Multivariate Analysis of Ecological Communities

214 Pages·1987·12.435 MB·English

by  P. G. N. Digby, R. A. Kempton (auth.)

Population and Community Biology
MULTIVARIATE ANALYSIS OF
ECOLOGICAL COMMUNITIES
Population and Community Biology
Series Editors
M. B. Usher
Reader, University ofYork, UK
M. L. Rosenzweig
Professor, Department ofEcology and Evolutionary
Biology, University ofArizona, USA
The study ofboth populations and communities
is central to the science ofecology. This series of
books will explore many facets ofpopulation
biology and the processes that determine the
structure and dynamics ofcommunities.
Although individual authors andeditors have
freedom to develop their subjects in their own
way, these books will all be scientifically rigorous
and often utilize a quantitative approach to
analysing population and community
phenomena.
MULTIVARIATE
ANALYSIS OF
ECOLOGICAL
COMMUNITIES
P. G. N. Digby
Department ofStatistics
Rothamsted Experimental Station
Harpenden
and
R. A. Kempton
Scottish Agricultural Statistics Service
(formerly Agriculturaland Food Research Council Unit ofStatistics)
University ofEdinburgh
LONDON NEW YORK
CHAPMAN AND HALL
First published in 1987 by
Chapman and Hall Ltd
11 New Fetter Lane, London EC4P 4EE
Published in the USA by
Chapman and Hall
29 West 35th Street, New York NY 10001
© 1987 P. G. N. Digby and R. A. Kempton
This title is available in both hardbound and
paperback editions. The paperback edition is
sold subject to the condition that it shall not,
by way of trade or otherwise, be lent, resold,
hired out, or otherwise circulated without the
publisher's prior consent in any form of binding
or cover other than that in which it is published
and without a similar condition including this
condition being imposed on the subsequent
purchaser.
All rights reserved. No part of this book may
be printed, or reproduced or utilized in any
form or by any electronic, mechanical or other
means, now known or hereafter invented,
including photocopying and recording, or in
any information storage and retrieval system,
without permission in writing from the publisher.
British Library Cataloguing in Publication Data
Digby, P. G. N.
Multivariate analysis of ecological
communities.-{Population and community
biology)
1. Biotic communities 2. Multivariate
analysis
I. Title II. Kempton, R. A. III. Series
574.5'247'01519535 QH541.l5.M34
ISBN-13: 978-0-412-24650-0 e-ISBN-13: 978-94-009-3135-0
DOl; 10.1007/978-94-009-3135-0
Library of Congress Cataloging in Publication Data
Digby, P.G.N.
Multivariate analysis of ecological communities.
(Population and community biology)
Bibliography: p.
Includes index.
1. Biotic communities-Statistical methods.
2. Ecology-Statistical methods. 3. Multivariate
analysis. I. Kempton, R.A., 1946- II. Title.
III. Series.
QH541.15.S72D54 1987 574.5'247 86-17583
Contents
Preface vii
1 Ecologicaldata 1
1.1 Types ofdata 7
1.2 Forms ofdata 9
1.3 Standardization and transformation ofdata 12
1.4 Constructingassociation data 15
2 Preliminary inspection ofdata 27
2.1 Displayingdata values 27
2.2 Mapping 32
2.3 Displayingdistributions ofvariables 34
2.4 Bivariateand multivariate displays 42
3 Ordination 49
3.1 Direct gradient analysis 49
3.2 Principal components analysis 55
3.3 Correspondence analysis 70
3.4 Ordination methods when rows or columns are grouped 76
3.5 Principal coordinates analysis 83
3.6 The horseshoeeffect 93
3.7 Non-metric ordination 97
3.8 Case studies 103
4 Methodsfor comparingordinations 112
4.1 Procrustes rotation 112
4.2 Generalized Procrustes analysis 117
4.3 Comparing ordination methods by multiple Procrustes
analysis 121
5 Classification 124
5.1 Agglomerative hierarchical methods 125
5.2 Divisive hierarchical methods 129
5.3 Non-hierarchical classification 131
5.4 Visual displays for classification 137
5.5 Case study 142
5.6 Methods for comparing classifications 147
Contents
VI
6 Analysisofasymmetry 150
6.1 Rowand column plots 151
6.2 Skew-symmetry analysis 155
6.3 Case studies 159
6.4 Aproofofthe triangle-area theorem 173
7 Computing 176
7.1 Computing options 176
7.2 Examples ofGenstat programs 180
7.3 Handling missing values 185
7.4 Conclusion 185
7.5 List ofsoftware 185
References 187
Appendix Matrix algebra 193
A.1 Matrices and vectors 193
A.2 Particularforms ofmatrices 195
A.3 Simple matrix operations 196
A.4 Simple geometry and some special matrices 197
A.5 Matrix inversion 198
A.6 Scalarfunctions ofmatrices 198
A.7 Orthogonal matrices 201
A.8 Matrix decompositions 202
A.9 Conclusion 203
Index 204
Preface
The last ten years have seen an enormous increase in the development and
application of multivariate methods in ecology; indeed the perceived
importance of these methods for elucidating the complex interactions
observed in community studies is shown by the number of recent books
devoted to introducing the more common multivariate techniques to
ecologists (Williams, 1976; Orloci, 1978; Whittaker, 1978a, b; Gauch, 1982;
Legendreand Legendre, 1983;Pielou, 1984)andbythechaptersaddedtonew
editionsofmoregeneraltextsonquantitativeecology(e.g.Greig-Smith, 1983;
Kershaw and Looney, 1985).
