Categorical Data Analysis

(1)

(2)

Categorical Data Analysis

(3)

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Peter Bloomfield, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Louise M. Ryan, David W. Scott, Adrian F. M. Smith, Jozef L. Teugels

Editors Emeriti: Vic Barnett, J. Stuart Hunter, David G. Kendall

A complete list of the titles in this series appears at the end of this volume.

p&s-cp.qxd 12/4/02 10:27 AM Page 1

(4)

Categorical Data Analysis

Second Edition

ALAN AGRESTI University of Florida Gainesville, Florida

(5)

⬁ This book is printed on acid-free paper. "

Copyright

䊚

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, 222 Rosewood Drive, Danvers, MA

Ž . Ž .

01923, 978 750-8400, fax 978 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New

Ž . Ž .

York, NY 10158-0012, 212 850-6011, fax 212 850-6008, E-Mail: PERMREQ@WILEY.COM.

For ordering and customer service, call 1-800-CALL-WILEY.

Library of Congress Cataloging-in-Publication Data Is A©ailable

ISBN 0-471-36093-7

Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

(6)

To Jacki

(7)

Preface

The explosion in the development of methods for analyzing categorical data that began in the 1960s has continued apace in recent years. This book provides an overview of these methods, as well as older, now standard, methods. It gives special emphasis to generalized linear modeling techniques, which extend linear model methods for continuous variables, and their extensions for multivariate responses.

Today, because of this development and the ubiquity of categorical data in applications, most statistics and biostatistics departments offer courses on categorical data analysis. This book can be used as a text for such courses.

The material in Chapters 1᎐7 forms the heart of most courses. Chapters 1᎐3 cover distributions for categorical responses and traditional methods for two-way contingency tables. Chapters 4᎐7 introduce logistic regression and related logit models for binary and multicategory response variables. Chap- ters 8 and 9 cover loglinear models for contingency tables. Over time, this model class seems to have lost importance, and this edition reduces some- what its discussion of them and expands its focus on logistic regression.

In the past decade, the major area of new research has been the development of methods for repeated measurement and other forms of clustered categorical data. Chapters 10᎐13 present these methods, including marginal models and generalized linear mixed models with random effects. Chapters 14 and 15 present theoretical foundations as well as alternatives to the maximum likelihood paradigm that this text adopts. Chapter 16 is devoted to a historical overview of the development of the methods. It examines contri- butions of noted statisticians, such as Pearson and Fisher, whose pioneering effortsᎏand sometimes vocal debatesᎏbroke the ground for this evolution.

Every chapter of the first edition has been extensively rewritten, and some substantial additions and changes have occurred. The major differences are:

䢇 A new Chapter 1 that introduces distributions and methods of inference for categorical data.

䢇 A unified presentation of models as special cases of generalized linear models, starting in Chapter 4 and then throughout the text.

xiii

(14)

PREFACE

xiv

䢇 Greater emphasis on logistic regression for binary response variables and extensions for multicategory responses, with Chapters 4᎐7 introduc- ing models and Chapters 10᎐13 extending them for clustered data.

䢇 Three new chapters on methods for clustered, correlated categorical data, increasingly important in applications.

䢇 A new chapter on the historical development of the methods.

䢇 More discussion of ‘‘exact’’ small-sample procedures and of conditional logistic regression.

In this text, I interpret categorical data analysis to refer to methods for categorical response variables. For most methods, explanatory variables can be qualitative or quantitative, as in ordinary regression. Thus, the focus is intended to be more general than contingency table analysis, although for simplicity of data presentation, most examples use contingency tables. These examples are often simplistic, but should help readers focus on understand- ing the methods themselves and make it easier for them to replicate results with their favorite software.

Special features of the text include:

䢇 More than 100 analyses of ‘‘real’’ data sets.

䢇 More than 600 exercises at the end of the chapters, some directed towards theory and methods and some towards applications and data analysis.

䢇 An appendix that shows, by chapter, the use of SAS for performing analyses presented in this book.

䢇 Notes at the end of each chapter that provide references for recent research and many topics not covered in the text.

Appendix A summarizes statistical software needed to use the methods described in this text. It shows how to use SAS for analyses included in the

Ž .

text and refers to a web site www.stat.ufl.edur;aarcdarcda.html that

Ž . Ž

contains 1 information on the use of other software such as R, S-plus, . Ž .

SPSS, and Stata , 2 data sets for examples in the form of complete SAS programs for conducting the analyses, 3 short answers for many of theŽ . odd-numbered exercises, 4 corrections of errors in early printings of theŽ . book, and 5 extra exercises. I recommend that readers refer to this ap-Ž . pendix or specialized manuals while reading the text, as an aid to implement- ing the methods.

I intend this book to be accessible to the diverse mix of students who take graduate-level courses in categorical data analysis. But I have also written it with practicing statisticians and biostatisticians in mind. I hope it enables them to catch up with recent advances and learn about methods that sometimes receive inadequate attention in the traditional statistics curricu- lum.

(15)

PREFACE xv The development of new methods has influencedᎏand been influenced byᎏthe increasing availability of data sets with categorical responses in the social, behavioral, and biomedical sciences, as well as in public health, human genetics, ecology, education, marketing, and industrial quality control. And so, although this book is directed mainly to statisticians and biostatisticians, I also aim for it to be helpful to methodologists in these fields.

