Table Of ContentA S U R V E Y
M U L T I V A R I A T E M E T H O D S
F O R S Y S T E M A T I C S
Nancy A. Neff
Leslie F. Marcus
for a workshop:
Numerical Methods in Systematic Mammalogy
American S ociety of Mammalogists
Annual Meeting 9 June 1980
.mi
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THE LIBRARY
A S U R V E Y 0 F
M U L T I V A R I A T E M E T H O D S
F O R S Y S T E M A T I C S
Nancy A. Neff
American Museum of Natural History, and
City College of the City University of New Yor k
Leslie F. Marcus
American Museum o f Natural History, and
Queens College of the City University of New Yor k
printed at the American Museum o f Natural History, New York
supported by National Science Foundation Research Grant DEB 79-17382
copyright @ 1980
by Nancy A. Neff and Leslie F. Marcus
All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any
means, electronic, mechanical, photocopying, recording, or otherwise,
without the prior written permission of the copyright owners.
Nancy A. Neff
Leslie F# Marcus
Dept. Vertebrate Paleontology Dept. Invertebrates
American Museum of Natural History
Central Park West at 79th Street
New York, New York 10024 USA
This publication may be cited:
Neff, N. A. and L. F. Marcus. 1980. A Survey of Multivariate Methods
for Systematics. New York: privately published
This manual was prepared for distribution at a workshop on Numerical
Methods in Systematic Mammalogy, held on 9 June 1980 during the 1980
Annual Meeting of the American Society of Mammalogists in Kingston
Rhode Island.
If additional copies are still available, they may be obtained by
sending:
1. a self-addressed mailing label, and
2. a check or money order for $1.00 made out to
the American Museum of Natural History, for
postage and handling,
to one of the authors at the above address.
T A B L E O F C O N T E N T S
Preface v
Acknowledgments vii
Table of Abbreviations viii
Table of Symbols ix
I N T R O D U C T I O N 1
Background for Use of this Manual 9
Design of Study
Assumptions 15
1) Independence and random sampling 21
2) Multivariate normality 22
3) Central Limit Theorem 24
4) Transformations 27
5) Homogeneity of variability 28
6) Robustness of estimators and tests 29
7) Validity of estimates and jackknife estimators 31
8) Randomization tests 34
Sample size 36
Missing Data 39
Implementation
Errors and accuracy ^2
Iteration ^
Interpretation and publication 45
M E T H O D S 4 7
Principal Components Analysis 51
Principal Coordinates Analysis 70
Other Principal Components Related Methods
Correspondence Analysis 81
Biplot 83
Nonmetric Multidimensional Scaling 86
Factor Analysis 94
Andrews Plots 115
Multiple Regression 118
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NEFF and MARCUS, 1980
Canonical Correlation Analysis 132
Multidimensional Contingency Tables 141
Discriminant Analysis 145
Multivariate Analysis of Variance 152
Canonical Variates Analysis 157
Discrimination 164
Summary of terminology and recommendations 171
P U R P O S E S
Data Screening 177
Data Reduction 181
Exploration: Looking for Structure 187
Cluster Analysis, Numerical Cladistics, and
Tree Analysis 193
Size and Shape 199
Biorthogonal transformation grids 205
Fourier analysis 207
Statistical Inference 209
B I B L I O G R A P H Y 2 1 3
Appendix I: Publication of the Results of Multivariate
Studies —by Michael A. Bogan 237
Appendix II: Statistical Packages and Computer Programs,
—with David J. Schmidly and Mark D. Engstrom 239
iv
P R E F A C E
That the biological world is multivariate has been widely
recognized. The importance of quantifying one's observations for
analysis and for communication has also been recognized. These facts,
plus widespread computer availability has led to the near exponential
growth in recent years in the applications of multivariate analyses in
systernatics and other areas of biology. Multivariate analyses are
numerical techniques used to study and describe the covariation
between variables, individuals, or both together. In systernatics, the
variables are usually characters measured on a set of individuals.
Tne g oals are varied: some possible purposes include data or
dimension reduction, searching for structure, testing the fit of one's
data to a model, or discriminating between populations. The strength
of multivariate analysis lies in the ability it confers on the user to
examine many variates simultaneously and quantitatively.
Multivariate methods have not yet fulfilled all expectations, and
will not without further development of the statistical methodology.
Demands by the practitioners for more relevant methodology will
encourage the statisticians. Also there is much for most of us to
learn about the potential of the methods already developed. All of
the questions have not been asked and all informative structures and
models have not yet been explored.
Our manual, and the American Society of Mammalogists workshop which
it accompanied, was c onceived in response to the growth and interest
in this methodology. We trust that this initial effort will shortly
be made o bsolete by more complete discussions reflecting the results
of continued growth and exploration.
