An Introduction to Generalized

1 An Introduction to Generalized Linear ModelsThird 14/8/08 9:54:27 AMAnalysis of Failure and Survival DataP. J. SmithThe Analysis of Time Series An Introduction , Sixth EditionC. ChatfieldApplied Bayesian Forecasting and Time SeriesAnalysisA. Pole, M. West and J. HarrisonApplied Nonparametric Statistical Methods,Fourth EditionP. Sprent and SmeetonApplied Statistics Handbook of Snell and H. SimpsonApplied Statistics Principles and Cox and SnellBayesian Data Analysis, Second EditionA. Gelman, Carlin, Sternand RubinBayesian Methods for Data Analysis,Third Carlin and LouisBeyond ANOVA Basics of Miller, Multivariate Analysis,Fourth Afifi and ClarkA Course in Categorical Data AnalysisT. LeonardA Course in Large Sample FergusonData Driven Statistical MethodsP. SprentDecision Analysis A Bayesian SmithElementary Applications of ProbabilityTheory, Second TuckwellElements of MorganEpidemiology Study Design andData Analysis, Second EditionM.

2 WoodwardEssential Statistics, Fourth ReesExtending the Linear Model with R: Generalized Linear, Mixed Effects andNonparametric Regression FarawayA First Course in Linear Model TheoryN. Ravishanker and DeyGeneralized Additive Models:An Introduction with RS. WoodInterpreting Data A First Coursein AndersonAn Introduction to GeneralizedLinear Models, Third Dobson and BarnettIntroduction to Multivariate AnalysisC. Chatfield and CollinsIntroduction to Optimization Methods andTheir Applications in EverittIntroduction to Probability with RK. BaclawskiIntroduction to Randomized ControlledClinical Trials, Second MatthewsIntroduction to Statistical Methods forClinical TrialsThomas D. Cook and David L. DeMetsLarge Sample Methods in Sen and J. da Motta SingerLinear Models with FarawayMarkov Chain Monte Carlo Stochastic Simulation for BayesianInference, Second EditionD. Gamerman and LopesMathematical StatisticsK. KnightCHAPMAN & HALL/CRCT exts in Statistical Science SeriesSeries EditorsBradley P.

3 Carlin, University of Minnesota, USAJ ulian J. Faraway, University of Bath, UKMartin Tanner, Northwestern University, USAJim Zidek, University of British Columbia, 24/8/08 9:54:27 AMModeling and Analysis of Stochastic SystemsV. KulkarniModelling Binary Data, Second EditionD. CollettModelling Survival Data in Medical Research,Second EditionD. CollettMultivariate Analysis of Variance and RepeatedMeasures A Practical Approach forBehavioural Hand and TaylorMultivariate Statistics A Practical ApproachB. Flury and H. RiedwylPractical Data Analysis for YandellPractical Longitudinal Data Hand and M. CrowderPractical Statistics for Medical AltmanA Primer on Linear MonahanProbability Methods and MeasurementA. O HaganProblem Solving A Statistician s Guide,Second EditionC. ChatfieldRandomization, Bootstrap andMonte Carlo Methods in Biology,Third ManlyReadings in Decision AnalysisS. FrenchSampling Methodologies with RaoStatistical Analysis of Reliability Crowder, Kimber, Sweeting, and SmithStatistical Methods for Spatial Data AnalysisO.

4 Schabenberger and GotwayStatistical Methods for SPC and TQMD. BissellStatistical Methods in Agriculture andExperimental Biology, Second EditionR. Mead, Curnow, and HastedStatistical Process Control Theoryand Practice, Third Wetherill and BrownStatistical Theory, Fourth LindgrenStatistics for AccountantsS. LetchfordStatistics for JewellStatistics for Technology A Course inApplied Statistics, Third EditionC. ChatfieldStatistics in Engineering A Practical MetcalfeStatistics in Research and Development,Second EditionR. CaulcuttSurvival Analysis Using S Analysis ofTime-to-Event DataM. Tableman and KimThe Theory of Linear ModelsB. J rgensenTime Series AnalysisH. 34/8/08 9:54:27 44/8/08 9:54:27 AMTexts in Statistical ScienceAnnette J. DobsonUniversity of QueenslandHerston, AustraliaAdrian G. BarnettQueensland University of TechnologyKelvin Grove, AustraliaAn Introduction to Generalized Linear ModelsThird 54/8/08 9:54:28 AMChapman & Hall/CRCT aylor & Francis Group6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa businessNo claim to original Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-58488-950-2 (Softcover)This book contains information obtained from authentic and highly regarded sources.

5 Reasonable efforts have been made to publish reliable data and information, but the author and publisher can-not 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 as permitted under 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 permission to photocopy or use material electronically from this work, please access ( ) or contact the Copyright Clearance Center, Inc.