Readers should possess a background that includes regression and analysis of variance models, as well as maximum likelihood methods of statistical theory. Those not having much theory background should be able to follow most methodological discussions. Sections and subsections marked with an asterisk are less important for an overview. Readers with mainly applied interests can skip most of Chapter 4 on the theory of generalized linear models and proceed to other chapters. However, the book has distinctly higher technical level and is more thorough and complete than my lower-level

Ž .

text, An Introduction to Categorical Data Analysis Wiley, 1996 .

I thank those who commented on parts of the manuscript or provided help of some type. Special thanks to Bernhard Klingenberg, who read several chapters carefully and made many helpful suggestions, Yongyi Min, who constructed many of the figures and helped with some software, and Brian Caffo, who helped with some examples. Many thanks to Rosyln Stone and Brian Marx for each reviewing half the manuscript and Brian Caffo, I-Ming Liu, and Yongyi Min for giving insightful comments on several chapters.

Thanks to Constantine Gatsonis and his students for using a draft in a course at Brown University and providing suggestions. Others who provided comments on chapters or help of some type include Patricia Altham, Wicher Bergsma, Jane Brockmann, Brent Coull, Al DeMaris, Regina Dittrich, Jianping Dong, Herwig Friedl, Ralitza Gueorguieva, James Hobert, Walter Katzenbeisser, Harry Khamis, Svend Kreiner, Joseph Lang, Jason Liao, Mojtaba Ganjali, Jane Pendergast, Michael Radelet, Kenneth Small, Maura Stokes, Tom Ten Have, and Rongling Wu. I thank my co-authors on various projects, especially Brent Coull, Joseph Lang, James Booth, James Hobert, Brian Caffo, and Ranjini Natarajan, for permission to use material from those articles. Thanks to the many who reviewed material or suggested examples for the first edition, mentioned in the Preface of that edition.

Thanks also to Wiley Executive Editor Steve Quigley for his steadfast encouragement and facilitation of this project. Finally, thanks to my wife Jacki Levine for continuing support of all kinds, despite the many days this work has taken from our time together.

ALANAGRESTI Gaines®ille, Florida

No®ember 2001

(16)

C H A P T E R 1

Introduction: Distributions and Inference for Categorical Data

From helping to assess the value of new medical treatments to evaluating the factors that affect our opinions and behaviors, analysts today are finding myriad uses for categorical data methods. In this book we introduce these methods and the theory behind them.

Statistical methods for categorical responses were late in gaining the level of sophistication achieved early in the twentieth century by methods for continuous responses. Despite influential work around 1900 by the British statistician Karl Pearson, relatively little development of models for categorical responses occurred until the 1960s. In this book we describe the early fundamental work that still has importance today but place primary emphasis on more recent modeling approaches. Before outlining the topics covered, we describe the major types of categorical data.

1.1 CATEGORICAL RESPONSE DATA

A categorical ®ariable has a measurement scale consisting of a set of categories. For instance, political philosophy is often measured as liberal, moderate, or conservative. Diagnoses regarding breast cancer based on a mammo- gram use the categories normal, benign, probably benign, suspicious, and malignant.

The development of methods for categorical variables was stimulated by research studies in the social and biomedical sciences. Categorical scales are pervasive in the social sciences for measuring attitudes and opinions. Cate- gorical scales in biomedical sciences measure outcomes such as whether a medical treatment is successful.

Although categorical data are common in the social and biomedical sciences, they are by no means restricted to those areas. They frequently 1

(17)

INTRODUCTION:DISTRIBUTIONS AND INFERENCE FOR CATEGORICAL DATA

2

occur in the behavioral sciences e.g., type of mental illness, with the cate-Ž gories schizophrenia, depression, neurosis , epidemiology and public health. Že.g., contraceptive method at last intercourse, with the categories none,

. Ž .

condom, pill, IUD, other , genetics type of allele inherited by an offspring , zoology e.g., alligators’ primary food preference, with the categories fish,Ž

. Ž

invertebrate, reptile , education e.g., student responses to an exam question,

. Ž

with the categories correct and incorrect , and marketing e.g., consumer preference among leading brands of a product, with the categories brand A, brand B, and brand C . They even occur in highly quantitative fields such as. engineering sciences and industrial quality control. Examples are the classification of items according to whether they conform to certain standards, and subjective evaluation of some characteristic: how soft to the touch a certain fabric is, how good a particular food product tastes, or how easy to perform a worker finds a certain task to be.

Categorical variables are of many types. In this section we provide ways of classifying them and other variables.

1.1.1 Response–Explanatory Variable Distinction

Ž .

Most statistical analyses distinguish between response ordependent ®ariables

Ž .

and explanatory or independent ®ariables. For instance, regression models describe how the mean of a response variable, such as the selling price of a house, changes according to the values of explanatory variables, such as square footage and location. In this book we focus on methods for categorical response variables. As in ordinary regression, explanatory variables can be of any type.

1.1.2 Nominal–Ordinal Scale Distinction

Categorical variables have two primary types of scales. Variables having categories without a natural ordering are called nominal. Examples are religious affiliation with the categories Catholic, Protestant, Jewish, Muslim,Ž

. Ž

other , mode of transportation to work automobile, bicycle, bus, subway,

. Ž .

walk , favorite type of music classical, country, folk, jazz, rock , and choice of

Ž .

residence apartment, condominium, house, other . For nominal variables, the order of listing the categories is irrelevant. The statistical analysis does not depend on that ordering.

