Biology Reference
In-Depth Information
FIGURE 9.1
Principal components of symmetric and fluctuating asymmetric variation of alpine chipmunk mandi-
ble shape. (A) PC1 of the symmetric component of variation; (B) PC2 of the symmetric component of variation; (C) PC1
of the fluctuating asymmetric component of variation; (D) PC2 of the fluctuating asymmetric component of variation.
be tested using permutation methods as discussed earlier and, if the independent slopes
model is significant, no further testing of the significance of the factor is warranted.
If the test for independent slopes is not significant, then the next step would be to test
the common slope model against the reduced model with no covariate,
ð
SS
common slopes
2
SS
zero slopes
Þ=ð
1
Þ
F
common slopes
(9.54)
5
ð
SS
residuals
=
independent slope model
Þ=ð
n
J
1
Þ
2
2
If the common slope model is significant, then the variance due to the common slope is
removed and the analysis of
A
proceeds as in the single-factor case discussed earlier. If neither
the common nor independent slope models are significant, then one also proceeds to test
A
.
MOD
ELS WITH MULTIPLE FACTORS AND A COVAR
IATE
In a situation where there are multiple factors
A
and
B
, as well as the covariate
X
, the
full model would be
Y
A
B
A
B
M
ð
A
B
Þ
X
(9.55)
5
1
1
1
3
1ε
with slopes dependent on the interactions of
A
and
B
. One might test this against both
possible reduced models (as implied by the discussion in
Rencher and Schaalje, 2008
)
Y
A
B
A
B
M
ð
A
Þ
X
(9.56)
5
1
1
3
1
1ε
Y
A
B
A
B
M
ð
B
Þ
X
(9.57)
5
1
1
3
1
1ε
