Geometric Morphometrics for Biologists (eBook)

A Primer

H. David Sheets, Donald Swiderski, Miriam Zelditch (Autoren)

eBook Download: EPUB

2012 | 2. Auflage
488 Seiten
Elsevier Science (Verlag)
9780123869043 (ISBN)

Infos

The first edition of Geometric Morphometrics for Biologists has been the primary resource for teaching modern geometric methods of shape analysis to biologists who have a stronger background in biology than in multivariate statistics and matrix algebra. These geometric methods are appealing to biologists who approach the study of shape from a variety of perspectives, from clinical to evolutionary, because they incorporate the geometry of organisms throughout the data analysis. The second edition of this book retains the emphasis on accessible explanations, and the copious illustrations and examples of the first, updating the treatment of both theory and practice. The second edition represents the current state-of-the-art and adds new examples and summarizes recent literature, as well as provides an overview of new software and step-by-step guidance through details of carrying out the analyses. - Contains updated coverage of methods, especially for sampling complex curves and 3D forms and a new chapter on applications of geometric morphometrics to forensics - Offers a reorganization of chapters to streamline learning basic concepts - Presents detailed instructions for conducting analyses with freely available, easy to use software - Provides numerous illustrations, including graphical presentations of important theoretical concepts and demonstrations of alternative approaches to presenting results

Dr. Miriam Zelditch is Associate Research Scientist at the University of Michigan's Museum of Paleontology. She obtained her PhD in Zoology from Michigan State University and conducted her NSF Postdoctoral Fellowship at the University of Michigan. Her research interests broadly include zoology, paleobiology, evolutionary biology, comparative biology, and morphology. She has co-edited both prior editions of Geometric Morphometrics for Biologists.