Many categorical variables do have ordered categories. Such variables are called ordinal. Examples are size of automobile Žsubcompact, compact,

. Ž .

midsize, large , social class upper, middle, lower , political philosophy Žliberal, moderate, conservative , and patient condition good, fair, serious,. Ž critical . Ordinal variables have ordered categories, but distances between. categories are unknown. Although a person categorized as moderate is more liberal than a person categorized as conservative, no numerical value de- scribes how much more liberal that person is. Methods for ordinal variables utilize the category ordering.

(18)

CATEGORICAL RESPONSE DATA 3 Aninter®al®ariableis one that doeshave numerical distances between any two values. For example, blood pressure level, functional life length of television set, length of prison term, and annual income are interval variables. An internal variable is sometimes called aŽ ratio ®ariable if ratios of values are also valid..

The way that a variable is measured determines its classification. For example, ‘‘education’’ is only nominal when measured as public school or private school; it is ordinal when measured by highest degree attained, using the categories none, high school, bachelor’s, master’s, and doctorate; it is interval when measured by number of years of education, using the integers 0, 1, 2, . . . .

A variable’s measurement scale determines which statistical methods are appropriate. In the measurement hierarchy, interval variables are highest, ordinal variables are next, and nominal variables are lowest. Statistical methods for variables of one type can also be used with variables at higher levels but not at lower levels. For instance, statistical methods for nominal variables can be used with ordinal variables by ignoring the ordering of categories. Methods for ordinal variables cannot, however, be used with nominal variables, since their categories have no meaningful ordering. It is usually best to apply methods appropriate for the actual scale.

Since this book deals with categorical responses, we discuss the analysis of nominal and ordinal variables. The methods also apply to interval variables

Ž .

having a small number of distinct values e.g., number of times married or for which the values are grouped into ordered categories e.g., educationŽ measured as -10 years, 10᎐12 years, )12 years ..

1.1.3 Continuous–Discrete Variable Distinction

Variables are classified as continuousor discrete, according to the number of values they can take. Actual measurement of all variables occurs in a discrete manner, due to precision limitations in measuring instruments. The continuous᎐discrete classification, in practice, distinguishes between variables that take lots of values and variables that take few values. For instance, statisticians often treat discrete interval variables having a large number of values Žsuch as test scores as continuous, using them in methods for continuous. responses.

This book deals with certain types of discretely measured responses: 1Ž .

Ž . Ž .

nominal variables, 2 ordinal variables, 3 discrete interval variables having relatively few values, and 4 continuous variables grouped into a smallŽ . number of categories.

1.1.4 Quantitative–Qualitative Variable Distinction

Nominal variables are qualitati®eᎏdistinct categories differ in quality, not in quantity. Interval variables are quantitati®eᎏdistinct levels have differing amounts of the characteristic of interest. The position of ordinal variables in

(19)

4

the quantitative᎐qualitative classification is fuzzy. Analysts often treat them as qualitative, using methods for nominal variables. But in many respects, ordinal variables more closely resemble interval variables than they resemble nominal variables. They possess important quantitative features:

Each category has a greater or smaller magnitude of the characteristic than another category; and although not possible to measure, an underlying continuous variable is usually present. The political philosophy classification Žliberal, moderate, conservative crudely measures an inherently continuous. characteristic.

Analysts often utilize the quantitative nature of ordinal variables by assigning numerical scores to categories or assuming an underlying continuous distribution. This requires good judgment and guidance from researchers who use the scale, but it provides benefits in the variety of methods available for data analysis.

1.1.5 Organization of This Book

The models for categorical response variables discussed in this book resemble regression models for continuous response variables; however, they assume binomial, multinomial, or Poisson response distributions instead of normality. Two types of models receive special attention, logistic regression and loglinear models. Ordinary logistic regression models, also called logit

Ž .

models, apply with binary i.e., two-category responses and assume a binomial distribution. Generalizations of logistic regression apply with multicategory responses and assume a multinomial distribution. Loglinear models apply with count data and assume a Poisson distribution. Certain equivalences exist between logistic regression and loglinear models.

The book has four main units. In the first, Chapters 1 through 3, we summarize descriptive and inferential methods for univariate and bivariate categorical data. These chapters cover discrete distributions, methods of inference, and analyses for measures of association. They summarize the non-model-based methods developed prior to about 1960.

In the second and primary unit, Chapters 4 through 9, we introduce models for categorical responses. In Chapter 4 we describe a class of generalized linear modelshaving models of this text as special cases. We focus on models for binary and count response variables. Chapters 5 and 6 cover the most important model for binary responses, logistic regression. In Chap- ter 7 we present generalizations of that model for nominal and ordinal multicategory response variables. In Chapter 8 we introduce the modeling of multivariate categorical response data and show how to represent association and interaction patterns by loglinear models for counts in the table that cross-classifies those responses. In Chapter 9 we discuss model building with loglinear and related logistic models and present some related models.

In the third unit, Chapters 10 through 13, we discuss models for handling repeated measurement and other forms of clustering. In Chapter 10 we

(20)

DISTRIBUTIONS FOR CATEGORICAL DATA 5 present models for a categorical response with matched pairs; these apply, for instance, with a categorical response measured for the same subjects at two times. Chapter 11 covers models for more general types of repeated categorical data, such as longitudinal data from several times with explanatory variables. In Chapter 12 we present a broad class of models, generalized linear mixed models, that use random effects to account for dependence with such data. In Chapter 13 further extensions and applications of the models from Chapters 10 through 12 are described.

