Generalized Additive Models An Introduction with R SECOND EDITION CHAPMAN & HALL/CRC Texts in Statistical Science Series Series Editors Joseph K. Blitzstein, Harvard University, USA Julian J. Faraway, University of Bath, UK Martin Tanner, Northwestern University, USA Jim Zidek, University of British Columbia, Canada Statistical Theory: A Concise Introduction Problem Solving: A Statistician’s Guide, F. Abramovich and Y. Ritov Second Edition C. Chatfield Practical Multivariate Analysis, Fifth Edition A. Afifi, S. May, and V.A. Clark Statistics for Technology: A Course in Applied Statistics, Third Edition Practical Statistics for Medical Research C. Chatfield D.G. Altman Analysis of Variance, Design, and Regression : Interpreting Data: A First Course Linear Modeling for Unbalanced Data, in Statistics Second Edition A.J.B. Anderson R. Christensen Introduction to Probability with R K. Baclawski Bayesian Ideas and Data Analysis: An Introduction for Scientists and Statisticians Linear Algebra and Matrix Analysis for R. Christensen, W. Johnson, A. Branscum, Statistics and T.E. Hanson S. Banerjee and A. Roy Modelling Binary Data, Second Edition Modern Data Science with R D. Collett B. S. Baumer, D. T. Kaplan, and N. J. Horton Modelling Survival Data in Medical Research, Mathematical Statistics: Basic Ideas and Third Edition Selected Topics, Volume I, D. Collett Second Edition P. J. Bickel and K. A. Doksum Introduction to Statistical Methods for Clinical Trials Mathematical Statistics: Basic Ideas and T.D. Cook and D.L. DeMets Selected Topics, Volume II P. J. Bickel and K. A. Doksum Applied Statistics: Principles and Examples Analysis of Categorical Data with R D.R. Cox and E.J. Snell C. R. Bilder and T. M. Loughin Multivariate Survival Analysis and Competing Statistical Methods for SPC and TQM Risks D. Bissell M. Crowder Introduction to Probability Statistical Analysis of Reliability Data J. K. Blitzstein and J. Hwang M.J. Crowder, A.C. Kimber, T.J. Sweeting, and R.L. Smith Bayesian Methods for Data Analysis, Third Edition An Introduction to Generalized B.P. Carlin and T.A. Louis Linear Models, Third Edition A.J. Dobson and A.G. Barnett Second Edition R. Caulcutt Nonlinear Time Series: Theory, Methods, and Applications with R Examples The Analysis of Time Series: An Introduction, R. Douc, E. Moulines, and D.S. Stoffer Sixth Edition C. Chatfield Introduction to Optimization Methods and Their Applications in Statistics Introduction to Multivariate Analysis B.S. Everitt C. Chatfield and A.J. Collins Extending the Linear Model with R: Graphics for Statistics and Data Analysis with R Generalized Linear, Mixed Effects and K.J. Keen Nonparametric Regression Models, Second Mathematical Statistics Edition K. Knight J.J. Faraway Introduction to Functional Data Analysis Linear Models with R, Second Edition P. Kokoszka and M. Reimherr J.J. Faraway Introduction to Multivariate Analysis: A Course in Large Sample Theory Linear and Nonlinear Modeling T.S. Ferguson S. Konishi Multivariate Statistics: A Practical Nonparametric Methods in Statistics with SAS Approach Applications B. Flury and H. Riedwyl O. Korosteleva Readings in Decision Analysis Modeling and Analysis of Stochastic Systems, S. French Second Edition Discrete Data Analysis with R: Visualization V.G. Kulkarni and Modeling Techniques for Categorical and Exercises and Solutions in Biostatistical Theory Count Data L.L. Kupper, B.H. Neelon, and S.M. O’Brien M. Friendly and D. Meyer Exercises and Solutions in Statistical Theory Markov Chain Monte Carlo: L.L. Kupper, B.H. Neelon, and S.M. O’Brien Stochastic Simulation for Bayesian Inference, Design and Analysis of Experiments with R Second Edition J. Lawson D. Gamerman and H.F. Lopes Design and Analysis of Experiments with SAS Bayesian Data Analysis, Third Edition J. Lawson A. Gelman, J.B. Carlin, H.S. Stern, D.B. Dunson, A. Vehtari, and D.B. Rubin A Course in Categorical Data Analysis T. Leonard Multivariate Analysis of Variance and Repeated Measures: A Practical Approach for Statistics for Accountants Behavioural Scientists S. Letchford D.J. Hand and C.C. Taylor Introduction to the Theory of Statistical Practical Longitudinal Data Analysis Inference D.J. Hand and M. Crowder H. Liero and S. Zwanzig Logistic Regression Models Statistical Theory, Fourth Edition J.M. Hilbe B.W. Lindgren Richly Parameterized Linear Models: Stationary Stochastic Processes: Theory and Additive, Time Series, and Spatial Models Applications Using Random Effects G. Lindgren J.S. Hodges Statistics for Finance Statistics for Epidemiology E. Lindström, H. Madsen, and J. N. Nielsen N.P. Jewell The BUGS Book: A Practical Introduction to Stochastic Processes: An Introduction, Bayesian Analysis Second Edition