Biology Reference
In-Depth Information
that effect (
Figure 8.6B
), magnified fivefold, shows that the impact of size is particularly pro-
nounced on the orientation and width of the coronoid process, curvature and length of the
angular process, position of the masseteric fossa, and curvature of the incisor alveolus.
ONE-WAY ANOVA/MANOVA
Having considered a case in which there are only two groups, we now extend the anal-
ysis to three or more. The categorical variable in the analysis of variance (ANOVA) is typi-
cally called a factor or, in the case of experimental studies, a treatment; the different
values it takes are called the levels of the factor. In the case above, there were only two
levels of the factor sex. We will now extend the analysis to factors that have more than
two levels, although in this chapter we still consider only one factor. The factor could be
species, with one level per species, so for nine species we would have nine levels of the
factor. In an experimental case, we might have three treatments, such as three diets (liquid,
soft, hard), in which case we have three levels of the treatment. The question we typically
ask in such cases is whether the factor influences the dependent variable. Classically, this
is called a single factor analysis of variance (ANOVA) or multivariate analysis of variance
(MANOVA), or a one-way ANOVA/MANOVA. This is also a simple example of a
General Linear Model, which encompasses a wide range of analyses, including regression,
ANOVA, MANOVA (with one or more than factors), and Analysis of Covariance
(ANCOVA/MANCOVA), which contain a mixture of categorical factors and continuous
variables (covariates), plus a number of other models. A lot of details about the nature of
factors, and the design of experiments, become critical once we have more than a single
factor. These are the subject of the next chapter. In this one, we consider only a simple one
factor (one-way) ANOVA/MANCOVA.
In the discussion of regression and of Goodall's F-test, we have seen how to character-
ize the variance in a data set using Procrustes distances, and to measure the variance
explained by a model. In both linear regression and the pairwise comparison of means, we
saw that a permutation test could be used to determine the significance of an F-ratio based
on summed squared Procrustes distances. There are other approaches to charactering vari-
ance that use what is called the sum of squares and cross products matrices (SSCP, which
are linearly related to the variance
covariance matrices) and there are analytic statistical
tests available based on SSCP, which we discuss in the next chapter. In many ways, it is
much easier to use and explain Procrustes distances and permutation tests than to work
with SSCP matrices. We thus begin by discussing a univariate ANOVA as a way of intro-
ducing the basic ideas of the method, and to explain how sums of squares are formed. The
extension of these ideas and explanation to Procrustes distances and to permutation tests
is then relatively straightforward.
Univariate ANOVAWith One Factor
In this section, we examine the development of a general linear model (GLM) for a uni-
variate variable (Y) that is hypothesized to depend on a single fixed factor (A). We will