The fourth and final unit is more theoretical. In Chapter 14 we develop asymptotic theory for categorical data models. This theory is the basis for large-sample behavior of model parameter estimators and goodness-of-fit statistics. Maximum likelihood estimation receives primary attention here and throughout the book, but Chapter 15 covers alternative methods of estimation, such as the Bayesian paradigm. Chapter 16 stands alone from the others, being a historical overview of the development of categorical data methods.

Most categorical data methods require extensive computations, and statistical software is necessary for their effective use. In Appendix A we discuss software that can perform the analyses in this book and show the use of SAS for text examples. See the Web sitewww.stat.ufl.edur;aarcdarcda.html to download sample programs and data sets and find information about other software.

Chapter 1 provides background material. In Section 1.2 we review the key distributions for categorical data: the binomial, multinomial, and Poisson. In Section 1.3 we review the primary mechanisms for statistical inference, using maximum likelihood. In Sections 1.4 and 1.5 we illustrate these by presenting significance tests and confidence intervals for binomial and multinomial parameters.

1.2 DISTRIBUTIONS FOR CATEGORICAL DATA

Inferential data analyses require assumptions about the random mechanism that generated the data. For regression models with continuous responses, the normal distribution plays the central role. In this section we review the three key distributions for categorical responses: binomial, multinomial, and Poisson.

1.2.1 Binomial Distribution

Many applications refer to a fixed number n of binary observations. Let y₁,y₂, . . . , y_n denote responses for n independent and identical trials such

Ž . Ž .

that P Y_is1 s␲ and P Y_is0 s1y␲. We use the generic labels

‘‘success’’ and ‘‘failure’’ for outcomes 1 and 0. Identical trials means that the probability of success ␲ is the same for each trial. Independent trials means

(21)

6

4

that the Y_i are independent random variables. These are often called Bernoulli trials. The total number of successes, YsÝⁿis1Yi, has the binomial

Ž .

distributionwith index n and parameter␲, denoted by bin n,␲ . The probability mass function for the possible outcomes y for Y is

n y nyy

p yŽ .s

ž /

^y ␲ Ž1y␲. , ys0, 1, 2, . . . ,n, Ž1.1.

n w Ž .x Ž . Ž 2.

where the binomial coefficient

ž /

^y sn!r y! nyy ! . Since E Y_i sE Y_i

Ž .

s1=␲q0= 1y␲ s␲,

E YŽ _i.s␲ and varŽY_i.s␲Ž1y␲.. The binomial distribution for YsÝ_iY_i has mean and variance

␮sE YŽ .sn␲ and ␴²svarŽY.s n␲Ž1y␲..

3 3

Ž . Ž .

'

The skewness is described by E Yy␮ r␴ s 1y2␲ r n␲Ž1y␲.. The distribution converges to normality as n increases, for fixed ␲.

There is no guarantee that successive binary observations are independent or identical. Thus, occasionally, we will utilize other distributions. One such case is sampling binary outcomes without replacement from a finite population, such as observations on gender for 10 students sampled from a class of size 20. The hypergeometric distribution, studied in Section 3.5.1, is then relevant. In Section 1.2.4 we mention another case that violates these binomial assumptions.

1.2.2 Multinomial Distribution

Some trials have more than two possible outcomes. Suppose that each of n independent, identical trials can have outcome in any of c categories. Let y_{i j}s1 if trial i has outcome in category j and y_{i j}s0 otherwise. Then

Ž .

y_is y_i1,y_i2, . . . , y_{i c} represents a multinomial trial, with Ý_j y_{i j}s1; for

Ž .

instance, 0, 0, 1, 0 denotes outcome in category 3 of four possible categories.

Note that y_{i c} is redundant, being linearly dependent on the others. Let n_jsÝ_iy_{i j} denote the number of trials having outcome in category j. The

Ž .

counts n₁,n₂, . . . ,n_c have the multinomial distribution.

Ž .

Let ␲_jsP Y_{i j}s1 denote the probability of outcome in category j for each trial. The multinomial probability mass function is

n! _n _n _n

1 2 c

p nŽ ₁,n₂, . . . ,n_cy1.s

ž

ⁿ₁^!ⁿ₂^!⭈⭈⭈ n_c!

/

␲ ␲ ⭈⭈⭈ ␲1 2 c . Ž1.2.

(22)

DISTRIBUTIONS FOR CATEGORICAL DATA 7

Ž . Ž

Since Ýjnjsn, this is cy1 -dimensional, with ncsny n1q⭈⭈⭈

.

qn_cy1 . The binomial distribution is the special case with cs2.

For the multinomial distribution,

E nŽ j.sn␲j, varŽnj.sn␲j

Ž

1y␲j

.

, covŽnj,nk.s yn␲ ␲j k. 1.3

Ž .

We derive the covariance in Section 14.1.4. The marginal distribution of each n_j is binomial.

1.2.3 Poisson Distribution

Sometimes, count data do not result from a fixed number of trials. For instance, if ysnumber of deaths due to automobile accidents on motorways in Italy during this coming week, there is no fixed upper limit nfor yŽas you are aware if you have driven in Italy . Since. y must be a nonnegative integer, its distribution should place its mass on that range. The simplest such distribution is the Poisson. Its probabilities depend on a single parameter,

Ž .

the mean ␮. The Poisson probability mass function Poisson 1837, p. 206 is e^y^␮␮^y

p yŽ .s , ys0, 1, 2, . . . . Ž1.4. y!

Ž . Ž .

It satisfies E Y svar Y s␮. It is unimodal with mode equal to the

3 3

Ž .

'

integer part of ␮. Its skewness is described by E Yy␮ r␴ s1r ␮. The distribution approaches normality as ␮ increases.

The Poisson distribution is used for counts of events that occur randomly over time or space, when outcomes in disjoint periods or regions are independent. It also applies as an approximation for the binomial when nis large and␲ is small, with ␮sn␲. So if each of the 50 million people driving in Italy next week is an independent trial with probability 0.000002 of dying in a

Ž .

fatal accident that week, the number of deathsY is a bin 50000000, 0.000002

Ž .

variate, or approximately Poisson with␮sn␲s50,000,000 0.000002 s100.

A key feature of the Poisson distribution is that its variance equals its mean. Sample counts vary more when their mean is higher. When the mean number of weekly fatal accidents equals 100, greater variability occurs in the weekly counts than when the mean equals 10.

1.2.4 Overdispersion

In practice, count observations often exhibit variability exceeding that predicted by the binomial or Poisson. This phenomenon is called o®erdispersion.

We assumed above that each person has the same probability of dying in a fatal accident in the next week. More realistically, these probabilities vary,

(23)

8

due to factors such as amount of time spent driving, whether the person wears a seat belt, and geographical location. Such variation causes fatality counts to display more variation than predicted by the Poisson model.

Ž < .

Suppose that Y is a random variable with variance var Y ␮ for given ␮, but ␮ itself varies because of unmeasured factors such as those just de-

Ž .

scribed. Let ␪sE ␮ . Then unconditionally,

< < <

E YŽ .sE E YŽ ␮. , varŽY.sE varŽY ␮. qvar E YŽ ␮. .

Ž . Ž . Ž .

WhenY is conditionally Poisson given␮ , for instance, then E Y sE ␮ s

Ž . Ž . Ž . Ž .

␪ and var Y sE ␮ qvar ␮ s␪qvar ␮ )␪.

Assuming a Poisson distribution for a count variable is often too simplistic, because of factors that cause overdispersion. The negati®e binomial is a related distribution for count data that permits the variance to exceed the mean. We introduce it in Section 4.3.4.

Ž .

Analyses assuming binomial or multinomial distributions are also sometimes invalid because of overdispersion. This might happen because the true distribution is a mixture of different binomial distributions, with the parameter varying because of unmeasured variables. To illustrate, suppose that an experiment exposes pregnant mice to a toxin and then after a week observes the number of fetuses in each mouse’s litter that show signs of malformation.

Let n_i denote the number of fetuses in the litter for mouse i. The mice also vary according to other factors that may not be measured, such as their weight, overall health, and genetic makeup. Extra variation then occurs because of the variability from litter to litter in the probability␲ of malformation. The distribution of the number of fetuses per litter showing malfor- mations might cluster near 0 and near n_i, showing more dispersion than expected for binomial sampling with a single value of ␲. Overdispersion could also occur when␲ varies among fetuses in a litter according to some

Ž .

distribution Problem 1.12 . In Chapters 4, 12, and 13 we introduce methods for data that are overdispersed relative to binomial and Poisson assumptions.

1.2.5 Connection between Poisson and Multinomial Distributions

In Italy this next week, let y₁snumber of people who die in automobile accidents, y₂snumber who die in airplane accidents, and y₃snumber who

Ž .

die in railway accidents. A Poisson model for Y₁,Y₂,Y₃ treats these as

Ž .

independent Poisson random variables, with parameters ␮₁,␮₂,␮₃. The 4

joint probability mass function for Y_i is the product of the three mass

Ž .

functions of form 1.4 . The total nsÝY_i also has a Poisson distribution, with parameter Ý␮_i.

With Poisson sampling the total count nis random rather than fixed. If we 4

assume a Poisson model but condition on n, Y_i no longer have Poisson 4

distributions, since each Y_i cannot exceed n. Given n, Y_i are also no longer independent, since the value of one affects the possible range for the others.

(24)

STATISTICAL INFERENCE FOR CATEGORICAL DATA 9 Ž .

For c independent Poisson variates, with E Y_i s␮_i, let’s derive their conditional distribution given thatÝYisn. The conditional probability of a

4

set of counts n_i satisfying this condition is

P ŽY1sn1,Y2sn2, . . . ,Ycsnc.

Ý

Yjsn P YŽ ₁sn₁,Y₂sn₂, . . . ,Y_csn_c. s P

Ž

ÝYjsn

.

n_i

Ł_i expŽy␮ ␮_i. _irn_i! n! _n

s ⁿ s

Ł

␲ii, Ž1.5. Ł n!

exp

Ž

yÝ␮j

. Ž

Ý␮j

.

rn! i i ⁱ

Ž .4 Ž 4.

where ␲_is␮_irÝ␮_j . This is the multinomial n, ␲_i distribution, charac-

4

terized by the sample size n and the probabilities ␲_i.

Many categorical data analyses assume a multinomial distribution. Such analyses usually have the same parameter estimates as those of analyses assuming a Poisson distribution, because of the similarity in the likelihood functions.

