Page 7 - Asylv_molars_Pmax_revised_2014_01

7

Statistics

Differences
between
groups
(sexes,
localities,
geographic
groups)
were
investigated
using
analyses
of

variance
(ANOVA)
completed
by
their
non-­‐parametric
analogue
(Kruskall
Wallis
test,
KW)
for

univariate
variables,
and
multivariate
analysis
of
variance
(MANOVA)
for
the
set
of
shape
variables.

The
pattern
of
shape
variation
was
assessed
using
a
principal
component
analysis
(PCA)
on
the

variance-­‐covariance
(VCV)
matrix
based
on
the
14
shape
variables
(FCs).
This
provided
synthetic
axes

(principal
components
(PCs),
e.g.
eigenvectors)
summarizing
the
most
important
directions
of

variance
within
the
data
set.

Relationships
between
the
set
of
shape
variables
considered
as
dependent
matrix,
and
another

quantitative
variable
(e.g.
size
or
latitude)
were
investigated
using
multivariate
regressions.

Directions
of
main
variance
and
comparison
of
vectors

Pmax
was
calculated
as
the
first
eigenvector
of
a
PCA
on
the
intra-­‐population
VCV
matrix
of
the
14

shape
variables.
Directions
of
evolution
were
calculated
as
the
vector
of
difference
between
two

populations
(e.g.,
an
insular
population
and
its
putative
mainland
relative)
or
as
direction
of
main

phenotypic
variance
among
a
set
of
populations
(first
eigenvector
based
on
the
VCV
matrix
among

population
means).

Similarity
between
vectors
(Pmax
and
evolutionary
directions)
was
assessed
by
calculating
their

correlation
R,
i.e.
the
arccosine
of
the
inner
product
of
the
two
vector
elements,
ranging
between
-­‐1

(vectors
pointing
in
totally
opposite
directions)
and
+1
(vectors
perfectly
pointing
in
the
same

direction).
This
observed
correlation
was
compared
to
the
distribution
of
R
from
fifty
thousand

simulated
random
vectors.
Each
random
vector
was
calculated
as
follow:
each
of
its
14
elements
was

drawn
from
a
uniform
probability
distribution
in
the
range
from
+1
to
-­‐1,
and
the
vector
was
then

normed
to
unity.
The
distribution
of
these
random
vectors
provided
the
following
significance

threshold
values
for
the
absolute
value
of
R
(a
significant
probability
meaning
that
the
observed
R
is

larger
than
expected
based
on
the
distribution
of
R
between
random
vectors):
P
<
0.01,
R
=
0.651
(*);

P

<
0.001,
R
=
0.770
(**);
P
<
0.0001,
R
=
0.860
(***).
Note
that
the
absolute
value
of
R
was

considered,
because
the
+/-­‐
direction
of
Pmax
(and
of
any
eigenvector)
is
arbitrary.