Although the increase in use of numerical methods is sometimes
cited as evidence of increasing objectivity in systernatics, the use of
measurements for data, or mathematical models for hypotheses, does not
obviate a continuing need for close observation of the specimens
themselves and clear thinking about the biological hypotheses being
investigated. Numerical methods are tools only—powerful tools, yes,
but only means to an end. We have written this manual in the hope
that concentration on how n umerical methods can be used in systernatics
will help the system3tists be demanding of the methodology and search
for the numerical methods to fit the biological questions, and never
vice versa.
We believe that numerical methods, even in their present state of
development, have a great potential for usefulness in systernatics.
The rapid increase in the employment of multivariate studies indicate
that others are of this opinion also. As interaction between
systematists and statisticians increases, we expect even more useful
design of methods, permitting closer tailoring to the biological
structures and relationships we study, and fitting more e xactly the
models and hypotheses which systematists wish to test.
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NEFF and MARCUS, 1980
Just as drawing a specimen usually increases the amount one sees on
it, so defining and taking measurements can often be instructive
because of the time spent thinking hard about and closely looking at
the specimens themselves Unexpected relationships among values or
0
individuals should lead one back to the specimens for a fresh look.
In sum, numerical methods are certainly not an alternative to looking
at the specimens themselves!
This manual is not about numerical taxonomy. We ha ve discussed
methods of multivariate analyses with respect to a diversity of goals
in systematics. We h ave written from a fairly strongly held
philosophical position about how s cientific studies are most
effectively done, but not with a narrow definition of appropriate
goals or topics of study. Numerical taxonomy, in our use of the term,
includes the philosophical positions well discussed in Sneath and
Sokal [1973]; in contrast, we wished to survey methodology in a
neutral fashion, divorced from a specific choice of goals. Also,
because of our interest in morphology, we have emphasized ordination
and descriptive techniques, rather than clustering techniques which
are well covered by s everal books to date.
One final point about our approach may be made e xplicit: what was
included in this survey, and the nature of the discussion, was largely
motivated by initially strong feelings on the part of at least one of
us about how numerical methods are most wisely used in systematics.
The research and writing have not substantially changed the major
tenets of our philosophy, although some specific details of belief
have certainly changed. However, we have come to realize that this
methodology is in fact complex, and difficult to present and discuss
from the orientation of biologists or systematists rather than
statisticians. We a re not attempting to set standards for numerical
studies in systematics. We do urge anyone, inexperienced or
experienced, to practice a healthy scepticism towards the literature
and the recommendations therein, including this text.
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A C K N O W L E D G M E N T S
We have been aided in the preparation of this manual by numerous
and diverse people. Dr. F. James Rohlf, Department of Ecology and
Evolution, SUNY at Stony Brook, gave generously of his time and
expertise. He critically read early drafts of most sections and
greatly improved the product. We are grateful for both his criticism
and his encouragement.
Tne National Science Foundation supported the production of this
manual, and the American Society of Mammalogists workshop at which it
was distributed, through a research grant DEB 79—17332 to the authors.
We thank the American Museum of Natural History, most especially
Ms. Diane Menditto, for help with and review of the grant application,
and subsequent administration of the grant. The staff of the museum
library was, as ever, cheerfully helpful. The Department of
Invertebrates, in which one of us (LFM) is a Research Associate, was
extremely helpful in supplying workspace and use of a CRT and Diablo
terminals; we especially thank Ms. Julia Golden of that department for
permitting us incredible license in the use of her office. The
Department of Vertebrate Paleontology also supplied space, supplies
and support. The Print Shop of the American Museum, in the person of
Mr. Vincent Tumillo, performed heroically in printing this manual
under too short a time estimate from two inexperienced publishers!
This manual will accompany a workshop on 9 June 1980 at the Annual
Meeting of the American Society of Mammalogists. We thank Drs. F. J.
Rohlf, M. A. Bogan, D. J. Schmidly, D. A. Schlitter, and G.D. Schnell
for agreeing to serve as discussants on a panel at this workshop. We
appreciate ASM's interest, and especially the support and help of Dr.
J. Mary Taylor. Drs. Duane A. Schlitter and Sydney Anderson provided
early comments and encouragement.
We thank the City University of New York, University Computer
Center for making available to us text-editing (WYLBUR) and
word-processing (Waterloo SCRIPT) capabilities.
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NEFF and MARCUS, 1980
ABBREVIATIONS USED IN THIS TEXT
A NOVA analysis of variance
BMDP Biomedical Computer Programs, P-series
CCA canonical correlation analysis
CVA canonical variates analysis
MANOVA multivariate analysis of variance
ML maximum likelihood (factor analysis)
MST minimum spanning tree
NMDS nonmetric multidimensional scaling
NT numerical taxonomy
NTSYS Numerical Taxonomic System of Multivariate Statistical Programs
OTU operational taxonomic unit
PCA principal components analysis
PCORD principal coordinates analysis
PFA principal factor analysis; PFA(iter) refers to
the iterative solution to PFA
SAS Statistical Analysis System
SPSS Statistical Package for the Social Sciences
VIF variance inflation factor
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