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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.
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) Preface: This section simply stops
mid-sentence. A last word should be added at the end of the last
sentence: “understanding.” 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: SDM
should be SDMB,AM/font> 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 C. apella
with location of six facial landmarks used in growth analysis indicated.
Immature (top right) and adult female (bottom right) skulls of M.
fascicularis are given in the right panel. Landmarks include: 1,
nasale; 2, intradentale superior; 3, premaxillary-maxillary junction; 4,
zygomaxillare superior; 5, maxillary tuberosity; 6, posterior nasal spine.
The posterior nasal spine is located on the sagittal plan and cannot be
seen from this view. Its location in this drawing is therefore
approximate. Though both immature specimens are from the same
developmental age group, the immature M. fascicularis skull is of a
younger age than the immature C. apella skull, which accounts for
the size difference in these immature specimens. Scale is approximate.
Landmark names are given in Table 3.3b.” 11) Postlude is full of errors and typos. 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, explanatory
models for change in the geometry of biological forms. An additional
aspect that requires attention is the selection and validation of
particular models in experimental and/or natural settings. This includes
validation of the Matrix Normal distribution as a sensible perturbation
model, and validation of explanatory models proposed by future
researchers. Finally, the concept of variability, its role in biological
form change, and the use of
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.
So easy it seemed once found, which yet unfound most would have thought
impossible.
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:
Hypothesis testing for similarity in growth pattern In this example the bootstrap reference
sample is Macaca fascicularis (the denominator). The histogram
below provides the distribution of 1000 bootstrapped G statistics (each
value of a single bootstrapped G statistic is accounted for in the
histogram) as well as the placement of Gobs (Gobs
= 1.478) within this distribution. The probability is given as
0.006 and Gobs clearly falls outside of the central tendency of the distribution. We reject the null hypothesis that
facial growth of Cebus apella is similar to facial growth of
Macaca fascicularis. When the test is done using Cebus apella
as the reference sample with 1000 bootstraps, the general outcome of the
hypothesis test (i.e., rejection of the null hypothesis) is the same but
the probability value is 0.0000. The increase in significance level
reflects the smaller sample size of the Cebus apella samples and
the influence of the sample size in the composition of the bootstrap
samples. Remember that these are one-way tests and if sample size
permits, the test should be run twice, once using the numerator as the
reference sample and again using the denominator as the reference sample. List of typos and errors in the mathematical parts
of the book: 1)
Page 50, middle of the page: Should read
2)
Page 53, line 14: ‘Let 3)
Page 55,line 19: ‘…the following formula’ should read as ‘.. the
following formulae’. 4)
Page 56, line 21: Then 5)
Page 62: The quote should be at the beginning of the chapter, not
at the beginning of this section. 6)
Page 77, line 18: ‘…,where
7)
Page 103, lines 20 and 26, page 104, line 9:
8)
Page 115, line 6: 9)
Page 119, line 8: It should read as
10)
Page 169, line 25: 11)
Page 210, line 16: It should read as 12)
Page 227-228: The equation K(K-1)/2 is used many times in the
text but the format for the equation varies. Sometimes it is written as
just written above (Pg 227) and other times it is written using a large
horizontal line to separate numerator and denominator (Pg 228). Why not
use a consistent format? 13)
Page 254: There is a change in the font. 14)
Page 285, Postlude, line 23: ‘.. at least a chapter on explicit
models ..’ 15)
Page 285, postlude, line 24: Should read ‘An additional aspect ..’
16)
Page 285, postlude, line 27: ‘a sensible perturbation model and
validation of explanatory …’
An invariant approach to statistical
analysis of shapes
By S. Lele and JT Richtsmeier Interdisciplinary studies in statistics series.
2001 Chapman and Hall-CRC press, London.
|Preface |Book Reviews |
Errata |Data Sets|
From the Preface of Lele and Richtsmeier (2001)
Book Reviews
Journal: Biometrics Reviewer: Ian Dryden Journal: Journal of Human Evolution Reviewer: Dennis Slice Journal: Biometrics Journal Reviewer: D. Stoyan
Errata
in
the study of biological variability will play a pivotal role in the future
applications of our methodology.
5.9.1
Statistical testing of 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, Gobs can be calculated following methods
outlined above and detailed in Chapter 4 for the two-sample case.
, brackets are missing.
denote the ..’ . This should read ‘Let
denote the ..’.
… Remove the word ‘Then’.
’ should read as ‘…,where
’.
should read as
.
should be replaced by
.
should read
.
Data Sets
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: