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Quantitative study of form and form change comprises the field of morphometrics. This field has had a long history. Cuvier (1828) was probably the first biologist to verbalize the dictum “form follows function”. Charles Darwin’s work on 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. More recent work in this field has always reverted back to Thompson, either to clarify or to repudiate novel approaches and ideas. 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 bioengineers (Lew and Lewis, 1977; Lewis et al., 1980) and then applied to additional biological problems by Cheverud and his colleagues (Cheverud et 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 mathematics 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 carefully aboutthe 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.
This book is composed of six chapters. Each chapter has two parts. Chapters 1, 2 and 6 are written to be accessible to all readers. Part 1 of Chapters 3-5 contains notation and mathematical concepts, but is written to be accessible to the quantitative biologist. Part 2 of Chapters 3-5 is 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 understanding.
Journal: Biometrics Reviewer:
Journal: Journal of Human Evolution Reviewer:
Journal: Biometrics Journal Reviewer: We apologize to the readers of our book for the many
errors. The book was rushed to printing prematurely without our knowledge
and many errors were made during production. The book will be reprinted
with corrections in the near future, probably in 2002. The corrections
are numerous. Some of the more obvious are listed below. We will add
more detail as time goes on. Page numbers given below refer to page numbers in the
published book. General problems: 1) The use of “N” versus “n” for sample size. The
use is inconsistent and should be “N” throughout the book 2) 3) Many of the references are incomplete and
incorrect and contain misspellings of authors’ names. This is due to the
production staff using a working copy rather than the finished copy.
These errors are being updated and corrected. 4) The notation used on page 162 is incorrect: 5)Table 4.5 (page 186) contains a random line (the first line: –1.961) that should be deleted. The whole point of that table is that the confidence interval EXCLUDES the number 0. There are numerous errors in the text on page 186 that discuss this table that are also very misleading. That paragraph should read: “This confidence interval does not contain zero and therefore suggests that the two populations differ significantly in scale. Remember, however, that there is no single value that represents ‘size’ and that this result may change depending upon the chosen scaling factor. Since we found a difference between the samples for this particular measure of ‘size’, differences that have been previously estimated for these samples are considered differences in the shapes of the populations. With evidence for a difference in size, confidence intervals for the estimated shape difference matrix can be examined.” 6) Page 189-191. The form difference matrix that begins on page 189 is the same as the form difference matrix presented on page 190, except that the one that begins on page 189 is in vector format and the entries are sorted from minimum value to maximum value. Unfortunately, the first few lines of the vector are given on page 189 and then is interrupted by the form difference matrix on page 190. The last portion of the vector is given on page 191. The vector that starts on page 189 and is continued on page 191 (BUT NOT PAGE 190) is a single vector. The columns on pages 189 and 191 should be labeled: “Linear Distance”, “Estimate”. Page 190 gives the same form difference matrix but written in matrix format and should be presented isolated from the vector. 7) The p-value given at the bottom of page 191 goes with the table that is presented on page 192. 8) Confidence intervals that go from page 192 to 193 span two pages. The column headings on pages 192-193 should read: “Linear distance”, “Lower limit”, “Estimate”, “Higher Limit” 9) Sections 5.7 and 5.8 were switched in order. The published section 5.7 should be numbered 5.8 and should appear AFTER the section that is currently labeled 5.8. In other words Section 5.8 (now on pages 229-230) should be assigned the section number 5.7 and should be put in front of the section entitled, “Statistical analysis of form and shape difference due to growth” on page 226 which should be renumbered as 5.8. 10) There are multiple errors in the figure caption for Figure 5.3 on page 232. It should read:
“Figure 5.3
Immature (top left) and adult female (bottom left) skulls of 11) It should read as follows: POSTLUDE This monograph provides the foundations for quantitative analysis of landmark coordinate data based on the invariance principle. In addition to developing the statistical foundations for the study of forms and shapes as represented by landmark coordinate data, we provide descriptions and examples of various applications of our approach. The fields of application in our monograph ranged from Paleontology where the origins of morphometrics lay, to the modern subjects of reconstructive surgery, the phenotypes of genetically engineered animal models, and molecular structure. Where should we go from here? Although we are not visionaries like D’Arcy Thompson, we take this opportunity to speculate about the future of the field and the problems that need to be addressed for the field to progress.
There is tremendous potential
for the use of landmark data analysis in the fields of medicine, molecular
biology, pattern recognition, computer vision, and biomechanics. One area
of study of particular interest to us is the fusion of morphological data
and other kinds of data; e.g., behavioral, genetic, life history data.
This monograph discussed techniques that are most useful in exploratory or
descriptive research. Now is the time to go beyond description and
venture into explanation. To accomplish this task, we need to contemplate
the construction of models for various processes that might be responsible
for form change (e.g., growth, evolution, biomechanical properties,
disease, genetic mutations). We hope that the next edition of this
monograph will have at least a chapter on explicit, We cannot predict the future, but for us it has been an extraordinary journey through the morphometrics landscape. We close with the following quote that we feel particularly appropriate after almost ten years of collaboration.
John Milton 12) An early version of section 5.9.1 (that looked at a different data set!) was included in the book by mistake. The updated version follows: ## 5.9.1 Statistical testing of similarity in growth patternHypothesis testing for similarity in growth pattern ##
The range of the
elements of the growth difference matrix just presented (0.671 - 0.991)
indicates that the difference in facial growth in these two species is not
simply a matter of scale. Our previous discussion of the differences in
growth matrices suggests that there are significant differences in facial
growth patterns of these two species. If statistical testing for
difference in overall growth pattern is required or desired by the
investigator
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The following datasets are available for download. The trisomic and normal mice datasets come from work supported by the National Science Foundation under Grant No. 0049031: |