1.3 STATISTICAL INFERENCE FOR CATEGORICAL DATA

The choice of distribution for the response variable is but one step of data analysis. In practice, that distribution has unknown parameter values. In this section we review methods of using sample data to make inferences about the parameters. Sections 1.4 and 1.5 cover binomial and multinomial parameters.

1.3.1 Likelihood Functions and Maximum Likelihood Estimation

In this book we use maximum likelihood for parameter estimation. Under weak regularity conditions, such as the parameter space having fixed dimen- sion with true value falling in its interior, maximum likelihood estimators have desirable properties: They have large-sample normal distributions; they are asymptotically consistent, converging to the parameter as n increases;

and they are asymptotically efficient, producing large-sample standard errors no greater than those from other estimation methods.

Given the data, for a chosen probability distribution the likelihood function is the probability of those data, treated as a function of the unknown

Ž .

parameter. The maximum likelihood ML estimate is the parameter value that maximizes this function. This is the parameter value under which the data observed have the highest probability of occurrence. The parameter value that maximizes the likelihood function also maximizes the log of that function. It is simpler to maximize the log likelihood since it is a sum rather than a product of terms.

(25)

10

We denote a parameter for a generic problem by ␤ and its ML estimate

ˆ Ž .

by ␤. The likelihood function is

ll

␤ and the log-likelihood function is ˆ

Ž . w Ž .x Ž .

L ␤ slog

ll

␤ . For many models, L ␤ has concave shape and ␤ is the point at which the derivative equals 0. The ML estimate is then the solution

Ž .

of the likelihood equation, ⭸L ␤ r⭸␤s0. Often, ␤ is multidimensional, denoted by ␤, and ␤ˆ is the solution of a set of likelihood equations.

ˆ Žˆ.

Let SE denote the standard error of ␤, and let cov ␤ denote the

ˆ Ž

asymptotic covariance matrix of ␤. Under regularity conditions Rao 1973, ˆ

. Ž . Ž .

p. 364 , cov ␤ is the inverse of the information matrix. The j,k element of the information matrix is

⭸²LŽ␤.

yE

ž

^{⭸␤ ⭸␤}^j ^k

/

. Ž1.6. The standard errors are the square roots of the diagonal elements for the inverse information matrix. The greater the curvature of the log likelihood, the smaller the standard errors. This is reasonable, since large curvature implies that the log likelihood drops quickly as␤moves away from ˆ␤; hence, the data would have been much more likely to occur if␤took a value near ˆ␤ rather than a value far from ˆ␤.

1.3.2 Likelihood Function and ML Estimate for Binomial Parameter The part of a likelihood function involving the parameters is called the kernel. Since the maximization of the likelihood is with respect to the parameters, the rest is irrelevant.

Ž .

To illustrate, consider the binomial distribution 1.1 . The binomial coefficient

ž /

n^y has no influence on where the maximum occurs with respect to␲. Thus, we ignore it and treat the kernel as the likelihood function. The binomial log likelihood is then

y nyy

LŽ␲.slog ␲ Ž1y␲. sylogŽ␲.qŽnyy.log 1Ž y␲.. Ž1.7. Differentiating with respect to␲ yields

⭸LŽ␲.r⭸␲syr␲yŽnyy.rŽ1y␲.sŽyyn␲.r␲Ž1y␲.. Ž1.8. Equating this to 0 gives the likelihood equation, which has solution␲ˆsyrn, the sample proportion of successes for the ntrials.

2 Ž . ²

Calculating ⭸ L␲ r⭸␲ , taking the expectation, and combining terms, we get

2 2 2 2

yE ⭸ LŽ␲.r⭸␲ sE yr␲ qŽnyy.rŽ1y␲. snr ␲Ž1y␲. . 1.9

Ž .

(26)

STATISTICAL INFERENCE FOR CATEGORICAL DATA 11

Ž .

Thus, the asymptotic variance of␲ˆ is␲ 1y␲ rn. This is no surprise. Since

Ž . Ž . Ž .

E Y sn␲ and var Y sn␲ 1y␲ , the distribution of␲ˆsYrnhas mean and standard error

␲Ž1y␲. EŽ␲ˆ.s␲, ␴ ␲Žˆ.s

(

_n . 1.3.3 Wald–Likelihood Ratio–Score Test Triad

Three standard ways exist to use the likelihood function to perform large-sample inference. We introduce these for a significance test of a null hypothesis H₀: ␤s␤₀ and then discuss their relation to interval estimation.

They all exploit the large-sample normality of ML estimators.

With nonnull standard error SE of ␤ˆ, the test statistic zs

Ž

␤ˆy␤0

.

rSE

has an approximate standard normal distribution when ␤s␤₀. One refers z to the standard normal table to obtain one- or two-sided P-values. Equiva- lently, for the two-sided alternative, z² has a chi-squared null distribution

Ž .

with 1 degree of freedom df ; the P-value is then the right-tailed chi-squared probability above the observed value. This type of statistic, using the nonnull

Ž .

standard error, is called aWald statistic Wald 1943 .

The multivariate extension for the Wald test of H₀: ␤s␤₀ has test statistic

X y1

ˆ ˆ ˆ

Ws

Ž

␤y␤0

.

cov

Ž .

␤

Ž

␤y␤0

.

