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Front Matter-C0319 11/25/02 4:22 PM Page 1 ©2001 CRC Press LLC An Invariant Approach to Statistical Analysis of Shapes Subhash R. Lele Department of Mathematical Sciences University of Alberta Edmonton, AB T6G 2G1 CANADA and Joan T. Richtsmeier Department of Anthropology The Pennsylvania State University University Park, PA 16802 U.S.A. Front Matter-C0319 11/25/02 4:22 PM Page 2 ©2001 CRC Press LLC Dedicated to the memory of our mothers, Sarojini Lele and Mary Hill Richtsmeier Front Matter-C0319 11/25/02 4:22 PM Page 7 ©2001 CRC Press LLC Contents Preface Acknowledgments 1 Introduction 1.1 A brief history of morphometrics 1.2 Foundations for the study of biological forms 1.3 Description of the data sets 2 Morphometric Data 2.1 Types of morphometric data 2.2 Landmark homology and correspondence 2.3 Collection of landmark coordinates 2.4 Reliability of landmark coordinate data 2.5 Summary Part 2: Statistical and Mathematical Preliminaries for Landmark Coordinate Data 2.6 Introduction to matrix algebra 2.7 Matrix representation of landmark coordinate data 2.8 Statistical model and inference for the measurement error study 3 Statistical Models for Landmark Coordinate Data 3.1 Statistical models 3.2 Models for intra-group variability 3.3 Effect of nuisance parameters 3.4 Invariance and elimination of nuisance parameters 3.5 A definition of form 3.6 Coordinate system-free representation of form 3.7 The abilty to estimate the mean form and variance ©2001 CRC Press LLC Front Matter-C0319 11/25/02 4:22 PM Page 8 ©2001 CRC Press LLC 3.8 Analysis of example data sets 3.9 Some comments on EDMA vs. other morphometric methods 3.10 Summary Part 2: Statistical Theory for the Analysis of a Single Population 3.11 The perturbation model 3.12 Invariance and elimination of nuisance parameters 3.13 Estimation of parameters 3.14 Computational algorithms 4 Statistical Methods for Comparison of Forms 4.1 Introduction 4.2 Limiting factors in morphometrics 4.3 Comparing two forms: introduction to the problem 4.4 Superimposition-based approaches 4.5 Transformational grids and deformation-based approaches 4.6 The relationship between mathematical and scientific invariance 4.7 An invariant approach: Euclidean Distance Matrix Analysis (EDMA) 4.8 Statistical analysis of form and shape difference using EDMA 4.9 Statistical hypothesis testing for shape difference 4.10 Methods for exploring the form difference matrix 4.11 A graphical tool for the detection of influential landmarks 4.12 Analysis of example data sets: mouse mandibles 4.13 Summary Part 2: Statistical Theory for the Comparison of Two Forms 4.14 Deformation approach to form difference 4.15 Superimposition methods for comparison of forms 4.16 Matrix transformations, invariance, and identifiability issues 4.17 Form comparisons based on distances ©2001 CRC Press LLC Front Matter-C0319 11/25/02 4:22 PM Page 9 ©2001 CRC Press LLC 4.18 Form space based on Euclidean Distance Matrix representation 4.19 Statistical properties of the estimators of mean form, mean form difference, and mean shape difference matrices 4.20 Computational algorithms 5 The Study of Growth 5.1 Limiting factors in studying growth using morphometric approaches 5.2 Longitudinal vs. cross-sectional data 5.3 Assigning age and forming age-related groups 5.4 EDMA applied to the study of growth 5.5 Growth measured as relative form difference 5.6 Estimation of growth using EDMA 5.7 Statistical analysis of form and shape difference due to growth 5.8 Growth difference matrix analysis: comparing patterns of growth using growth matrices 5.9 Analysis of example data sets: differences in facial growth 5.10 Producing hypothetical morphologies from forms and growth patterns 5.11 Summary 6 Classification and Clustering Applications 6.1 Classification Analysis 6.2 Methods of classification 6.3 Dissimilarity measures for landmark coordinate data 6.4 A classification example 6.5 Cluster analysis 6.6 Clustering analysis example 6.7 Summary 7 Further Applications of EDMA 7.1 The study of asymmetry 7.2 Comparisons of molecular structures 7.3 Detection of phylogenetic signals 7.4 Summary ©2001 CRC Press LLC Front Matter-C0319 11/25/02 4:22 PM Page 10 ©2001 CRC Press LLC Postlude References ©2001 CRC Press LLC Front Matter-C0319 11/25/02 4:22 PM Page 3 ©2001 CRC Press LLC Preface “Faith is the bird that feels the light and sings while the dawn is still dark” Rabindranath Tagore Quantitative study of form and form change comprises the field of morphometrics.Cuvier (1828) was one of the first biologists to verbalize the dictum “form follows function”. Charles Darwin’s work on the theory of natural selection and evolution relied heavily on the study of form and especially variation in form (Darwin,1859).The seminal work of D’Arcy Thompson (Thompson, 1917) formulated the subject in detail. Substantial developments in both biological and statistical aspects of morphometrics occurred over the next several decades of the twentieth century.Work by Mahalanobis,Rao,and their colleagues initiated the use of multivariate statistical analysis for classification of organisms into groups. Julian Huxley (Huxley, 1932) formulated the field of allometry studying the relationship between size and shape of organisms. James Mosimann (1970) constructed a proper statistical foundation for the ideas of size,shape and allometry. The method of superimposition,particularly the Procrustes superimposition,was developed and introduced to the biological sciences by the famed anthropologist Franz Boaz and his student Eleanor Phelps (Boas, 1905;Phelps,1932;see Cole,1996).Later,Sneath (1967) initiated the use of explicit deformation functions for modeling