H. David Sheets

5 – Superimposition methods

Publisher Summary This chapter aims to explain some of the superimposition methods, compare them, and discuss their relative advantages and disadvantages. It also discusses in some detail the issue of interpreting the pictures of superimposed landmarks. At first sight, different methods may appear to suggest different interpretations of the shape differences, but to a large extent the differences are illusory—the pictures might look different, but they have the same meaning. Before discussing the alternative methods of superimposition, the chapter first explains why researchers would even want an alternative to Booksteins shape coordinates (BC). It then describes a superimposition method that is based on a very similar approach. This method, called sliding baseline registration (SBR), also involves a two-point registration, but the two points are not entirely fixed (they are allowed to “slide” along one axis). Next, it presents the most widely used method, Procrustes generalized least squares (GLS), followed by an alternative that is similar to it in some respects (variously called Procrustes resistant fit, or resistant fit theta-rho analysis, RFTRA). After presenting all of these methods, the chapter summarizes their similarities and differences, discusses the interpretation of their graphical results, and concludes with recommendations regarding their uses.

A re-examination of the contributions of biofacies and geographic range to extinction risk in Ordovician graptolites

A set of 137 Ordovician graptolite species were used to examine the associations among geographic range, sampling, biofacies and species longevity. Model-choice using general linear models combined with partial least-squares regression analysis found seven distinct predictive variables. The dominant factors were overall commonness, biofacies, geographic range and sampling in decreasing order of variance explained. However, the data-set is biased toward particularly well-sampled and widespread taxa. Region (represented as a set of discrete geographic areas) was a strong factor in extinction risk, whereas latitudinal range and endemicity were poor predictors. Results suggest that other factors besides just geographic range and biofacies need to be considered when understanding extinction dynamics.

Morphometrics and systematics

Systematists use morphometrics to answer three types of questions. The first, “taxonomic,” asks whether populations are drawn from multiple species, and, if so, by what variable(s) they are most effectively discriminated. The second, “phylogenetic,” asks about phylogenetic relationships among taxa. Although they cannot be used to construct cladograms, morphometric analyses might nonetheless be useful for finding informative characters. The third, “evolutionary,” asks about the evolutionary history of the feature of interest—which, for our purposes, is shape. These are all interrelated issues, but there are important distinctions that bear on choosing the appropriate analytic method. Most importantly, taxonomic discriminators are often not equivalent to phylogenetically informative characters, so finding discriminators is not equivalent to finding characters. Also, characters usually comprise a subset of features that evolve, so tracing characters on a cladogram does not fully reconstruct the evolution of shape. Unfortunately, of the three types of questions, only those relating to taxonomic discrimination are so straightforward that they require nothing more than standard morphometric tools. This does not mean taxonomic discrimination is easy; on the contrary, it can be very difficult.

Forensic Applications of Geometric Morphometrics

In the forensic sciences, as in biology, anthropology and paleontology, there is much interest in comparing the shapes of objects. There are several specific areas of forensics in which morphometric methods, both traditional and geometric, have been applied. In this chapter, we will first look at a morphometric approach that retains size information in the analysis (Procrustes size preserving, or Procrustes-SP). Next, we will discuss what it means to say that shape data from different sources “match” one another, which differs from the usual concerns of biological studies with differences or variation. We will then examine three research programs that have used geometric morphometrics to address forensic problems in different areas and which also introduce new conceptual material.

Computer-based statistical methods

This chapter presents a brief discussion of some of the basic statistical concepts that are needed to understand statistical methods in general (such as confidence intervals and hypothesis testing), as well as the more specialized concepts that are needed to understand computer-based statistical methods. Four classes of these methods are presented, including the bootstrap, jackknife, and permutation tests, and Monte Carlo simulations. To illustrate these methods, the chapter focuses on a few univariate statistical tests. The extension to multivariate statistics is not difficult, but it seems useful to focus on univariate statistics to develop an intuitive understanding of how computer-based methods work.

The thin-plate spline: Visualizing shape change as a deformation

The thin-plate spline provides a visually interpretable description of a deformation, with the same number of variables as there are statistical degrees of freedom, and it employs the Procrustes distance as a metric. Even if we were not concerned with the advantages of the spline for graphical analysis, we might still want to use it for purposes of statistical inference. Conversely, even if we were not concerned with the advantages of the spline for statistical analysis, we might still wish to use it for its graphical capabilities. The spline can be used to depict ones results, and one can use partial warps in statistical analyses without worrying that the mathematical details (and complexities) will have any impact on the results. The spline is a convenient tool for visual display and for obtaining variables with the correct degrees of freedom—it is nothing more (or less) than that. The chapter begins with a basic overview of the mathematical idea of a deformation. It then discusses the mathematical metaphor underlying one particular model of a deformation, the thin-plate spline, and how we can decompose it to yield variables. In general, the chapter presents a largely intuitive overview before delving more deeply into the mathematics.

