Categorical Data Analysis by Example

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Introduces the key concepts in the analysis of categoricaldata with illustrative examples and accompanying R code This book is aimed at all those who wish to discover how to analyze categorical data without getting immersed in complicated mathematics and without needing to wade through a large amount of prose. It is aimed at researchers with their own data ready to be analyzed and at students who would like an approachable alternative view of the subject. Each new topic in categorical data analysis is illustrated with an example that readers can apply to their own sets of data. In many cases, R code is given and excerpts from the resulting output are presented. In the context of log-linear models for cross-tabulations, two specialties of the house have been included: the use of cobweb diagrams to get visual information concerning significant interactions, and a procedure for detecting outlier category combinations. The R code used for these is available and may be freely adapted. In addition, this book: • Uses an example to illustrate each new topic in categorical data • Provides a clear explanation of an important subject • Is understandable to most readers with minimal statistical and mathematical backgrounds • Contains examples that are accompanied by R code and resulting output • Includes starred sections that provide more background details for interested readers Categorical Data Analysis by Example is a reference for students in statistics and researchers in other disciplines, especially the social sciences, who use categorical data. This book is also a reference for practitioners in market research, medicine, and other fields. GRAHAM J. G. UPTON is formerly Professor of Applied Statistics, Department of Mathematical Sciences, University of Essex. Dr. Upton is author of The Analysis of Cross-tabulated Data (1978) and joint author of Spatial Data Analysis by Example (2 volumes, 1995), both published by Wiley. He is the lead author of The Oxford Diction
CATEGORICAL DATA ANALYSIS BY EXAMPLE CATEGORICAL DATA ANALYSIS BY EXAMPLE GRAHAM J G. UPTON WILEY Copyright o 2017 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 in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, InC, 222 Rosewood Drive, Danvers, MA 01923, (978)750-8400, fax (978)750-4470, be addressed to the Permissions Department, John Wiley sons, Inc, Ill River Street, Hoboken, N 07030,(201)748-6011,fax(201)748-6008,oronlineat Limit of liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at(800)762-2974, outside the United States at (317)572-3993 or fax(317)572-4002 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data Names: Upton, Graham J G, author Title: Categorical data analysis by example/ Graham J.G. Upton Description Hoboken, New Jersey: John Wiley sons, 2016. Includes index Identifiers: LCCN 2016031847(print)I LCCn 2016045176(ebook)| isbn9781119307860(cloth) IsBN9781119307914( pdf)| ISBN9781119307938(epub) Subjects: LCSH: Multivariate analysis. I Log-linear models Classification: LCC QA278 U68 2016(print)I LCC QA278(ebook)I DDC 519.5/35-dc23 Lcrecordavailableat Printed in the united states of america 10987654321 CONTENTS PREFACE ACKNOWLEDGMENTS 1 NTRODUCTION 1.1 What are Categorical data? 1.2 A Typical Data set 2 1.3 Visualization and Cross-Tabulation 3 1.4 Samples, Populations, and random Variation 4 1.5 Proportion, Probability, and Conditional Probability 5 1.6 Probability distributions 6 1.6.1 The Binomial distribution 6 1. 6.2 The Multinomial distribution 7 1. 6. 