D. Lunn, C. Jackson, N. Best, A. Thomas, and P.W. Jones and P. Smith D. Spiegelhalter The Theory of Linear Models Introduction to General and Generalized B. Jørgensen Linear Models H. Madsen and P. Thyregod Pragmatics of Uncertainty J.B. Kadane Time Series Analysis H. Madsen Principles of Uncertainty J.B. Kadane Pólya Urn Models H. Mahmoud Randomization, Bootstrap and Monte Carlo Sampling Methodologies with Applications Methods in Biology, Third Edition P.S.R.S. Rao B.F.J. Manly A First Course in Linear Model Theory Statistical Regression and Classification: From N. Ravishanker and D.K. Dey Linear Models to Machine Learning Essential Statistics, Fourth Edition N. Matloff D.A.G. Rees Introduction to Randomized Controlled Stochastic Modeling and Mathematical Clinical Trials, Second Edition Statistics: A Text for Statisticians and J.N.S. Matthews Quantitative Scientists Statistical Rethinking: A Bayesian Course with F.J. Samaniego Examples in R and Stan Statistical Methods for Spatial Data Analysis R. McElreath O. Schabenberger and C.A. Gotway Statistical Methods in Agriculture and Bayesian Networks: With Examples in R Experimental Biology, Second Edition M. Scutari and J.-B. Denis R. Mead, R.N. Curnow, and A.M. Hasted Large Sample Methods in Statistics Statistics in Engineering: A Practical Approach P.K. Sen and J. da Motta Singer A.V. Metcalfe Spatio-Temporal Methods in Environmental Statistical Inference: An Integrated Approach, Epidemiology Second Edition G. Shaddick and J.V. Zidek H. S. Migon, D. Gamerman, and Decision Analysis: A Bayesian Approach F. Louzada J.Q. Smith Beyond ANOVA: Basics of Applied Statistics Analysis of Failure and Survival Data R.G. Miller, Jr. P. J. Smith A Primer on Linear Models Applied Statistics: Handbook of GENSTAT J.F. Monahan Analyses Stochastic Processes: From Applications to E.J. Snell and H. Simpson Theory Applied Nonparametric Statistical Methods, P.D Moral and S. Penev Fourth Edition Applied Stochastic Modelling, Second Edition P. Sprent and N.C. Smeeton B.J.T. Morgan Data Driven Statistical Methods Elements of Simulation P. Sprent B.J.T. Morgan Generalized Linear Mixed Models: Probability: Methods and Measurement Modern Concepts, Methods and Applications A. O’Hagan W. W. Stroup Introduction to Statistical Limit Theory Survival Analysis Using S: Analysis of A.M. Polansky Time-to-Event Data Applied Bayesian Forecasting and Time Series M. Tableman and J.S. Kim Analysis Applied Categorical and Count Data Analysis A. Pole, M. West, and J. Harrison W. Tang, H. He, and X.M. Tu Statistics in Research and Development, Elementary Applications of Probability Theory, Time Series: Modeling, Computation, and Second Edition Inference H.C. Tuckwell R. Prado and M. West Introduction to Statistical Inference and Its Essentials of Probability Theory for Applications with R Statisticians M.W. Trosset M.A. Proschan and P.A. Shaw Understanding Advanced Statistical Methods Introduction to Statistical Process Control P.H. Westfall and K.S.S. Henning P. Qiu Statistical Process Control: Theory and Epidemiology: Study Design and Practice, Third Edition Data Analysis, Third Edition G.B. Wetherill and D.W. Brown M. Woodward Generalized Additive Models: Practical Data Analysis for Designed An Introduction with R, Second Edition Experiments S. Wood B.S. Yandell Texts in Statistical Science Generalized Additive Models An Introduction with R SECOND EDITION Simon N. Wood University of Bristol, UK CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 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 Printed on acid-free paper Version Date: 20170307 International Standard Book Number-13: 978-1-4987-2833-1 (Hardback) 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 storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.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 provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Contents Preface xvii 1 LinearModels 1 1.1 Asimplelinearmodel 2 Simpleleastsquaresestimation 2 1.1.1 Samplingpropertiesofβˆ 3 1.1.2 Sohowoldistheuniverse? 4 1.1.3 Addingadistributionalassumption 7 Testinghypothesesaboutβ 7 Confidenceintervals 8 1.2 Linearmodelsingeneral 9 1.3 Thetheoryoflinearmodels 11 1.3.1 Leastsquaresestimationofβ 12 1.3.2 Thedistributionofβˆ 13 1.3.3 (βˆ β )/σˆ t 13 1.3.4 F-ria−tioriesulβtˆsiI∼ n−p 14 1.3.5 F-ratioresultsII 14 1.3.6 Theinfluencematrix 16 1.3.7 Theresiduals,ǫˆ,andfittedvalues,µˆ 16 1.3.8 ResultsintermsofX 17 1.3.9 The Gauss Markov Theorem: What’s special about least squares? 18 1.4 Thegeometryoflinearmodelling 19 1.4.1 Leastsquares 19 1.4.2 Fittingbyorthogonaldecompositions 20 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 34 1.5.5 Confidenceintervals 35 1.5.6 Prediction 36 1.5.7 Co-linearity,confoundingandcausation 36 ix