Inhalt

Front Cover 1
Geometric Morphometrics for Biologists: A Primer 4
Copyright Page 5
Contents 6
Contributors 8
Preface 10
1. Introduction 12
A Critical Overview of Measurement Theory 14
Shape and Size 22
Shape 22
Size 24
Methods of Data Analysis 25
Biological and Statistical Hypotheses 27
Organization of the Book 28
Software and Other Resources 30
References 31
1. Basics of Shape Data 32
2. Landmarks and Semilandmarks 34
Criteria for Choosing Landmarks 36
Homology 36
Adequate Coverage of the Form 39
Repeatability 40
Consistency of Relative Position 41
Coplanarity of Landmarks 43
Bookstein’s Typology of Landmarks 43
Examples: Applying Ideals to Actual Cases 46
Landmarks on the Lateral Surface of the Squirrel Scapula 46
Landmarks on the External Body of Piranhas 48
Landmarks on the Skull of Sigmodon fulviventer and Mus musculus domesticus 51
Three-Dimensional Landmarks on a Marmot Skull 53
Landmarks on a Squirrel Mandible 53
Designing Your Own Measurement Scheme 57
References 59
3. Simple Size and Shape Variables: Shape Coordinates 62
Shape Coordinates 62
Comparing Shapes of Two Triangles 64
Comparing Many Individual Triangles 65
Multiple Triangles on Each Individual 66
Choosing the Baseline 68
Size 70
Bookstein Shape Coordinates in Three Dimensions 71
Statistics of Shape Coordinates 72
Procrustes Superimposition 73
Procrustes Superimposition in Three-Dimensions 76
Semilandmark Sliding 76
Resistant-Fit Superimposition 78
Interpreting the Graphical Results 80
References 84
4. Theory of Shape 86
The Definition of Shape 87
Morphometric Spaces 89
The Configuration Matrix 89
Configuration Space 89
Position or Location of a Configuration Matrix 90
Size of a Configuration Matrix 91
Pre-Shape Space 92
The Shape of Pre-Shape Space 93
Fibers in Pre-Shape Space 94
Shape Spaces 96
Finding the Angle of Rotation That Minimizes the Euclidean Distance Between Two Shapes 100
The Spaces of Three-Dimensional Configurations 101
A Numerical Example For the Simplest Case 102
Tangent Spaces 108
Selecting the Reference Configuration 110
Dimensions and Degrees of Freedom 111
Summary 112
References 113
5. The Thin-plate Spline: Visualizing Shape Change as a Deformation 114
Modeling Shape Change as a Deformation 118
The Physical Metaphor 119
Uniform and Non-Uniform Components of a Deformation 120
Uniform (Affine) Components 122
Decomposing the Non-Uniform (Non-Affine) Component 124
An Intuitive Introduction to Partial Warps 124
An Algebraic Introduction to Partial Warps 127
Decomposing the Deformation of Three-Dimensional Data 130
Using the Thin-Plate Spline to Visualize Shape Change 132
Interpreting Changes Depicted by the Thin-Plate Spline 133
Using Bending Energy to Superimpose Semilandmarks 134
Appendix 137
Calculating the Shear and Compression/Dilation Terms 137
Conceptual Framework 137
Mathematical Derivations 138
Further Derivation of the Uniform Components 139
Calculating Uniform Components Based on Other Superimpositions 142
References 143
2. Analyzing Shape Variables 144
6. Ordination Methods 146
Principal Components Analysis 147
Geometric Description of PCA 148
Algebraic Description of PCA 152
A Formal Proof That Principal Components are Eigenvectors of the Variance–Covariance Matrix 156
Interpretation of Results 157
Canonical Variates Analysis 162
Groups and Grouping Variables 162
Geometric Description of CVA 163
Algebraic Description 166
Interpretation of Results 167
CVA of Rank-Deficient Data 174
Between Groups Principal Components Analysis 175
References 177
7. Partial Least Squares Analysis 180
Analyzing Covariances Between Blocks and Significance Testing 181
Mathematical Details of Two Block PLS 182
Three Block PLS 183
Using PLS to Compare Patterns of Covariance Between Blocks Across Groups 184
Comparing PLS to Other Methods 185
PLS Compared to Multiple Regression 185
PLS Compared to PCA 186
PLS Compared to Canonical Correlation Analysis 187
PLS compared to Canonical Variates/Discriminant Analysis 188
Applications of PLS 188
Using PLS as an Exploratory Tool to Characterize a Population: The Anterior Human Dentition 188
Using PLS to Examine Morphological Integration and Modularity 191
Using PLS to Relate Shape to Ecological Factors 194
Cichlid Body Shape and the Biotic Environment: The Relationship Between Body Shape and Trophic Morphology 194
Human Nasal Cavity Morphology and the Climatic Environment: Temperature and Vapor Pressure 196
Environmental Correlates of Geographic Variation in Skull and Mandible Shape of a Rodent, the Punaré Rat (Thrichomys apereoides) 197
References 197
8. Statistics 200
The Correlation Coefficient 204
Multivariate Regression 205
Distance-Based Methods of Hypothesis Testing 210
Examples: Testing the Null Hypothesis that X is Not a Linear Predictor of Y 211
Comparing Two Means 213
Testing the Difference Between Mean Shapes of Two Groups 214
Testing the Null Hypothesis that Chipmunk Jaw Shape is Not Sexually Dimorphic 215
One-Way ANOVA/MANOVA 217
Univariate ANOVA With One Factor 217
Extension of the Univariate ANOVA to Multivariate Shape Data 220
Appendix: An Overview of Randomization and Monte Carlo Methods 221
Resampling Statistics 221