7 – Ordination methods

Publisher Summary This chapter discusses two methods for describing the diversity of shapes in a sample: principal components analysis (PCA) and canonical variates analysis (CVA). The discussion of these methods draws heavily on expositions presented by Morrison (1967), Chatfield and Collins (1980), and Campbell and Atchley (1981). Both methods are used to simplify descriptions rather than to test hypotheses. PCA is a tool for simplifying descriptions of variation among individuals, whereas CVA is used for simplifying descriptions of differences between groups. Both analyses produce new sets of variables that are linear combinations of the original variables. They also produce scores for individuals on those variables, and these can be plotted and used to inspect patterns visually. Because the scores order specimens along the new variables, the methods are called “ordination methods.” The most important difference between PCA and CVA is that PCA constructs variables that can be used to examine variation among individuals within a sample, whereas CVA constructs variables to describe the relative positions of groups (or subsets of individuals) in the sample. PCA and CVA both serve a similar purpose, and the mathematical transformations performed in the two analyses are similar. The chapter describes PCA first because it is somewhat simpler, and because it provides a foundation for understanding the transformations performed in CVA. It begins the description of PCA with some simple graphical examples, and then presents a more formal exposition of the mathematical mechanics of PCA. This is followed by a presentation of an analysis of a real biological data set.

Simple Size and Shape Variables: Shape Coordinates

This chapter presents methods for obtaining shape variables. One is particularlysimple and easily understood, and we present it first because the method is so accessible.This method is sometimes called the “two-point registration” and it produces coordinatesthat are termed “Bookstein shape coordinates”, which can be used both for graphicaldisplays and formal statistical tests. The second method, the Procrustes superimpositionis perhaps less intuitive, but the method is the one most widely used, for reasons that willbecome apparent in the next chapter. It is the one that we will use throughout the rest ofthis book so we introduce it now rather than laying its theoretical foundations in the nextchapter and deriving the superimposition method from theory. We focus on the simplestpossible application of the methods, the analysis of shapes with only three two-dimensional landmarks (triangles). We also discuss how information about size can be restored (because it is removed in the course of obtaining shape coordinates). We then extend the analysis to three-dimensional landmarks and, in the case of the Procrustes superimposition, to semilandmarks, points along outlines or curves. As well as presenting the methods for obtaining the coordinates, we also discuss the graphical description of results because, to a large extent, it is the descriptive power of geometric morphometrics–the visualization of shape change–that makes these methods so useful. The graphical results can differ depending on the methods for obtaining the shape variables, so we show how apparent inconsistencies can be reconciled.

Evolutionary Developmental Biology(1): The Evolution of Ontogeny

Studies of evolving ontogenies are grounded in two important insights. The first is that all evolutionary change arises from changes in ontogeny and therefore we need to understand how ontogenies evolve in order to understand the origins of morphological diversity. The second is that organisms have time-extended phenotypes. An organism’s phenotype is not static – it changes from age to age in both form and function. To comprehend that dynamic form and function relationship, as it evolves, we need to understand how developing organisms negotiate the ontogenetic transformations in form and function. The concept central to all studies of evolving ontogenies is the “ontogenetic trajectory”. In this chapter, we discuss the range of hypotheses about the evolution of ontogenetic trajectories that can be tested and how to test them. This chapter begins with the review of the formalism for the analysis of allometry using traditional morphometric variables, then briefly recalls the analysis of allometry using geometric morphometric data, and then examines a series of hypotheses that can be tested about the evolution of ontogenetic trajectories. As well as comparing ontogenetic trajectories, we also incorporate analyses of disparity to dissect the developmental sources of disparity. Finally, we discuss the disparity of ontogeny itself.

Evolutionary Developmental Biology(1)

Studies of evolving ontogenies are grounded in two important insights. The first is that all evolutionary change arises from changes in ontogeny and therefore we need to understand how ontogenies evolve in order to understand the origins of morphological diversity. The second is that organisms have time-extended phenotypes. An organism’s phenotype is not static – it changes from age to age in both form and function. To comprehend that dynamic form and function relationship, as it evolves, we need to understand how developing organisms negotiate the ontogenetic transformations in form and function. The concept central to all studies of evolving ontogenies is the “ontogenetic trajectory”. In this chapter, we discuss the range of hypotheses about the evolution of ontogenetic trajectories that can be tested and how to test them. This chapter begins with the review of the formalism for the analysis of allometry using traditional morphometric variables, then briefly recalls the analysis of allometry using geometric morphometric data, and then examines a series of hypotheses that can be tested about the evolution of ontogenetic trajectories. As well as comparing ontogenetic trajectories, we also incorporate analyses of disparity to dissect the developmental sources of disparity. Finally, we discuss the disparity of ontogeny itself.