Two reasons can be put forward to explain this development. The first is
undoubtedly the increasing availability of cheap computing power which
makes it feasible to analyse the large data matrices involved in community
studies.Thesecond,perhapslesswidelyappreciated,isthechangeinemphasis
oftheoretical work on multivariate analysis, away from the development of
formal statisticalmodelsandassociateddistribution theory towardsdescrip
tive techniques for exploring pattern in data sets and providing succinct
summaries and displays. This new approach, termed 'pattern analysis' by
Williams(1976), has led to a range ofstatistical techniques which have been
enthusiastically taken up by ecologists to replace the collection of ad hoc
proceduresdeveloped over the years for analysingcommunity data.
This book brings togetherfor thefirst time many ofthe new techniquesof
multivariateanalysisappropriateforecologicaldata. The techniquesinclude
thefamiliar, andsomelessfamiliar, methodsforordinationandclassification
and also some special techniques; for example, methods for analysing
asymmetricassociation matricesand for comparingseveraldifferent ordina
tions. Two preliminary chapters introduce the different types and forms of
ecological community data and methods for preliminary inspection ofdata
using graphs and tables. Little attempt is made to review early methods of
analysis, many of which should now be of only historical interest: a good
review oftheseearly developments is given by Greig-Smith (1983).
A matrix approach is adopted for the theoretical development and this
formsa unifyingthreadwhichlinksapparentlydisparatemethodsofanalysis.
Itisexpectedthat readerswill havesomefamiliarity with matrixalgebra, but
the Appendix contains the basic mathematical definitions and properties
underlyingthemethodscovered,andisincludedforreferenceandasaconcise
refresher course. Suitable elementary primers in matrix algebra are given in
Vlll Preface
Williams(1976)and Pielou(1984), although both ofthese texts subsequently
covera very limited range ofmultivariate techniques.
Aparticularstrengthofthisbookliesinthenumerousillustrativeexamples.
Several data sets are subject to successive reanalysis throughout the book,
allowingcomparisontobemadeoftheresultsofdifferentmethodsofanalysis.
Detailed case studies are also discussed at the ends ofchapters, allowing a
more considered interpretation of the results of multivariate analysis and
leadinginsomecasestomoreformalmodellingandanalysis. Readerswhoare
used toseeingthesamefamiliarexamplesrepeated in many recent textbooks
will bepleasedtoknowthat, whilethedatasetsherearein mostcasesdrawn
from previously published sources, the analyses presented are nearly all
entirely new to this book. Most ofthese analyses were carried out using the
Genstatstatisticalcomputerpackageandafinalchapterincludesdetailsofthe
relevantmultivariatedirectivesandbriefexamplesoftheinputinstructionsfor
the program.
ThestimulationforthisbookoriginatedfromaWorkshoponMultivariate
MethodsforEcologistsorganizedjointlybytheBritishEcologicalSocietyand
Biometric Society at Edinburgh in September 1981. We are grateful to the
participants who contributed ideas and examples to that Workshop and
particularlyto Ron Smithwho was responsiblefor much oftheorganization
and contributed to ourinitial thoughtsabout theform and structure for the
book. We are also grateful to John Gower and Michael Usher who made
many helpfulcriticisms ofanearlydraft. Wealso thank especially Sue Land
for her careful and patient preparation ofthe typescript. Our debt to John
Gower, who has been a stimulating Rothamsted colleague for both ofus, is
particularlygreatand goesfar beyond the many references to his pioneering
work cited in this book.