Resampling-Based Methods 223
The Bootstrap 223
Permutation Tests 226
The Jackknife 228
Monte Carlo Methods 229
Example: Resampling Tests and Regression Models 230
Issues Common to All Resampling Methods 232
Statistical Power 232
How Many Repetitions? 233
Summary 234
References 234
9. General Linear Models 236
Factors and Experimental Design 238
Fixed and Random Factors 239
Crossed and Nested Factors 240
Main Effects and Interaction Terms 241
Decomposition of Variance 241
Balanced and Unbalanced Design 242
Design Matrices 243
The Form of a General Linear Model 245
F-tests and Mean Squares 246
Univariate Data with One Factor 246
Generalizing and Extending the Simple Univariate ANOVA 249
Models 249
Univariate Two Factor Balanced Design 249
Univariate Two Factor Nested Model 251
Univariate Three Factor Model 252
Models with Covariates 253
More Complex Designs 254
Unbalanced Designs and Sums of Squares 254
Type I Sums of Squares 255
Type II Sums of Squares 256
Type III Sums of Squares 257
Which Sums of Squares to Use? 258
Working with Multivariate Sum of Squares 258
Classical Analytic Approaches to Significance Testing of GLM Models 258
Permutation Approaches to General Linear Models 259
Expressing GLM Models in Terms of Distance Matrices A and B for Which We Can Compute Both AB and BA 260
Permutation Tests Based on the Distance Matrix 261
Types of Permutations 262
Models with Multiple Factors 262
Models with Covariates 264
Models with Multiple Factors and a Covariate 266
Analyzing Measurement Error 267
Implementing GLM 269
References 270
3. Applications 272
10. Ecological and Evolutionary Morphology 274
Incorporating Phylogeny in Comparative Studies 274
A Note on Phylogenetic “Signal” or “Constraint” 279
Evolutionary Allometry 280
Form and Function 281
Comparing Trajectories 285
Magnitude and Structure of Morphological Diversity 287
Disparity 287
Variation 289
Metrics for Disparity and Variance 290
Measuring Disparity 292
Partial Disparity 295
Measuring Variation 296
Analyzing the Structure of Disparity 297
Nearest-Neighbor Analysis 297
Cluster Analysis 301
References 304
11. Evolutionary Developmental Biology (1): The Evolution of Ontogeny 308
Why Allometry is Interesting in Its Own Right 315
Formalisms for the Analysis of Ontogenetic Allometry: Traditional Morphometric Data 316
Interpreting Allometric Coefficients 319
The Developmental Meaning of b and k 323
Revisiting Geometric Morphometric Analyses of Allometry 326
Hypotheses About the Evolution of Ontogenetic Trajectories 328
Heterochrony 329
Parallel Trajectories (Transpositional Allometry) 333
Divergent Ontogenies of Shape 334
Testing Hypotheses About the Evolution of Ontogeny 335
Framing Hypotheses About the Ontogeny of Shape in Terms of Size Variables 335
Calculating the Angle Between Two Vectors 336
Testing the Statistical Significance of the Angle 337
A Traditional Approach to Estimating the Contribution of Scaling Makes to Morphological Variation 338
Examples: Applying These Methods to Data 339
Re-evaluating the Traditional Method for Estimating the Variation Explained by Scaling 342
Dissecting the Developmental Basis of Disparity 343
Testing Hypotheses About the Evolution of Ontogeny 345
Example: Ontogeny of Shape and Disparity 349
Age-Based Comparisons of Growth and Developmental Rates and Timings 353
Disparity of Ontogeny 357
References 358
12. Evolutionary Developmental Biology (2): Variational Properties 364
Phenotypic Plasticity: Quantifying Norms of Reaction 366
Visualizing Norms of Reaction 371
Canalization: Quantifying Variation 371
Example: Ontogenetic Decrease in Variance of Skull Shape 373
Developmental Stability: Quantifying Developmental Noise 375
The Statistical Analysis of FA 375
Example: Fluctuating Asymmetry of Prairie Deer Mouse Mandibular and Cranial Shape 378
Measuring the Overall Magnitude of FA 379
Analyzing the Relationship Between Plasticity, Canalization and Developmental Stability 381
Examples: Comparing Phenotypic Variation to FA for Prairie Deer Mouse Mandibular and Cranial Shape 382
Predicting the Structure of Covariation: Morphological Integration and Modularity 384
A Brief Overview of Methods for Analyzing Modularity 388
Minimum Intermodular Covariance Method 390
Minimum Deviance Method 393
Distance-Matrix Method 394
Examples: Evaluating Four Hypotheses of Mandibular Modularity 398
“Minimum Intermodular Covariance Method” 399
“Minimum Deviance Method” 399
Distance-Matrix Method 401
What Do We Do Next to Interpret These Results? 403
References 404
13. Morphometrics and Systematics 410
Taxonomic Discrimination 412
Finding Characters 415
Defining the Problem 415
Why Not to Use Partial Warps as Characters 416
Using PCA to Find Characters 420
Using Comparisons Between Interspecific Vectors to Find Characters 424
Coding 425
Summary 427
References 428
14. Forensic Applications of Geometric Morphometrics 430
Size and Shape 431
What Does it Mean for Shapes to “Match”? 433
Matching Shapes in the Human Dentition 434
Sex Estimation in a Forensic Context 438
Likelihood Ratios and Fetal Alcohol Syndrome 441
References 444
Bibliography 446
Glossary 466
Index 482

