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Wiley Series in Probability and Statistics Foundations of Linear and Generalized Linear Models Alan Agresti Foundations of Linear and Generalized Linear Models WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A SHEWHART and SAMUEL S WILKS Editors: David J Balding, Noel A C Cressie, Garrett M Fitzmaurice, Geof H Givens, Harvey Goldstein, Geert Molenberghs, David W Scott, Adrian F M Smith, Ruey S Tsay, Sanford Weisberg Editors Emeriti: J Stuart Hunter, Iain M Johnstone, Joseph B Kadane, Jozef L Teugels A complete list of the titles in this series appears at the end of this volume Foundations of Linear and Generalized Linear Models ALAN AGRESTI Distinguished Professor Emeritus University of Florida Gainesville, FL Visiting Professor Harvard University Cambridge, MA Copyright © 2015 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored 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Cataloging-in-Publication Data Agresti, Alan, author Foundations of linear and generalized linear models / Alan Agresti pages cm – (Wiley series in probability and statistics) Includes bibliographical references and index ISBN 978-1-118-73003-4 (hardback) Mathematical analysis–Foundations Linear models (Statistics) I Title QA299.8.A37 2015 003′ 74–dc23 2014036543 Printed in the United States of America 10 To my statistician friends in Europe Contents Preface xi 1 Introduction to Linear and Generalized Linear Models 1.1 Components of a Generalized Linear Model, 1.2 Quantitative/Qualitative Explanatory Variables and Interpreting Effects, 1.3 Model Matrices and Model Vector Spaces, 10 1.4 Identifiability and Estimability, 13 1.5 Example: Using Software to Fit a GLM, 15 Chapter Notes, 20 Exercises, 21 Linear Models: Least Squares Theory 26 2.1 Least Squares Model Fitting, 27 2.2 Projections of Data Onto Model Spaces, 33 2.3 Linear Model Examples: Projections and SS Decompositions, 41 2.4 Summarizing Variability in a Linear Model, 49 2.5 Residuals, Leverage, and Influence, 56 2.6 Example: Summarizing the Fit of a Linear Model, 62 2.7 Optimality of Least Squares and Generalized Least Squares, 67 Chapter Notes, 71 Exercises, 71 Normal Linear Models: Statistical Inference 3.1 3.2 3.3 80 Distribution Theory for Normal Variates, 81 Significance Tests for Normal Linear Models, 86 Confidence Intervals and Prediction Intervals for Normal Linear Models, 95 vii viii CONTENTS 3.4 Example: Normal Linear Model Inference, 99 3.5 Multiple Comparisons: Bonferroni, Tukey, and FDR Methods, 107 Chapter Notes, 111 Exercises, 112 Generalized Linear Models: Model Fitting and Inference 120 4.1 4.2 4.3 Exponential Dispersion Family Distributions for a GLM, 120 Likelihood and Asymptotic Distributions for GLMs, 123 Likelihood-Ratio/Wald/Score Methods of Inference for GLM Parameters, 128 4.4 Deviance of a GLM, Model Comparison, and Model Checking, 132 4.5 Fitting Generalized Linear Models, 138 4.6 Selecting Explanatory Variables for a GLM, 143 4.7 Example: Building a GLM, 149 Appendix: GLM Analogs of Orthogonality Results for Linear Models, 156 Chapter Notes, 158 Exercises, 159 Models for Binary Data 165 5.1 Link Functions for Binary Data, 165 5.2 Logistic Regression: Properties and Interpretations, 168 5.3 Inference About Parameters of Logistic Regression Models, 172 5.4 Logistic Regression Model Fitting, 176 5.5 Deviance and Goodness of Fit for Binary GLMs, 179 5.6 Probit and Complementary Log–Log Models, 183 5.7 Examples: Binary Data Modeling, 186 Chapter