P. G. N. Digby and R. A. Kempton
Rothamsted and Cambridge
December 1985
1 Ecological data
The development ofstatistical methods in field biology has, until recently,
been dominated by the requirements ofagricultural scientists working with
highlycontrolledsystemsofexperiments. Here theexperimentalfactors (e.g.
plant variety or nutrient level or site) generally cover only a limited range,
while the variation in uncontrolled factors is made as small as possible, for
example, by spraying to control disease. In consequence, the range of
measured responses and the unexplained error in those responses are also
fairly small. This has led to the development ofa large statistical theory in
which the pattern ofresponse, possibly after transformation, is described by
the effects of additive factors and a residual response whose distribution
approximatestosomeknownmathematicalform(e.g.anormaldistribution).
In contrast, data from ecological communities rarely conform to these
modelassumptions,evenwhentheyderivefromformalexperiments.Consider
thetypicalexampleofTable1.1,whichwewillbeusingonseveraloccasionsto
illustrate the methods developed in this book. These data show the relative
yields ofmeadow species recorded between 1973 and 1975 in plots from the
famous Park Grass experiment laid out by John Bennett Lawes at
Rothamsted Experimental Station in 1856 (Williams, 1978). Here ofcourse
plantspeciesspanafarwiderrangeofgeneticdiversitythanthatusuallyfound
incroptrialsand, after 130years ofcontinuousfertilizer treatment, the plots
span a far wider range ofsoil nutrient status. In consequence the range of
individual yields is far larger than normally observed in agricultural
experiments;in particular, more than 50% ofthecellentriesinTable 1.1 are
represented byzeroyields,eventhoughanumberofrarespeciesfrom thesite
have beencompletely omitted. Furthermore, the interaction betweenspecies
andtreatments,andamongspecieswithinthesameplot,isverycomplex. For
these data, it is ofinterest to identify similarities in overall species response
among the different treatments as well as associations among the different
species themselves, but the standard procedures ofstatisticalestimation and
hypothesis testingareclearly inappropriate.
Thisbookisconcernedwithmethodsforinvestigatingthecomplexpatterns
whichariseinecologicaldataofthistype. However,beforewelookindetailat
thedifferentmethods, wewillfirst considerthecommonertypesandforms of
such data and how one form may be constructed from another.
5
973-7 13ganic c 18.815.35.727.8 2.7 0.7 13.80.11.3
ots1 Or d 32.316.19.47.0 2.2 3.2 20.3 0.3
Grasspl 18KNaMg c 51.8 23.01.6 0.5 14.1 0.81.5
rk N2 d 82.7 17.2 0.1 t
a
P
son 1Nt c 19.6 9.5 48.7 1.32.50.1
ment d 84.4 11.1 2.9
at
e g
fertilizertr treatment 14N1PKNaM da 13.436.80.3I37.039.8 2.41.3 2.71.4 2.72.04.2 0.40.44.65.6
differentcandd. dfertilizer 16KNaMg a 4.31.142.4 0.1 3.1 3.40.21.5 0.4t3.1
undersa,b, mberan NIP d 1.028.86.137.6 7.8 1.0 0.6 3.00.10.8
matteryield)ngtreatment Plotnu 17Ntt da 6.11.924.47.914.25.30.30.80.23.9 t5.011.0 7.813.40.30.416.7 2.73.8t0.20.1
totaldryferentlimi 7PKNaMg da 29.00.47.35.211.31.10.430.8 0.9t 5.03.0 0.314.70.50.10.3 2.56.50.61.30.21.1
ntspecies(%esplitfordif 8PNaMg da 4.64.02.81.09.613.12.06.10.11.6 1.91.5 15.611.70.33.60.312.0 16.75.60.20.90.41.2
ndanceofplamainplotsar0.05%, 3Unmanured da 15.52.32.51.07.26.50.20.21.02.0 0.12.22.2 33.213.6 0.38.5 4.31.50.20.90.4
Table1.1Relativeabu(Williams,1978).The14tindicatesabundance<•speciesnotrecorded. Species AgrostistenuisAlopecuruspratensisAnthoxanthumodoratumArrhenatherumelatiusBrizamediaBromusmollisCynosuruscristatusDactylisglomerataDeschampsiacaespitosaFestucarubraFestucapratensisHelictotrichonpubescensHolcuslanatusPoapratensisPoatrivialis