form change. In the last two decades,the idea of studying form change using superimposition and deformation approaches has been seriously considered and further developed by several individuals.While Bookstein considered the deformation approach,Kendall and his colleagues Mardia,Goodall,Small,and others concentrated on superimposition techniques. A particular deformation approach, Finite Element Scaling Analysis, was developed by bioengi- neers (Lew and Lewis,1977;Lewis et al.,1980) and then applied to addi- tional biological problems by Cheverud and his colleagues (Cheverud,et ©2001 CRC Press LLC Front Matter-C0319 11/25/02 4:22 PM Page 4 ©2001 CRC Press LLC al., 1983, 1991; Richtsmeier and Cheverud, 1986). However, finite element scaling analysis was never fully embraced by biologists.Some of the reluctance felt by biologists stemmed from the seemingly complex math- ematics that served as the foundation of the finite element method,but the lack of invariance of this method and other superimposition techniques was recognized (Moyers and Bookstein, 1982; Cheverud and Richtsmeier, 1987; Richtsmeier, 1990). Lele (1991) formalized a precise statement regarding the lack of invariance in morphometrics and provided the solution that is invariant to the arbitrary choice of coordinate system. This monograph summarizes and synthesizes the development of this solution in the context of significant scientific problems. This work is a collaborative effort between a statistician (SL) and a biologist (JTR),each one making the other think more deeply and care- fully about the problems and solutions.It is intended for both biologists and statisticians.We have strived to make discussions as mathematically and statistically precise as possible,while keeping “the science,”that is the scientific question posed at the top of our agenda. We feel it necessary to caution the reader on one aspect of this book: this is not a statistics textbook.Although some of the basic concepts in statistics are explained in the text at an intuitive level,these discussions are not,in any way,meant as a substitute or replacement for the study of a proper statistical textbook. Most chapters have two parts. Part 1 is intended to be accessible to biologists with some statistical training,thus making it more intuitive and unfortunately,somewhat more vague and less mathematically rigorous. The underlying mathematical concepts require equations.Part 2 of most chapters contains fully rigorous mathematical arguments. Critical readers of statistical methodology should concentrate on the part 2 of most chapters.We have chosen pedagogy over mathematical rigor in Part 1 of each chapter,knowing that mathematical rigor will be demonstrated properly in Part 2.We also have provided fair- ly detailed computational algorithms,when appropriate. This book is composed of seven chapters. Chapters 2, 3, and 4 have two parts. Chapters 1, 2, 5 and 6 are written to be accessible to all readers. Part 1 of Chapters 3 to 5 contain notation and mathematical concepts, but are written to be accessible to the quantitative biologist. Part 2 of Chapters 3 and 4 are targeted towards statisticians, or more advanced quantitative biologists. Included in Chapters 2 through 6 are detailed computational algorithms for the implementation of various methods. These are targeted towards statisticians,or more advanced quantitative biologists. The book is organized in this way so that the more difficult mathematical portions can be passed over without loss of continuity or of ©2001 CRC Press LLC Front Matter-C0319 11/25/02 4:22 PM Page 5 ©2001 CRC Press LLC Acknowledgments This work could never have been done without the help, advice and contributions of many individuals.First and foremost,we would like to express our gratitude to Stephanie Harding and Sharon Taylor, our editors,for keeping the faith and trusting that we would indeed finish the project while at the same time prodding us to actually “get the book done.”Their patience and trust will forever be appreciated.This project started under the editorship of John Kimmel and then was taken over by Mark Pollard, both of whom were extremely helpful in their roles as editors. We are also grateful to Bill Heyward Bob Stern, Maggie Mogck,and the production staff for their help and patience. There were several statistician colleagues whose encouragement and thoughts have helped us along. We would like to thank Noel Cressie,Bruce Lindsay,Charles E.McCulloch,George Casella,Yehuda Vardi,C.R.Rao,and an anonymous Associate Editor for the Journal of the American Statistical Association who asked the most insightful questions.We would like to thank our collaborators from the fields of Neurosurgery and Reconstructive and Plastic Surgery whose advice and generosity enabled the application of our methods to morphometric problems of true consequence. These colleagues include: Dr. Ben Carson, Dr.Alex Kane, Dr. Jeffrey Marsh, and Dr. Craig VanderKolk. Encouragement and penetrating questions from our biologist colleagues were also instrumental throughout the development of our methodology.We specifically thank Jim Cheverud, Bill Atchley, Roger Reeves, Craig VanderKolk,Tim Cole,Alex Kane, Brian Corner, Norm McLeod,J.D.Singh,and Bill Jungers.Students,post docs and technol- ogists of the Richtsmeier lab who suffered through the initial drafts of this monograph providing invaluable advice include:Anita Lubensky, Yizheng Li, Michael Zumpano, Kristina Aldridge, Jay Mussell, Frank Williams, Jideotor Aniukwu, Gail Krovitz, Valerie DeLeon, and Christopher Valeri. Bruce Latimer, Rich Sherwood, and Dana Duren took time from their busy schedules to provide photographs of the mon- ©2001 CRC Press LLC