
(type I error) at, say
5%
. Namely,
α
defines the
significance threshold for each association test.
α
controls the probability to make one
or more false discoveries among all hypotheses when performing multiple association
tests.
An advantage of the FLTM strategy relies on the fact that there are less variables in
the highest layers than in the lowest ones. Thus,
α
is expected to increase with the layer
level. That is the reason why a permutation procedure need be adapted to the calculation
of layer-specific thresholds
α
.
The procedure implemented to obtain the
α
threshold specific to each layer is de-
scribed in Algorithm 2. This procedure performs
n
p
permutations (line
4
). For each
permutation, and each layer
of the FLTM, independence tests are run for any variable
X
v
belonging to this layer and the target variable
Y
(line
9
). Then, for each permuta-
tion, the minimum of the p-values over all variables belonging to layer
is added to the
distribution of minimum p-values for this layer (line
12
). Given a specified family-wise
error rate
α
, this distribution then allows to extract the corresponding
α
threshold (line
16
). This
α
value, specific to each layer, is to be compared with the p-value resulting
Algorithm 2.
PermutationProcedure
(
X,D
X
,Y,D
Y
,n
p
,α
)
INPUT:
X
,
D
X
,asetof
n
v
candidate variables (observed or latent)
X
=
X
1
, ..., X
n
v
and the
corresponding data observed or imputed for
n
individuals,
Y
,
D
Y
, a target variable
Y
and the corresponding data observed for
n
individuals,
n
p
, the number of permutations,
α
, the family-wise error rate.
OUTPUT:
{α
(
1
)
, ..., α
(
n
)
}
,
the
set
of
per-test
error
rates
respectively
computed
for
layers
l
to
n
.
1:
for
=1
to
n
2:
distrib
minP V alues
(
)
←∅
3:
end for
4:
for
p
=1
to
n
p
5:
D
Y
p
← permuteLabels
(
D
Y
)
6:
for
=1
to
n
l
7:
pV alues
(
p,
)
←∅
8:
for
each variable
X
v
in layer
9:
pV alue
p, , v
← runAssociationTest
(
X
v
,D
Y
p
)
10:
pV alues
(
p,
)
← pV alues
(
p,
)
∪ pV alue
p, , v
11:
end for
12:
distrib
minP V alues
(
)
← distrib
minP V alues
(
)
∪ min
X
v
(
pV alues
(
p,
))
13:
end for
14:
end for
15:
for
=1
to
n
l
16:
α
(
)
← quantile
(
distrib
minP V alues
(
)
,α
)
17:
end for
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