3 The Poisson distribution 7 1. 6. 4 The normal distribution 7 1.6.5 The Chi-Squared(x)Distribution 8 1. 7 *The Likelihood 9 2 ESTIMATION AND INFERENCE FOR CATEGORICAL DATA 2.1 Goodness of fit 11 I CONTENTS 2.1.1 Pearson's x Goodness-of-Fit Statistic 11 2.1.2 *The Link between X and the poisson and x2-Distributions 12 2.1.3 The Likelihood-Ratio Goodness-of-Fit Statistic 2.1.4* Why the G and X Statistics Usually Have Similar Values 14 2.2 Hypothesis Tests for a Binomial Proportion Large Sample) 14 2. 2. 1 The Normal score Test 15 2.2.2 *Link to pearson's X- Goodness-of-Fit Test 15 2.2.3 G2 for a Binomial Proportion 15 2.3 Hypothesis Tests for A Binomial Proportion(Small Sample)16 2.3.1 One-Tailed Hypothesis Test 16 2.3.2 Two-Tailed Hypothesis Tests 18 2.4 Interval Estimates for A Binomial Proportion 18 2.4.1 Laplaces method 19 2.4.2 Wilsons Method 19 2.4.3 The Agresti-Coull Method 20 2.4.4 Small Samples and Exact Calculations 2( R ferenc 2 3 THE 2 X 2 CONTINGENCY TABLE 25 3.1 Introduction 25 3.2 Fisher's Exact Test(For Independence) 27 3.2.1 Derivation of the exact Test formula 28 3.3 Testing Independence with Large Cell Frequencies 29 3.3.1 USing Pearson,s Goodness-of-Fit Test 30 3.3.2 The Yates Correction 30 3.4 The 2x2 Table in a medical Context 32 3.5 Measuring Lack of Independence( Comparing Proportions )34 3.5.1 Difference of Proportions 35 3.5.2 Relative risk 36 3.5.3 Odds-Ratio 37 References 40 CONTENTS VIl 4 THEXJ CONTINGENCY TABLE 4.1 Notation 41 4.2 Independence in the l J Contingency Table 42 4.2.1 Estimation and degrees of freedom 42 4.2.2 Odds-Ratios and Independence 43 4.2.3 Goodness of fit and lack of fit of the Independence model 43 4.3 Partitioning 46 4.3.1 Additivity of G 46 4.3.2 Rules for partitioning 4 4.4 Graphical Displays 49 4.4.1 Mosaic Plots 49 4.4.2 Cobweb diagrams 50 4.5 Testing Independence with Ordinal Variables 52 References 54 5 THE EXPONENTIAL FAMILY 55 5.1 Introduction 55 5.2 The Exponential Family 56 5.2.1 The Exponential Dispersion Family 57 5.3 Components of a general Linear Model 57 5.4 Estimation 58 References 59 6 A MODEL TAXONOMY 61 6.1 Underlying Questions 61 6.1.1 Which Variables are of Interest? 61 6.1.2 What Categories should be used? 61 6.1.3 What is the Type of Each Variable?62 6.1. 4 What is the Nature of each variable? 62 6.2 Identifying the Type of Model 63 7 THE2 XJ CONTINGENCY TABLE 65 7.1 A Problem with X(and G-)65 7. 2 USing the logit 66 v CONTENTS 7.2.1 Estimation of the Logit 67 7. 2. 2 The null model 68 7.3 Individual data and grouped data 69 7.4 Precision Confidence Intervals and Prediction Intervals 73 7.4.1 Prediction intervals 74 7.5 Logistic Regression with a Categorical Explanatory Variable 76 7.5.1 Parameter Estimates with Categorical variables (>2)78 7.5.2 The dummy Variable Representation of a Categorical Variable 79 References 80 8 LOGISTIC REGRESSION WITH SEVERAL EXPLANATORY VARIABLES 81 8.1 Degrees of freedom when there are no interactions 81 8.2 Getting a Feel for the data 83 8.3 Models with two-Variable interactions 85 8.3.1 Link to the Testing of Independence between Two Variables 87 9 MODEL SELECTION AND DIAGNOSTICS 89 9.1 Introduction 89 9.1.1 Ockham's razor 90 9.2 Notation for Interactions and for models 91 9.3 Stepwise Methods for Model Selection Using G 92 9.3.1 Forward Selection 94 9.3.2 Backward Elimination 96 9.3.3 Complete stepwise 98 9.4 AIC and Related Measures 98 9.5 The Problem Caused by rare Combinations of Events 100 9.5.1 Tackling the Problem 101 9.6 Simplicity versus Accuracy 103 9.7 DFBETAS 105 References 107

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