Textauszug

Chapter 1

Introduction

Owing largely to developments in measurement theory over the past two decades, there has been remarkable progress in morphometrics. That progress resulted from first precisely defining “shape” and then pursuing the mathematical implications of that definition. We therefore now have a theory of measurement. We offer a critical overview of the recent history of measurement theory, presenting it first in terms of exemplary data sets and then in more general terms, emphasizing the core of the theory underlying geometric morphometrics – the definition of shape. We conclude the conceptual part of this Introduction with a brief discussion of methods of data analysis. The rest of the Introduction is concerned with the organization of this book and where you can find more information about available software and other resources for carrying out the analyses.

Keywords

geometric morphometrics, shape analysis, landmark coordinates, shape and size

Shape analysis plays an important role in many kinds of biological studies. A variety of biological processes produce differences in shape between individuals or their parts, such as disease or injury, mutation, ontogenetic development, adaptation to local geographic factors, or long-term evolutionary diversification. Differences in shape may signal differences in processes of growth and morphogenesis, different functional roles played by the same parts, different responses to the same selective pressures, or differences in the selective pressures themselves. Shape analysis is an approach to understanding those diverse causes of morphological variation and transformation.

Sometimes, differences in shape are adequately summarized by comparing the observed shapes to more familiar objects such as circles, kidneys or letters of the alphabet (or even, in the case of the Lower Peninsula of Michigan, a mitten). Organisms, or their parts, are then characterized as being more or less circular, reniform, C-shaped or mitten-like. Such comparisons can be extremely valuable because they help us to visualize unfamiliar organisms or to focus attention on biologically meaningful components of shape. However, they can also be vague, inaccurate or even misleading, especially when the shapes are complex and do not closely resemble familiar icons. Even under the best of circumstances, we still cannot say precisely how much more circular, reniform, or C-shaped or mitten-like one shape is than another. When we need that precision, we turn to measurement.

Morphometrics is a quantitative way of addressing the shape comparisons that have always interested biologists. This may not seem to be the case because the morphological approaches once typical of the quantitative literature appeared very different from the qualitative descriptions of morphology; whereas the qualitative studies produce pictures or detailed descriptions (in which analogies figure prominently), morphometric studies usually produced tables with disembodied lists of numbers. Those numbers seemed so highly abstract that we could not readily visualize them as descriptors of shape differences, and the language of morphometrics also seemed highly abstract and mathematical. As a result, morphometrics seemed closer to statistics or algebra than to morphology. In one sense that perception is entirely accurate: morphometrics is a branch of mathematical shape analysis. The way that we extract information from morphometric data involves mathematical operations rather than concepts rooted in biological intuition or classical morphology. Indeed, the pioneering work in modern geometric morphometrics (the focus of this book) had nothing at all to do with organismal morphology; the goal was to answer a question about the alignment of megalithic “standing stones” like Stonehenge (Kendall and Kendall, 1980). Nevertheless, morphometrics can be as much a branch of morphology as it is a branch of statistics. It is that when the tools of shape analysis are turned to organismal shapes, illustrating and even explaining shape differences that have been mathematically analyzed.

