Biology Reference
In-Depth Information
FIGURE 8.1 Checking the
assumption of a linear relation-
ship between shape and the inde-
pendent variable: (A) using a
single variable plotted on cen-
troid size; (B) using the
Procrustes distance of each speci-
men from the shape having the
smallest size, plotted on centroid
size.
transform of centroid size, repeating the two analyses to check for linearity ( Figure 8.2 ).
Both plots now suggest a nearly linear relationship between shape and log centroid size.
Thus, we would use log centroid size as our independent variable.
To this point, we have talked about the assumption of linearity as that is usually stated in
context of bivariate regression. However, multivariate studies make another assumption of
linearity, which is that all components of the dependent variable be linearly related to the
independent variable. In other words, we are assuming that all the components of shape are
linearly related to each other, as we have assumed they are all linearly related to the inde-
pendent variable. This assumption will not hold if some components of shape are linearly
related to the independent variable but some others are non-linearly related to it. The com-
ponents of shape cannot be linearly related to each other if different ones fit differently
shaped curves. Because this departure from the assumption of non-linearity is specific to
multivariate data, it may not be intuitively obvious what the assumption means. What it
means is that the slope of the relationship between shape and the independent variable is
constant the values {m 1 , m 2 , m 3 ,
m P } are not functions of the independent variable.
In some cases, such as in studies of ontogeny, the shape variable correlated with age
may change from age to age. If that is the case, we cannot model the ontogeny of shape by
a single vector of slope coefficients because that vector would change with time. The onto-
genetic trajectory of shape is then a curving path in shape space, not a straight line. The
assumption of multivariate linearity can be checked in two ways although, again, neither
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