General Linear Models - Geometric Morphometrics for Biologists - page 248

TABLE 9.13 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

Region 0.0057994 1 0.0057995 13.91 0.106 0.001

Sex 0.001604 1 0.0016040 3.85 0.0291 0.001

Region 3 Sex 0.000679 1 0.0006790 1.65 0.0123 0.075

Residuals 0.047114 113 0.0004169

Total 0.055197 116 1

Region entered as the first term in the model.

TABLE 9.14 A Comparison Between the Results Using Sequential (Type I) and Marginal (Type III) Sums of
Squares for the Impact of Sex and Region on Alpine Chipmunk Jaw Shape Fitted to the Matrix of Pairwise
Procrustes Distance, and Tested by Permutations

Sequential

R 2 Source SS F P

Sex 0.0020 4.88 0.0369 0.001

Region 0.0054 12.88 0.0973 0.001

Sex Region 0.0007 1.635 0.0123 0.068

3

Marginal

R 2

Source SS F P

Sex 0.0020 4.40 0.0368 0.0020

Region 0.0058 13.50 0.1051 0.0020

Sex Region 0.0009 2.01 0.0171 0.0260

3

For the analysis using sequential sums of squares, sex is entered first.

what Klingenberg calls a “Procrustes Anova” ( Klingenberg and McIntyre, 1998;
Klingenberg and Zaklan, 2000 ). Table 9.15 shows the results and, in this case, it is the
interaction term (which is tested again measurement error) that is of greatest interest.
That interaction term is highly significant. Figure 9.1 shows the first two principal com-
ponents of the symmetric ( Figure 9.1A,B ) and fluctuating symmetric ( Figure 9.1 C,D)
components of alpine chipmunk jaw shape.

MODELS WITH COVARIATES

If we examine a simple model with a single factor A and a single covariate X , it will
have the general form