The tools of geometric shape analysis have a tremendous advantage when it comes to these purposes: not only because it offers precise and accurate description, but also because it enables rigorous statistical analyses and serves the important purposes of visualization, interpretation and communication of results. Geometric morphometrics allows us to visualize differences among complex shapes with nearly the same facility as we can visualize differences among circles, kidneys and letters of the alphabet (and mittens).

In emphasizing the biological component of morphometrics, we do not discount the importance of its mathematical component. Mathematics provides the models used to analyze data, both the general linear models exploited in statistical analyses and the algebraic models underlying exploratory methods such as principal components analysis. Additionally, mathematics provides a theory of measurement that we use to obtain the data in the first place. It may not be obvious that any theory governs measurement because very little theory (if any) underlays traditional measurement approaches. Asked the question “What are you measuring?”, we could give many answers based on our biological motivation for measurement – such as (1) “functionally important characters”; (2) “systematically important characters”; (3) “developmentally important characters”; or (4) “size and shape”. However, when asked “what do you mean by “character” and how is that related either mathematically or conceptually to what you are measuring?” or if asked “what do you mean by “size and shape”?”, it was difficult to provide coherent answers. A great deal of experience and tacit knowledge went into devising measurement schemes, but that knowledge and experience had very little to do with any general theory of measurement. Rather than being grounded in a general theory of measurement, each study appeared to devise its approach to measurement according to the biological questions at hand, as guided by the particular tradition within which that question arose. There was no general theory of shape nor were there any analytic methods adapted to the characteristics of shape data.

Owing largely to developments in measurement theory over the past two decades, there has been remarkable progress in morphometrics. That progress resulted from first precisely defining “shape” and then pursuing the mathematical implications of that definition. We therefore now have a theory of measurement. Below we offer a critical overview of the recent history of measurement theory, presenting it first in terms of exemplary data sets and then in more general terms, emphasizing the core of the theory underlying geometric morphometrics – the definition of shape. We conclude the conceptual part of this Introduction with a brief discussion of methods of data analysis. The rest of the Introduction is concerned with the organization of this book and where you can find more information about available software and other resources for carrying out morphometric analyses.

A Critical Overview of Measurement Theory

Traditionally, morphometric data were measurements of length, depth and width, such as those shown in Figure 1.1, based on a scheme presented in a classic ichthyology text (Lagler et al., 1962). Such a data set contains relatively little information about shape and some of it is fairly ambiguous. These kinds of data sets contain less information than they appear to hold because many of the measurements overlap or run in similar directions. What may be most obvious is that several measurements radiate from a single point so that their values cannot be completely independent; any error in locating that point affects all of these measurements. Such a data set contains less information than could have been collected with no greater effort because some directions are measured redundantly and many measurements overlap. For example, there are many measurements of length along the anteroposterior body axis and most of them cross some part of the head, whereas there are only two measurements along the dorsoventral axis and both are of post-cranial dimensions. In addition, because most of the measurements are long, it is difficult to localize shape differences to any region, such as any change in the proportions of the pre- and postorbital head or the position of the dorsal fin relative to the back of the head. Also, some of the information that is missing from this type of measurement scheme, but which is necessary for morphological analysis, concerns the spatial relationships among measurements. That information might be in the descriptions of the measurements, i.e. the line segments, but it is not captured by the data. The data consist solely of a list of observed values of those lengths. Finally, the measurements may not sample homologous features of the organism, making it difficult to interpret the results. For example, body depth can be measured by a line extending between two well-defined points (e.g. the anterior base of the dorsal fin to the anterior base of the anal fin), but it can also be measured wherever the body is deepest, yielding a measurement of “greatest body depth” wherever that occurs. That second measurement of depth might not be comparable anatomically from species to species, or even from specimen to specimen, so it provides almost no useful information (except for maximal depth). Considering these many limitations of traditional measurements, it is clear...

Produkt-Details

• Erscheint lt. Verlag: 24.9.2012
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Technik
• ISBN-13: 9780123869043 / 9780123869043