ŽThe prime on a vector or matrix denotes the transpose.. The nonnull ˆ

Ž .

covariance is based on the curvature 1.6 of the log likelihood at ␤. The asymptotic multivariate normal distribution for ␤ˆ implies an asymptotic

ˆ Ž .

chi-squared distribution forW. The df equal the rank of cov ␤, which is the number of nonredundant parameters in ␤.

A second general-purpose method uses the likelihood function through the ratio of two maximizations: 1 the maximum over the possible parameterŽ . values under H₀, and 2 the maximum over the larger set of parameterŽ . values permitting H0 or an alternative Ha to be true. Let

ll

0 ^{denote the}

maximized value of the likelihood function under H₀, and let

ll

₁denote the

Ž .

maximized value generally i.e., under H₀jH_a . For instance, for parameter

Ž .

vector␤s ␤₀,␤₁ ⬘and H₀:␤₀s0,

ll

₁is the likelihood function calculated at the ␤ value for which the data would have been most likely;

ll

₀ is the likelihood function calculated at the ␤₁ value for which the data would have been most likely, when ␤₀s0. Then

ll

₁ is always at least as large as

ll

₀^{, since}

ll

₀ results from maximizing over a restricted set of the parameter values.

(27)

12

The ratio ⌳s

ll

₀^r

ll

₁of the maximized likelihoods cannot exceed 1. Wilks Ž1935, 1938 showed that. y2 log⌳ has a limiting null chi-squared distribution, as n™⬁. The df equal the difference in the dimensions of the parameter spaces under H0jHa and under H0. The likelihood-ratio test statisticequals

y2 log⌳s y2 log

Ž ll

0^r

ll

1

.

^{s y}²ŽL0yL1., where L₀ and L₁denote the maximized log-likelihood functions.

The third method uses the score statistic, due to R. A. Fisher and C. R.

Rao. The score test is based on the slope and expected curvature of the Ž .

log-likelihood function L ␤ at the null value ␤₀. It utilizes the size of the score function

uŽ␤.s⭸LŽ␤.r⭸␤,

ˆ

Ž .

evaluated at ␤0. The value u ␤0 tends to be larger in absolute value when ␤ w ² Ž . ²x Ž . is farther from ␤₀. Denote yE ⭸ L ␤ r⭸␤ i.e., the information evalu-

Ž . Ž .

ated at ␤₀ by␫ ␤₀ . The score statistic is the ratio of u ␤₀ to its null SE, w Ž .x¹^r²

which is ␫ ␤₀ . This has an approximate standard normal null distribution. The chi-squared form of the score statistic is

2 2

uŽ␤₀. ⭸LŽ␤.r⭸␤₀

s ₂ ₂ ,

␫ ␤Ž ₀. yE ⭸ LŽ␤.r⭸␤₀

where the partial derivative notation reflects derivatives with respect to ␤ that are evaluated at ␤₀. In the multiparameter case, the score statistic is a quadratic form based on the vector of partial derivatives of the log likelihood with respect to ␤ and the inverse information matrix, both evaluated at the

Ž .

H₀ estimates i.e., assuming that ␤s␤₀ .

Ž .

Figure 1.1 is a generic plot of a log-likelihood L ␤ for the univariate case. It illustrates the three tests of H₀: ␤s0. The Wald test uses the

ˆ ˆ ²

Ž . Ž .

behavior of L ␤ at the ML estimate ␤, having chi-squared form ␤rSE .

ˆ Ž . ˆ

The SE of␤ depends on the curvature of L ␤ at ␤. The score test is based Ž .

on the slope and curvature of L ␤ at ␤s0. The likelihood-ratio test ˆ

Ž .

combines information about L ␤ at both ␤ and ␤₀s0. It compares the log-likelihood values L₁ at ␤ˆ and L₀ at ␤₀s0 using the chi-squared

Ž .

statistic y2 L₀yL₁ . In Figure 1.1, this statistic is twice the vertical dis- ˆ

Ž .

tance between values of L ␤ at ␤ and at 0. In a sense, this statistic uses the most information of the three types of test statistic and is the most versatile.

As n™⬁, the Wald, likelihood-ratio, and score tests have certain asymp-

Ž .

totic equivalences Cox and Hinkley 1974, Sec. 9.3 . For small to moderate sample sizes, the likelihood-ratio test is usually more reliable than the Wald test.

(28)

STATISTICAL INFERENCE FOR CATEGORICAL DATA 13

FIGURE 1.1 Log-likelihood function and information used in three tests of H₀:␤s0.

1.3.4 Constructing Confidence Intervals

In practice, it is more informative to construct confidence intervals for parameters than to test hypotheses about their values. For any of the three test methods, a confidence interval results from inverting the test. For instance, a 95% confidence interval for ␤ is the set of ␤₀ for which the test of H₀:␤s␤₀ has a P-value exceeding 0.05.

Let z_a denote the z-score from the standard normal distribution having

Ž .

right-tailed probabilitya; this is the 100 1ya percentile of that distribution.

2Ž . Ž .

Let ␹_df a denote the 100 1ya percentile of the chi-squared distribution

Ž .

with degrees of freedom df. 100 1y␣ % confidence intervals based on asymptotic normality use z_␣_r2, for instance z_0.025s1.96 for 95% confidence.

< ˆ <

The Wald confidence interval is the set of␤₀ for which ␤y␤₀ rSE-z_␣_r2.

ˆ Ž .

