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less significant after the insertion of some variables. Finally, the algorithm ends at
step 3, when it is not possible to further improve the model by adding or remov-
ing variables. The minimum values of significance for introducing and for dropping
variables in the model are called
P
in
and
P
out
, respectively. They correspond to sig-
nificance levels of the partial
F
-test. They do not need to be equal, but they must
be set very carefully because if
P
in
P
out
, then a loop may occur where a variable
enters the model, then leaves it, then reenters, and so on.
>
The stepwise regression algorithm is based on the following steps:
1. Start from a user defined
k
-variable multiple regression model (possibly,
k
0).
2. Compute the partial
F
-test for each variable not yet included in the regres-
sion model.
3. Check if there is at least one variable whose p-value is less than the user
defined threshold value
P
in
(this is the same of checking for those variable
whose partial
F
-statistic is greater than the threshold value
F
[
P
in
;1
,
n
−
(
k
+
1
)]
).
If there are some variables which fulfill the requirement, add to the model
the most significant one (i.e. the one with smaller p-value). If there is no
variable which can be added, the regression algorithm stops.
4. Compute the partial
F
-test for all variables currently in the regression
model.
5. Check if there is at least one variable whose p-value is greater than the user
defined threshold value
P
out
. If there are some variables which fulfill the
requirement, remove from the model the less significant one (i.e. the one
with greater p-value).
6. Restart from step 2.
=
7.8
Trees and Graphs
Trees and graphs are everywhere in discrete mathematics and in computer science.
Moreover, a huge number of discrete models in biology are based on trees and/or
graphs. Here we present some basic concepts about them (see [224, 217, 220] for
more advanced concepts).
A
(rooted) tree
T
is specified by a structure
T
,where
N
is a set of
n
odes, (or
vertices
), including a node
r
, called the
root
of
T
,and
f
is a function,
assigning a
father
node to any node of
N
different from
r
, such that, for any node
x
different from
r
,
f
n
=(
N
,
r
,
f
)
r
for some natural number
n
(
f
n
(
x
)=
(
x
)
denotes the
n
-fold
of
f
to
x
,
f
0
x
,and
f
1
application
f
). The termi-
nology about trees is inspired by the many different contexts where they occur, es-
pecially genealogy and botanic. For example:
root, node, leaf, parent, father, child,
son, brother, ancestor, descendant, branch, generation, path, and depth
.
(
...
f
(
f
(
x
))
...
)
(
x
)=
(
x
)=
f
(
x
)