Notes, 193 Exercises, 194 Multinomial Response Models 202 6.1 Nominal Responses: Baseline-Category Logit Models, 203 6.2 Ordinal Responses: Cumulative Logit and Probit Models, 209 6.3 Examples: Nominal and Ordinal Responses, 216 Chapter Notes, 223 Exercises, 223 Models for Count Data 7.1 7.2 Poisson GLMs for Counts and Rates, 229 Poisson/Multinomial Models for Contingency Tables, 235 228 442 Projection matrix (Continued) null model, 42 one-way layout, 45, 86 orthogonal, 34, 72 two-way layout, 48 Propensity score, 194 Proportional hazards model, 234 Proportional odds model cumulative logit, 209–212, 223 testing fit, 213–214 Proportional reduction in variation, 147 Purposeful selection, 145 Pythagoras’s theorem, 39–41 Q-Q plot, 57, 101 QR decomposition, 72 Qualitative variable, response, 203–209 Quantile regression, 384–385, 387 Quantitative variable, Quasi-complete separation, 179, 187, 348, 374 Quasi-likelihood methods, 268–285 binomial overdispersion, 270–278, 282 GEE for clustered data, 316–318 Poisson overdispersion, 247–250, 269–270, 282 R (software) aod package, 277 arm package, 346 biglm function, 15 cond package, 132, 188 confint function, 132 gam package, 384 gee package, 281, 321 glm function, 15, 148, 154, 178, 187, 235, 244, 255, 271, 275, 348, 371, 374, 383 glmmML package, 320 glmnet package, 369, 372 hmmm package, 319 lars package, 369 lm function, 15, 64, 101–106 lme4 package, 298 logistf package, 374 MASS package, 154, 164, 255, 271, 366, 369 MCMCglmm package, 357 SUBJECT INDEX MCMCpack package, 345, 349 nlme package, 301 nnet package, 217 ProfileLikelihood package, 132, 220 pscl package, 256 ridge package, 369 ROCR package, 190 truncreg package, 116 VGAM package, 217, 223, 254, 256, 276, 383 R-squared measures, 54–56, 71 F statistic, 114 adjusted, 55, 64, 71, 114 binomial models, 172 GLM, 147 multinomial models, 212 Random explanatory variables, 20 Random-effects models, see Generalized linear mixed models Random-intercept model, 289 Randomized block design, 47 Rank, 11 projection matrix, 35 Rasch model, 307 Rate data, 233–235 Regression model, 2, 5, 7, 17 OLS with ordinal data, 216 Regression sum of squares, 51–56 comparing models, 52, 74 Regression toward the mean, 30 Regularization methods, 33, 366–378, 386–387 REML, 306–307, 323, 324 Residual ML, see REML Residual sum of squares, 51–52, 54, 88–89 Residuals, 32 data=fit+residuals, 40 deviance, 137, 182 GLMs, 136–138 Pearson, 136, 181, 244 plots, 56–57, 74, 103, 116 Pythagoras’s theorem, 39 standardized, 137, 158, 182, 244 uncorrelated with fitted values, 56, 135–136 Retrospective studies, 170 Ridge regression, 367, 386 Robust regression, 128, 365–374, 386 SUBJECT INDEX Sandwich covariance matrix, 279–282, 317–318 Saturated model, 74, 132 loglinear, 241 Scheff´e method, 112 Score function, 129 Score test, 158 comparing GLMs, 158 confidence interval, 131 goodness of fit of GLM, 135 Pearson chi-squared statistic, 135, 158 Selection bias, 116, 147 Sensitivity, 171–172 Sequential sum of squares, 52–54 Shrinkage estimator, 68, 341, 366, 368, 369, 373 Bayesian, 338, 343, 347, 353–357, 378 Simpson’s paradox, 105, 328 Simultaneous confidence intervals, 107–110 Simultaneous testing, 197 Small-area estimation, 304, 323 Small-dispersion asymptotics, 128, 133, 136, 137, 166, 181 Smoothing, 387 generalized additive model, 378 kernel, 379–380 penalized likelihood, 374 Sparse structure, 376 Spatial data, 323, 387 Specificity, 171–172 Spectral decomposition, 68, 85 Spline function, 380, 387 SSE (sum of squared