Analysis of variance and covariance: How to choose and construct models for the life sciences

303 Pages·2007·1.361 MB·English

by  C. Patrick Doncaster, Andrew J. H. Davey

Analysis of Variance and Covariance: How to Choose and Construct
Models for the Life Sciences
Analysisofvariance(ANOVA) isacoretechnique foranalysingdatainthe life
sciences. Thisreferencebookbridges the gapbetweenstatisticaltheory and
practicaldataanalysisbypresentingacomprehensivesetoftablesforallstandard
modelsof analysisof varianceand covariance withupto threetreatment
factors.The bookwill serveas a tool tohelp post-graduatesandprofessionals
definetheir hypotheses, designappropriate experiments, translate themintoa
statisticalmodel,validate the outputfrom statistics packagesand verifyresults.
Thesystematic layoutmakesiteasyfor readersto identifywhich types of
modelbest fit thethemes theyareinvestigating, andto evaluatethe strengths
andweaknessesof alternative experimentaldesigns. Inaddition, a concise
introductiontotheprinciplesofanalysisofvarianceandcovarianceisprovided,
alongsideworkedexamples illustrating issues anddecisions facedbyanalysts.
patrick doncaster is a Reader in Ecology in the School of Biological
Sciencesat the Universityof Southampton.
andrew davey is an Environmental Scientist at the UK Water Research
centre(WRc).

Analysis of Variance and Covariance
How to Choose and Construct Models
for the Life Sciences
C. PATRICK DONCASTER and ANDREW J.H. DAVEY
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521865623
© C. P. Doncaster and A. J. H. Davey 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13 978-0-511-35593-6 eBook (NetLibrary)
ISBN-10 0-511-35593-9 eBook (NetLibrary)
ISBN-13 978-0-521-86562-3 hardback
ISBN-10 0-521-86562-X hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
Contents
Preface page ix
Introduction to analysis of variance 1
What is analysis of variance? 1
How to read and write statistical models 2
General principles of ANOVA 7
Assumptions of ANOVA 14
How to distinguish between fixed and random factors 16
Nested and crossed factors, and the concept
of replication 21
Uses of blocking, split plots and repeated measures 25
Uses of covariates 29
How F-ratios are constructed 35
Use of post hoc pooling 38
Use of quasi F-ratios 40
Introduction to model structures 42
Notation 43
Allocation tables 43
Examples 46
Worked example 1: Nested analysis of variance 47
Worked example 2: Cross-factored analysis
of variance 49
Worked example 3: Split-plot, pooling and
covariate analysis 51
Key to types of statistical models 57
How to describe a given design with a
statistical model 58
v
vi Contents
1 One-factor designs 61
1.1 One-factor model 62
2 Nested designs 67
2.1 Two-factor nested model 68
2.2 Three-factor nested model 72
3 Fully replicated factorial designs 76
3.1 Two-factor fully cross-factored model 78
3.2 Three-factor fully cross-factored model 86
3.3 Cross-factored with nesting model 98
3.4 Nested cross-factored model 109
4 Randomised-block designs 115
4.1 One-factor randomised-block model 121
4.2 Two-factor randomised-block model 128
4.3 Three-factor randomised-block model 134
5 Split-plot designs 141
5.1 Two-factor split-plot model (i) 146
5.2 Three-factor split-plot model (i) 150
5.3 Three-factor split-plot model (ii) 154
5.4 Split-split-plot model (i) 158
5.5 Split-split-plot model (ii) 163
5.6 Two-factor split-plot model (ii) 167
5.7 Three-factor split-plot model (iii) 170
5.8 Split-plot model with nesting 173
5.9 Three-factor split-plot model (iv) 176
6 Repeated-measures designs 179
6.1 One-factor repeated-measures model 187
6.2 Two-factor repeated-measures model 190
6.3 Two-factor model with repeated measures
on one cross factor 195
6.4 Three-factor model with repeated measures
on nested cross factors 200
6.5 Three-factor model with repeated measures
on two cross factors 205
6.6 Nested model with repeated measures on
a cross factor 214
6.7 Three-factor model with repeated measures
on one factor 220
Contents vii
7 Unreplicated designs 229
7.1 Two-factor cross factored unreplicated model 230
7.2 Three-factor cross factored unreplicated model 232
Further Topics 237
Balanced and unbalanced designs 237
Restricted and unrestricted mixed models 242
Magnitude of effect 244
A priori planned contrasts and post hoc unplanned
comparisons 245
Choosing experimental designs 248
Statistical power 248
Evaluating alternative designs 250
How to request models in a statistics package 258
Best practice in presentation of the design 260
Troubleshooting problems during analysis 264
Glossary 271
References 281
Index of all ANOVA models with up to three factors 284
Index 286
Categories of model 288

Preface
Hypothesis testing in the life sciences often involves comparing samples
ofobservations,andanalysisofvarianceisacoretechniqueforanalysing
such information. Parametric analysis of variance, abbreviated as
‘ANOVA’,encompassesagenericmethodologyforidentifyingsourcesof
variation in continuous data, from the simplest test of trend in a single
sample, or difference between two samples, to complex tests of multiple
interacting effects. Whilst simple one-factor models may suffice for
closely controlled experiments, the inherent complexities of the natural
world mean that rigorous tests of causality often require more sophisti-
cated multi-factor models. In many cases, the same hypothesis can be
tested using several different experimental designs, and alternative
designs must be evaluated to select a robust and efficient model. Text-
booksonstatisticsareavailabletoexplaintheprinciplesofANOVAand
statistics packages will compute the analyses. The purpose of this book
is to bridge between the texts and the packages by presenting a com-
prehensive selection of ANOVA models, emphasising the strengths
and weaknesses of each and allowing readers to compare between
alternatives.
Our motivation for writing the book comes from a desire for a more
systematic comparison than is available in textbooks, and a more con-
sidered framework for constructing tests than is possible with generic
software. The obvious utility of computer packages for automating
otherwise cumbersome analyses has a downside in their uncritical pro-
duction of results. Packages adopt default options until instructed
otherwise, which will not suit all types of data. Numerous problems can
arisefromincautioususeofanystatisticspackage,beitofthesimplestor
ix