For Bonnie and Jesse (again)

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1 SECOND EDITION A P P L I E D R E G R E S S I O N A N A L Y S I S a n d G E N E R A L I Z E D L I N E A R M O D E L S

2 For Bonnie and Jesse (again)

3 SECOND EDITION A P P L I E D R E G R E S S I O N A N A L Y S I S a n d G E N E R A L I Z E D U N E A R M O D E L S J o h n F o x McMaster University, Hamilton, Ontario, Canada DATHQC THAI NGUYEN TRUNG TAM HOC LIEU ( D S A G E Los Angeles London New Delhi Singapore

4 Copyright 2008 by Sage Publications, Inc. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. For information: 9 SAGE Publications, Inc Teller Road Thousand Oaks, California order@sagepub.com SAGE Publications Ltd. 1 Oliver's Yard 55 City Road London EC1Y ISP United Kingdom SAGE Publications India Pvt. Ltd. B 1/1 1 Mohan Cooperative Industrial Area Mathura Road, New Delhi India SAGE Publications Asia-Pacific Pte. Ltd. 33 Pekin Street #02-01 Far East Square Singapore Printed in the United States of America Library of Congress Cataloging-in-Publication Data Fox, John, Applied regression analysis and generalized linear models/john Fox. 2nd ed. p. cm. Rev. ed. of: Applied regression analysis, linear models, and related methods. cl997. Includes bibliographical references and index. ISBN (cloth) 1. Regression analysis. 2. Linear models (Statistics) 3. Social sciences Statistical methods. I. Fox, John, 1947-Applied regression analysis and generalized linear models. II. Title. HA31.3.F ' dc Printed on acid-free paper Acquisitions Editor: Vicki Knight Associate Editor: Sean Connelly Editorial Assistant: Lauren Habib Production Editor: Cassandra Margaret Seibel Copy Editor: QuADS Prepress (P) Ltd. Typesetter: C&M Digitals (P) Ltd. Proofreader: Kevin Gleason Cover Designer: Candice Harman Marketing Manager: Stephanie Adams

5 C o n t e n t s Preface xiv 1 Statistical Models and Social Science Statistical Models and Social Reality Observation and Experiment Populations and Samples 8 Exercise 9 Summary 9 Recommended Reading 10 PART I DATA CRAFT 11 2 What Is Regression Analysis? Preliminaries Naive Nonparametric Regression Local Averaging 21 Exercise 24 Summary 25 3 Examining Data Univariate Displays Histograms Nonparametric Density Estimation Quantile-Comparison Plots Boxplots Plotting Bivariate Data Plotting Multivariate Data Scatterplot Matrices Coded Scatterplots Three-Dimensional Scatterplots Conditioning Plots 46 Summary 47 Recommended Reading 49

6 4 Transforming Data The Family of Powers and Roots Transforming Skewness Transforming Nonlinearity Transforming Nonconstant Spread Transforming Proportions Estimating Transformations as Parameters* 68 Exercises 71 Summary 72 Recommended Reading 72 PART II LINEAR MODELS AND LEAST SQUARES 75 5 Linear Least-Squares Regression Simple Regression Least-Squares Fit Simple Correlation Multiple Regression Two Explanatory Variables Several Explanatory Variables Multiple Correlation Standardized Regression Coefficients 94 Exercises 96 Summary 98 6 Statistical Inference for Regression Simple Regression The Simple-Regression Model Properties of the Least-Squares Estimator Confidence Intervals and Hypothesis Tests Multiple Regression The Multiple-Regression Model Confidence Intervals and Hypothesis Tests Empirical Versus Structural Relations Measurement Error in Explanatory Variables* 112 Exercises 115 Summary Dummy-Variable Regression A Dichotomous Factor Polytomous Factors Coefficient Quasi-Variances* Modeling Interactions Constructing Interaction Regressors The Principle of Marginality Interactions With Polytomous Factors 135

7 7.3.4 Interpreting Dummy-Regression Models With Interactions Hypothesis Tests for Main Effects and Interactions A Caution Concerning Standardized Coefficients 140 Exercises 140 Summary Analysis of Variance One-Way Analysis of Variance Two-Way Analysis of Variance Patterns of Means in the Two-Way Classification The Two-Way ANOVA Model Fitting the Two-Way ANOVA Model to Data Testing Hypotheses in Two-Way ANOVA Equal Cell Frequencies Some Cautionary Remarks Higher-Way Analysis of Variance The Three-Way Classification Higher-Order Classifications Empty Cells in ANOVA Analysis of Covariance Linear Contrasts of Means 176 Exercises 180 Summary Statistical Theory for Linear Models* Linear Models in Matrix Form Dummy Regression and Analysis of Variance Linear Contrasts Least-Squares Fit Properties of the Least-Squares Estimator The Distribution of the Least-Squares Estimator The Gauss-Markov Theorem Maximum-Likelihood Estimation Statistical Inference for Linear Models Inference for Individual Coefficients Inference for Several Coefficients General Linear Hypotheses Joint Confidence Regions Multivariate Linear Models Random Regressors Specification Error 212 Exercises 213 Summary 217 Recommended Reading 219

