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models do not require a main effect to be present in the model when its interaction term
is included. This may induce alaising, and therefore the proposed methodology herein
ensures main effects are in the model prior to any interaction effects. A further restric-
tion on previous methods is that candidate models can only include loci that are near to
loci in the current model. Our methodology does not require this restrriction and allows
more flexibility.
Once the posterior model probabilities have been computed activation probabilities
can be used to assess the impact of loci
X
j
and can be computed via:
|
D
)=
|M|
P
(
β
j
=0
P
(
β
j
=0
|
D
,M
c
)
P
(
M
c
|
D
)
.
(8)
c
=1
Activation probabilities are different from the traditional p-value in that large values
indicate significance versus small values. In addition, activation probabilities do not
depend on a specific model as do p-values. The activation probabilities can be calculated
via
MC
3
as defined in section 2.3.
Activation probabilities will have a problem detecting two-way interactions when the
main effect terms are required to be in the model in order for the two-way interaction
term to be present. This induces the following inequalities:
P
(
β
jk
|
D
)
≤
P
(
β
j
|
D
)
(9)
P
(
β
jk
|
D
)
≤
P
(
β
k
|
D
)
.
Hence, using the standard activation probabilities for two-way interaction effects will
produce probabilities that are damped. In order to amplify the activation probabilities
of the two-way interaction effects one can use conditional activation probabilities. Con-
ditional activation probabilities can also be obtained by:
P
(
β
jk
=0
|
β
j
=0
,β
k
=0
,
D
)
(10)
=
P
(
β
jk
=0
,β
j
=0
,β
k
=0
|
D
)
,
P
(
β
j
=0
,β
k
=0
|
D
)
provided that
P
(
β
j
D
)
>
0
. In practice one should only consider condi-
tional activation probabilities when both
P
(
β
j
|
=0
,β
k
=0
|
D
)
and
P
(
β
k
|
D
)
are considerably large.
In cases where
P
(
β
j
|
D
)
are small then unreasonably large inflations to the
conditional activation probabilities will occur and hence the result in incorrect infer-
ences.
D
)
or
P
(
β
k
|
2.3
Restricted Model Space
A simple approach to defining the neighborhoods of a model
M
c
is to include all mod-
els that add an additional term or drop an existing term. However, this violates a model
that require both main effect terms need to be present in the model in order for the cor-
responding two-way interaction to be added. Furthermore, the model need not contain
all interaction terms possible. Notice this creates a large model space. For the first order
models with
p
predictors the size of the model space is
2
p
. However with the addition
