Generalized additive models : an introduction with R

404 Pages·2006·1.7 MB·English

by  Wood

Texts in Statistical Science
Generalized Additive
Models
An Introduction with R
J
CHAPMAN & HALL/CRC
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Texts in Statistical Science
Generalized Additive
Models
An Introduction with R
Simon N. Wood
J
Boca Raton London New York
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2006 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20110713
International Standard Book Number-13: 978-1-4200-1040-4 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable
efforts have been made to publish reliable data and information, but the author and publisher cannot
assume responsibility for the validity of all materials or the consequences of their use. The authors and
publishers have attempted to trace the copyright holders of all material reproduced in this publication
and apologize to copyright holders if permission to publish in this form has not been obtained. If any
copyright material has not been acknowledged please write and let us know so we may rectify in any
future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced,
transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or
hereafter invented, including photocopying, microfilming, and recording, or in any information stor-
age or retrieval system, without written permission from the publishers.
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right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222
Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-
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Contents
Preface xv
1 LinearModels 1
1.1 Asimplelinearmodel 2
Simpleleastsquaresestimation 3
1.1.1 Samplingpropertiesofβˆ 3
1.1.2 Sohowoldistheuniverse? 5
1.1.3 Addingadistributionalassumption 7
Testinghypothesesaboutβ 7
Confidenceintervals 9
1.2 Linearmodelsingeneral 10
1.3 Thetheoryoflinearmodels 12
1.3.1 Leastsquaresestimationofβ 12
1.3.2 Thedistributionofβˆ 13
1.3.3 (βˆ β )/σˆ t 14
i− i βˆi ∼ n−p
1.3.4 F-ratioresults 15
1.3.5 Theinfluencematrix 16
1.3.6 Theresiduals,(cid:15)ˆ,andfittedvalues,µˆ 16
1.3.7 ResultsintermsofX 17
1.3.8 The Gauss Markov Theorem: What’s special about least
squares? 17
1.4 Thegeometryoflinearmodelling 18
1.4.1 Leastsquares 19
1.4.2 Fittingbyorthogonaldecompositions 20
vii
viii CONTENTS
1.4.3 Comparisonofnestedmodels 21
1.5 Practicallinearmodelling 22
1.5.1 Modelfittingandmodelchecking 23
1.5.2 Modelsummary 28
1.5.3 Modelselection 30
1.5.4 Anothermodelselectionexample 31
Afollow-up 35
1.5.5 Confidenceintervals 36
1.5.6 Prediction 36
1.6 Practicalmodellingwithfactors 37
1.6.1 Identifiability 38
1.6.2 Multiplefactors 39
1.6.3 ‘Interactions’offactors 40
1.6.4 UsingfactorvariablesinR 41
1.7 GenerallinearmodelspecificationinR 44
1.8 Furtherlinearmodellingtheory 45
1.8.1 ConstraintsI:Generallinearconstraints 46
1.8.2 ConstraintsII:‘Contrasts’andfactorvariables 46
1.8.3 Likelihood 48
1.8.4 Non-independentdatawithvariablevariance 49
1.8.5 AICandMallow’sstatistic 51
1.8.6 Non-linearleastsquares 53
1.8.7 Furtherreading 55
1.9 Exercises 55
2 GeneralizedLinearModels 59
2.1 ThetheoryofGLMs 60
2.1.1 Theexponentialfamilyofdistributions 62
2.1.2 Fittinggeneralizedlinearmodels 63
2.1.3 The IRLS objective is a quadratic approximation to the
log-likelihood 66
CONTENTS ix
2.1.4 AICforGLMs 68
2.1.5 Largesampledistributionofβˆ 69
2.1.6 Comparingmodelsbyhypothesistesting 69
Deviance 70
Modelcomparisonwithunknownφ 71
2.1.7 φˆandPearson’sstatistic 71
2.1.8 Canonicallinkfunctions 72
2.1.9 Residuals 73
Pearsonresiduals 73
Devianceresiduals 73
2.1.10 Quasi-likelihood 74
2.2 GeometryofGLMs 76
2.2.1 ThegeometryofIRLS 77
2.2.2 GeometryandIRLSconvergence 78
2.3 GLMswithR 81
2.3.1 Binomialmodelsandheartdisease 81
2.3.2 APoissonregressionepidemicmodel 87
2.3.3 Log-linearmodelsforcategoricaldata 93
2.3.4 SoleeggsintheBristolchannel 97
2.4 Likelihood 102
2.4.1 Invariance 102
2.4.2 Propertiesoftheexpectedlog-likelihood 103
2.4.3 Consistency 106
2.4.4 Largesampledistributionofθˆ 107
2.4.5 Thegeneralizedlikelihoodratiotest(GLRT) 108
2.4.6 Derivationof2λ χ2underH 109
∼ r 0
2.4.7 AICingeneral 111
2.4.8 Quasi-likelihoodresults 113
2.5 Exercises 115