8 10 The Vector Geometry of Linear Models* Simple Regression Variables in Mean-Deviation Form Degrees of Freedom Multiple Regression Estimating the Error Variance Analysis-of-Variance Models 233 Exercises 235 Summary 236 Recommended Reading 238 PART III LINEAR-MODEL DIAGNOSTICS Unusual and Influential Data Outliers, Leverage, and Influence Assessing Leverage: Hat-Values Detecting Outliers: Studentized Residuals Testing for Outliers in Linear Models Anscombe's Insurance Analogy Measuring Influence Influence on Standard Errors Influence on Collinearity Numerical Cutoffs for Diagnostic Statistics Hat-Values Studentized Residuals Measures of Influence Joint Influence Added-Variable Plots Forward Search Should Unusual Data Be Discarded? Some Statistical Details* Hat-Values and the Hat-Matrix The Distribution of the Least-Squares Residuals Deletion Diagnostics Added-Variable Plots and Leverage Plots 263 Exercises 264 Summary 265 Recommended Reading Diagnosing Non-Normality, Nonconstant Error Variance, and Nonlinearity Non-Normally Distributed Errors Confidence Envelopes by Simulated Sampling* Nonconstant Error Variance Residual Plots Weighted-Least-Squares Estimation* Correcting OLS Standard Errors for Nonconstant Variance* How Nonconstant Error Variance Affects the OLS Estimator* 276

9 12.3 Nonlinearity Component-Plus-Residual Plots Component-Plus-Residual Plots for Models With Interactions When Do Component-Plus-Residual Plots Work? Discrete Data Testing for Nonlinearity ("Lack of Fit") Testing for Nonconstant Error Variance Maximum-Likelihood Methods* Box-Cox Transformation of Y Box-Tidwell Transformation of the Xs Nonconstant Error Variance Revisited Structural Dimension 298 Exercises 301 Summary 305 Recommended Reading Collinearity and Its Purported Remedies Detecting Collinearity Principal Components* Generalized Variance Inflation* Coping With Collinearity: No Quick Fix Model Respecification Variable Selection Biased Estimation Prior Information About the Regression Coefficients Some Comparisons 329 Exercises 330 Summary 331 PART IV GENERALIZED LINEAR MODELS Logit and Probit Models for Categorical Response Variables Models for Dichotomous Data The Linear-Probability Model Transformations of n: Logit and Probit Models An Unobserved-Variable Formulation Logit and Probit Models for Multiple Regression Estimating the Linear Logit Model* Models for Polytomous Data The Polytomous Logit Model Nested Dichotomies Ordered Logit and Probit Models Comparison of the Three Approaches Discrete Explanatory Variables and Contingency Tables The Binomial Logit Model* 372 Exercises 375 Summary 377 Recommended Reading 378

10 15 Generalized Linear Models The Structure of Generalized Linear Models Estimating and Testing GLMs Generalized Linear Models for Counts Models for Overdispersed Count Data Loglinear Models for Contingency Tables Statistical Theory for Generalized Linear Models* Exponential Families Maximum-Likelihood Estimation of Generalized Linear Models Hypothesis Tests Effect Displays Diagnostics for Generalized Linear Models Outlier, Leverage, and Influence Diagnostics Nonlinearity Diagnostics 415 Exercises 417 Summary 421 Recommended Reading 424 PART V EXTENDING LINEAR AND GENERALIZED LINEAR MODELS Time-Series Regression and Generalized Least Squares* Generalized Least-Squares Estimation Serially Correlated Errors The First-Order Autoregressive Process Higher-Order Autoregressive Processes Moving-Average and Autoregressive-Moving-Average Processes Partial Autocorrelations GLS Estimation With Autocorrelated Errors Empirical GLS Estimation Maximum-Likelihood Estimation Diagnosing Serially Correlated Errors Concluding Remarks 444 Exercises 446 Summary 449 Recommended Reading Nonlinear Regression Polynomial Regression A Closer Look at Quadratic Surfaces* Piece-Wise Polynomials and Regression Splines Transformable Nonlinearity Nonlinear Least Squares* Minimizing the Residual Sum of Squares An Illustration: U.S. Population Growth 467 Exercises 469 Summary 474 Recommended Reading 475

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