errors), 51 SSR (regression sum of squares), 51 Standardized regression coefficients, 21 Standardized residuals, 57–58 binomial GLM, 182, 196 GLM, 137, 158 loglinear model, 244 Poisson, 158 Stepwise procedures, 143–146, 376 Stochastic ordering ordinal response, 213 Studentized range distribution, 109 Studentized residual, 58 Subject-specific effect binary matched pairs, 291–293 generalized linear mixed model, 293 443 Sum of squared errors, see Residual sum of squares Survival model, 226, 233–235, 261, 323 t distribution, 82 approximation of logistic, 351 noncentral, 83 Threshold model, 166–167, 198 ordinal response, 211 Time series, 294, 322 Toeplitz correlation structure, 302 Tolerance distribution, 198 Total sum of squares, 47, 51 Transforming data, 6, 20, 229–230 Transition model, 294, 323 Truncated discrete model, 252–254, 260 Truncated regression, 116 TSS (total sum of squares), 51 Tukey multiple comparisons, 109–110, 112 Two-way layout, 22–23, 47–49 ANOVA, 113–114 Unbiased estimating function, 278, 282, 284 Utility model, 198 Variable selection, 143–156, 159, 375–377 Bayesian, 358, 387 high-dimensional, 375–378 Variance Bayesian inference, 343–345 estimating in linear model, 49–50, 70 estimating using REML, 306–307 inflated variance function, 269–273 modeling, 282 Variance components, 295, 296 Variance inflation factor, 148 Variance-stabilizing transformations, 6, 20, 229–230 Vector space, 11 Wald statistic, 129 aberrant behavior for binary GLM, 174, 195 confidence interval, 131 dependence on parameterization, 174 Weight matrix, 142 Weighted least squares, 69, 140–142, 284, 381 444 Wilks’ lambda, 315 Wishart distribution, 351 Within-groups sum of squares, 46 Within-subject effects, 287, 293 Yule’s parameter notation, 10, 60–62 SUBJECT INDEX Zero count infinite estimates, 179 Zero-inflated negative binomial model, 252, 260 Zero-inflated Poisson model, 251–252, 256–257, 260 Zero-truncated model, 254, 260 WILEY SERIES IN PROBABILITY AND STATISTICS established by Walter A Shewhart and Samuel S Wilks Editors: David J Balding, Noel A C Cressie, Garrett M Fitzmaurice, Geof H Givens, Harvey Goldstein, Geert Molenberghs, David W Scott, Adrian F M Smith, Ruey S Tsay, Sanford Weisberg Editors Emeriti: J Stuart Hunter, Iain M Johnstone, Joseph B Kadane, Jozef L Teugels The Wiley Series in Probability and Statistics is well established and authoritative It covers many topics of current research interest in both pure and applied statistics and probability theory Written by leading statisticians and institutions, the titles span both state-of-the-art developments in the field and classical methods Reflecting the wide range of current research in statistics, the series encompasses applied, methodological and theoretical statistics, ranging from applications and new techniques made possible by advances in computerized practice to rigorous 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Introduction to Linear and Generalized Linear Models 1.1 Components of a Generalized Linear Model, 1.2 Quantitative/Qualitative Explanatory Variables and Interpreting Effects, 1.3 Model Matrices and Model... Bayesian approach for linear models and generalized linear models, which treats the model parameters as random variables having their Foundations of Linear and Generalized Linear Models, First... in Chapter CHAPTER NOTES Section 1.1 : Components of a Generalized Linear Model 1.1 1.2 1.3 1.4 GLM: Nelder and Wedderburn (1972) introduced the class of GLMs and the algorithm for fitting them,