Biology Reference
In-Depth Information
TABLE 9.15
A Two-Factor Mixed Model for Shape Using a Generalized Goodall's Test: Fluctuating
Asymmetry of Chipmunk Jaw Shape
R
2
Source
SS
df
MS
F
P
Individuals
0.1479
9435
0.0013
6.71
0.83
,
0.0001
Sides
0.00034
111
3.0843E-06
1.45
0.002
,
0.0001
Individual
3
Sides
0.02204
9435
1.9847E-04
5.90
0.124
,
0.0001
Measurement error
0.00755
19092
6.8018E-05
0.042
Fluctuating asymmetry is estimated by the “Individual
3
Sides” term.
Y
A
M
ð
A
Þ
X
(9.49)
5
1
1ε
where
Y
is the centered matrix of the dependent data,
X
is the centered matrix of covariate
values,
A
is the factor,
M(A)
is a matrix of coefficients (or slopes), one per variable in
Y
. The
coefficients in
M(A)
are assumed to depend on the levels in
A,
and
is again the residual or
error term. This would be the “full” model because it contains all the interactions of interest.
We would like to compare it against a model in which
M
is independent of
A
ε
Y
A
M
0
X
(9.50)
5
1
1 ε
which is a reduced model relative to the model with
M(A
). We would probably also con-
sider the model in which there is no dependence of
Y
on
X
Y
A
5
1ε
(9.51)
If the second model, which turns out to be significant, and the first is not, we might
also consider the model with no dependence of
Y
on
A
.
Y
M
0
X
(9.52)
5
1ε
As discussed above in the context of univariate analyses, the typical approach is to com-
pute the regression slopes for each level of
A
and then to test the null hypothesis that the
slopes do not differ by comparing values of the derived univariate slopes. A multivariate
approach, presented by
Rencher and Schaalje (2008)
, tests the hypothesis that the slopes
are all equal using an F-test, in which they compute the variance explained by the full
model (of independent slopes) beyond that explained by the reduced model (of homoge-
neous slopes) as the sums of squares in the numerator, and use the sums of squares of the
error term of the full model (independent slopes). This produces an F-ratio of the form:
F
independent slopes
5
ð
SS
independent slopes
2
SS
common slopes
Þ=ð
J
1
Þ
2
(9.53)
ð
SS
residuals
=
independent slope model
Þ=ð
n
2J
Þ
2
noting that the numerator has df
1) because the independent slopes model has
J slopes estimated for J levels in
A
, whereas the common slope model estimates only one
slope. The residuals of the independent slope model have J estimated slopes, and J esti-
mated means, leaving n
(J
5
2
2
2J degrees of freedom remaining for n specimens. This ratio can