6 (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro-vides 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 Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to of Congress Cataloging-in-Publication DataDobson, Annette J., 1945-An Introduction to Generalized linear models / Annette J. Dobson and Adrian G. Barnett. -- 3rd cm. -- (Chapman & Hall/CRC texts in statistical science series ; 77)Includes bibliographical references and 978-1-58488-950-2 (alk. paper)1. Linear models (Statistics) I. Barnett, Adrian G. II. Title. III. 2008013034 Visit the Taylor & Francis Web site the CRC Press Web site 64/8/08 9:54:28 AMContentsPreface1 Distributions related to the Normal Quadratic Exercises152 Model Some principles of statistical Notation and coding for explanatory Exercises403 Exponential Family and Generalized Linear Exponential family of Properties of distributions in the exponential Generalized linear Exercises554 Example.

7 Failure times for pressure Maximum likelihood Poisson regression Exercises695 Sampling distribution for score Taylor series Sampling distribution for Log-likelihood ratio Sampling distribution for the Hypothesis Exercises876 Normal Linear Basic Multiple linear Analysis of Analysis of General linear Exercises1187 Binary Variables and Logistic Probability Generalized linear Dose response General logistic regression Goodness of fit Other Example: Senility and Exercises1438 Nominal and Ordinal Logistic Multinomial Nominal logistic Ordinal logistic General Exercises1639 Poisson Regression and Log-Linear Poisson Examples of contingency Probability models for contingency Log-linear Inference for log-linear Numerical Exercises18310 Survival Survivor functions and hazard Empirical survivor Model Example: Remission Exercises20211 Clustered and Longitudinal Example: Recovery from Repeated measures models for Normal Repeated measures models for non-Normal Multilevel Stroke example Exercises22512 Bayesian Frequentist and Bayesian Distributions and hierarchies in Bayesian WinBUGS software for Bayesian Exercises24113 Markov Chain Monte Carlo Why standard inference Monte Carlo Markov Bayesian Diagnostics of chain Bayesian model fit.

8 The Exercises26214 Example Bayesian Binary variables and logistic Nominal logistic Latent variable Survival Random Longitudinal data Some practical tips for Exercises288 Appendix291 Software293 References295 Index303 PrefaceThe original purpose of the book was to present a unified theoretical andconceptual framework for statistical modelling in a way that was accessible toundergraduate students and researchers in other second edition was expanded to include nominal and ordinal logisticregression, survival analysis and analysis of longitudinal and clustered relied more on numerical methods, visualizing numericaloptimization andgraphical methods for exploratory data analysis and checking model fit. Thesefeatures have been extended further in this new third edition contains three new chapters on Bayesian analysis. Thefundamentals of Bayesian theory were written long before the development ofclassical theory but practical Bayesian analysis has only recently become avail-able.

9 This availability is mostly thanks to Markov chain Monte Carlo methodswhich are introduced in Chapter 13. The increased availability of Bayesiananalysis means that more people with a classical knowledge of statistics aretrying Bayesian methods for Generalized linear models. Bayesian analysis offerssignificant advantages over classical methods because of the ability formallyto incorporate prior information, greater flexibility and an ability to solvecomplex edition has also been updated with Stata and R code, which shouldhelp the practical application of Generalized linear models. The chapters onBayesian analyses contain R and WinBUGS data sets and outline solutions of the exercises are available on thepublisher s website: are grateful to colleagues and students at the Universities of Queens-land and Newcastle, Australia, and those taking postgraduate courses throughthe Biostatistics Collaboration of Australia for their helpful suggestions andcomments about the Dobson and Adrian BarnettBrisbaneCHAPTER BackgroundThis book is designed to introduce the reader to generalizedlinear models;these provide a unifying framework for many commonly used statistical tech-niques.

10 They also illustrate the ideas of statistical reader is assumed to have some familiarity with classical statisticalprinciples and methods. In particular, understanding the concepts of estima-tion, sampling distributions and hypothesis testing is necessary. Experience inthe use of t-tests, analysis of variance, simple linear regression and chi-squaredtests of independence for two-dimensional contingency tables is assumed. Inaddition, some knowledge of matrix algebra and calculus is reader will find it necessary to have access to statistical computingfacilities. Many statistical programs, languages or packages can now performthe analyses discussed in this book. Often, however, they doso with a differentprogram or procedure for each type of analysis so that the unifying structureis not programs or languages which have procedures consistent with theapproach used in this book areStata,R,S-PLUS,SASandGenstat. ForChapters 13 to 14 programs to conduct Markov chain Monte Carlo methodsare needed and WinBUGS has been used here.