Biology Reference
In-Depth Information
clearly cannot
covariance matrix. However, we can reduce the
dimensionality of the data using principal components analysis. To ensure that we have
four times as many individuals as variables, we can use the first 30 principal components,
which explain 95.4% of the variance. The results are shown in
Table 9.11
; both main effects
are statistically significant (although only sex had a significant effect on jaw size, see
Table 9.5
). The interaction term is not statistically significant. As an alternative approach,
we can use the distance-matrix permutation method; those results, based on the sequential
(Type I) sums of squares, are shown in
Table 9.12
. Again, both main factors are statistically
significant and the interaction term is not. Because we are using sequential sums of
squares, we might also wish to look at the results after reordering the terms in the model.
Table 9.13
shows the results, now with region entered first. Again, the two main effects
are statistically significant and the interaction term is not, but the proportion of variation
explained by the terms does change, albeit slightly. We might also wish to examine the
marginal (Type III) sums of squares and, in this case, as shown in
Table 9.14
, there is one
important difference
invert
the variance
the interaction term is statistically significant. That is important
because it means that we cannot generalize about the “main effects”. The effects of each
factor are not general because the impact of each one depends on the level of the other.
This ambiguity is the consequence of the unbalanced design.
The second case that we will consider is the model for fluctuating asymmetry,
a two-factor mixed model. The right and left side of each individual's jaw is measured
twice; the random factor is individual and the fixed factor is side. The interaction term
(individual
3
side) is the estimate of fluctuating asymmetry. This is usually tested by
TABLE 9.11
A Two-Factor Model for Alpine Chipmunk Jaw Shape
Source
df
Pillai Approx
F
df
1
df
2
P
Sex
1
0.39386
1.8194
30
84
0.0173
Region
1
0.80366
11.4607
30
84
,
2e
2
16*
Sex
3
Region
1
0.30464
1.2267
30
84
0.231
Residuals
113
TABLE 9.12
A Two-Factor Model for Alpine Chipmunk Jaw Shape, Fitted to the Matrix of Pairwise
Procrustes Distance, and Tested by Permutations Using Sequential (Type I) Sums of Squares
R
2
Source
SS
df
MS
F
P
Sex
0.002034
1
0.0020338
4.88
0.0369
0.001
Region
0.005370
1
0.0053697
12.88
0.0973
0.001
Sex
Region
0.000679
1
0.0006790
1.635
0.0123
0.068
3
Residuals
0.047114
113
0.0004169
Total
0.055197
116
1
Sex entered as the first term in the model.
