Statistics - Geometric Morphometrics for Biologists - page 192

FIGURE 8.2 Checking the
assumption of a linear relationship
between shape and the indepen-
dent variable: (A) using a single
variable plotted on ln centroid size;
(B) using the Procrustes distance of
each specimen from the shape hav-
ing the smallest size, plotted on ln
centroid size.

method is ideal. One is to conduct a principal components analysis (PCA) of the data, and
check for a statistical relationship between multiple PCs and the independent variable
( Figure 8.3 ). In the example shown in Figure 8.3A , there is a substantial deviation from lin-
earity not only is PC1 correlated with age (which is expected) but PC2 and PC3 also are,
with PC2 and PC3 describing the deviations from the linear trend represented by PC1.
The assumption can also be checked by regressing several shape variables on each other
( Figure 8.3B ) because, if the relationship among these variables is non-linear, we must
reject the assumption of multivariate linearity.
When shape data violate the assumption of multivariate linearity, there is no easy way to
transform them. They are not individual variables that can be individually transformed
because all of them, taken together, represent a single variable shape. If we log transform
some of the components, we thereby alter the meaning of “shape”. Also, whenever the
dependent variable is transformed, the error structure of the data is also affected. That is not
the case when the independent variable is transformed, because that variable is presumed
to be measured without error. The non-linear dynamics of the shape variable are not just a
nuisance, they are biologically interesting but they do complicate statistical analyses.
Presuming that the assumption of linearity actually is met, we can go forward with the
analysis and test the hypothesis that our independent variable predicts shape. The classic
analytic approach represents the variance (which includes total variance, the variance
explained by the model and the residual) by variance covariance matrices instead of