Biology Reference
In-Depth Information
yields Bookstein's shape coordinates, which are the easiest to understand; next Procrustes
superimposition, the most widely used method; and third, resistant fit Procrustes superim-
position. The main reason for presenting all three is so that you can see how the same
three operations that remove non-shape information from the data are applied. We discuss
these first because doing so allows us to discuss a number of general issues (including the
interpretation of results) before presenting the more abstract theory of shape analysis in
Chapter 4. That theory provides the framework for generating (as well as analyzing) shape
variables. After reviewing the basic theory, we introduce the thin-plate spline (Chapter 5),
an interpolation function useful for depicting results by means of a deformed grid (as in
Figures 1.11
1.13
). All the chapters in this first section are about the data.
The second section of the topic is about the methods for analyzing the data. These
methods produce the biologically interesting variables
the ones that covary with the
biological factors of interest. Unlike the variables produced by the methods of the first
section, the variables produced by these analytic methods have a biological meaning.
They answer such fundamental questions as “What impact does size have on shape?”,
or “By how much, and in what way, do these species differ in their ontogenies?”, or “Do
these populations vary along a single latitudinal gradient?”, or even “What shape has the
highest fitness in this population?” Each of these questions is answered in terms of a shape
variable
the vector of coordinates that covaries with size or age, or that covaries with
latitude or fitness, or that expresses the difference between the means of two groups.
When we do not know what factors might be present in the data, a common problem in
studies that analyze the variation within and between natural populations, we can explore
the data algebraically, using methods of matrix algebra to determine if any interesting pat-
terns emerge. Principal components analysis (PCA) is one example of this kind of explor-
atory technique; the main purpose is to determine what varies and to look for biological
explanations for that variation. Canonical variates analysis (CVA) and between-group
principal components analysis (BGPCA) are also exploratory methods, but they presume
that you are asking questions about differences between groups, so there is a factor of
interest: your grouping variable.
Because many biologists begin a study by exploring patterns in the data, the section on
analytic methods begins with an overview of the exploratory methods (Chapter 6). These
are useful for extracting simple patterns from complex multidimensional data because
they provide a space of relatively low dimensionality that captures most of the variation
among specimens (PCA), or most of the differences among groups (canonical variates
analysis, CVA and between-groups PCA). We explain the algebra underlying these meth-
ods, compare them, and discuss when each is appropriate in light of particular biological
questions. We also discuss a method for analyzing the covariance between two multivari-
ate blocks of data, partial least squares, PLS (Chapter 7).
The two following chapters cover methods of hypothesis testing. We begin with simple
statistical models (Chapter 8), regression of shape onto a single continuous independent
variable (multivariate regression) or a single independent categorical variable (one-way
multivariate analysis of variance, MANOVA). We discuss how the hypotheses are formu-
lated and tested using both analytic tests and resampling-based methods (bootstrapping
and permutations). The next chapter (Chapter 9) introduces the General Linear Model and
discusses more complex hypotheses, those that include multiple factors, both continuous