This gives the interval ␤"z_␣_r2 SE . The likelihood-ratio-based confidence

ˆ ²

w Ž . Ž .x Ž . w interval is the set of ␤₀ for which y2 L ␤₀ yL ␤ -␹ ␣₁ . Recall

2Ž . ² x

that ␹ ␣₁ sz_␣r2.

When ␤ˆ has a normal distribution, the log-likelihood function has a

Ž .

parabolic shape i.e., a second-degree polynomial . For small samples with categorical data, ␤ˆmay be far from normality and the log-likelihood function can be far from a symmetric, parabolic-shaped curve. This can also happen with moderate to large samples when a model contains many parameters. In such cases, inference based on asymptotic normality of ␤ˆ may have inadequate performance. A marked divergence in results of Wald and likelihood- ratio inference indicates that the distribution of ␤ˆ may not be close to normality. The example in Section 1.4.3 illustrates this with quite different confidence intervals for different methods. In many such cases, inference can

(29)

14

instead utilize an exact small-sample distribution or ‘‘higher-order’’ asymptotic methods that improve on simple normality e.g., Pierce and PetersŽ 1992 ..

The Wald confidence interval is most common in practice because it is simple to construct using ML estimates and standard errors reported by statistical software. The likelihood-ratio-based interval is becoming more widely available in software and is preferable for categorical data with small to moderate n. For the best known statistical model, regression for a normal response, the three types of inference necessarily provide identical results.

1.4 STATISTICAL INFERENCE FOR BINOMIAL PARAMETERS

In this section we illustrate inference methods for categorical data by presenting tests and confidence intervals for the binomial parameter ␲, based on y successes in n independent trials. In Section 1.3.2 we obtained the likelihood function and ML estimator␲ˆsyrn of␲.

1.4.1 Tests about a Binomial Parameter

Consider H₀: ␲s␲₀. Since H₀ has a single parameter, we use the normal rather than chi-squared forms of Wald and score test statistics. They permit tests against one-sided as well as two-sided alternatives. The Wald statistic is

␲ˆy␲0 ␲ˆy␲0

z_Ws s . Ž1.10.

SE

'

␲ˆŽ1y␲ˆ.rn

Ž . Ž .

Evaluating the binomial score 1.8 and information 1.9 at␲₀ yields

y nyy n

uŽ␲₀.s y , ␫ ␲Ž ₀.s .

␲₀ 1y␲₀ ␲₀Ž1y␲₀. The normal form of the score statistic simplifies to

uŽ␲0. yyn␲0 ␲ˆy␲0

z_Ss ₁_r₂ s s . Ž1.11.

n␲ 1y␲ ␲ 1y␲ rn

'

^Ž ^.

'

^Ž ^.

␫ ␲Ž 0. 0 0 0 0

Whereas the Wald statistic zW uses the standard error evaluated at␲ˆ, the score statistic z_S uses it evaluated at ␲₀. The score statistic is preferable, as it uses the actual null SE rather than an estimate. Its null sampling distribution is closer to standard normal than that of the Wald statistic.

Ž .

The binomial log-likelihood function 1.7 equals L₀sylog␲₀ q Žnyy.log 1Ž y␲0. under H0 and L1sylog␲ˆqŽnyy.log 1Ž y␲ˆ. more

(30)

STATISTICAL INFERENCE FOR BINOMIAL PARAMETERS 15 generally. The likelihood-ratio test statistic simplifies to

␲ˆ 1y␲ˆ

y2ŽL₀yL₁.s2

ž

ylog␲0 qŽnyy.log1y␲0

/

. Expressed as

y nyy

y2ŽL₀yL₁.s2

ž

ylogn␲0 qŽnyy.lognyn␲0

/

,

Ž .

it compares observed success and failure counts to fitted i.e., null counts by observed

2

Ý

observed log . Ž1.12.

fitted

We’ll see that this formula also holds for tests about Poisson and multinomial parameters. Since no unknown parameters occur under H₀ and one occurs

Ž .

under H_a, 1.12 has an asymptotic chi-squared distribution with df s1.

1.4.2 Confidence Intervals for a Binomial Parameter

A significance test merely indicates whether a particular ␲ value such asŽ

␲s0.5 is plausible. We learn more by using a confidence interval to. determine the range of plausible values.

Inverting the Wald test statistic gives the interval of ␲₀ values for which

<z_W< -z_␣_r2, or

␲ˆŽ1y␲ˆ.

␲ˆ"z^␣^r²

(

_n . Ž1.13.

Historically, this was one of the first confidence intervals used for any

Ž .

parameter Laplace 1812, p. 283 . Unfortunately, it performs poorly unless n

Ž .

is very large e.g., Brown et al. 2001 . The actual coverage probability usually falls below the nominal confidence coefficient, much below when␲ is near 0

1 2

or 1. A simple adjustment that adds 2z_␣r2 observations of each type to the

Ž .

sample before using this formula performs much better Problem 1.24 .

< <

The score confidence interval contains ␲₀ values for which z_S -z_␣r2. Its endpoints are the␲₀ solutions to the equations

␲y␲ r

'

␲ ^Ž1y␲ ^.rn s"z .

Ž

_ˆ ₀

.

₀ ₀ _␣r2

(31)

16

Ž .

These are quadratic in ␲₀. First discussed by E. B. Wilson 1927 , this interval is

n 1 z_␣²r2

␲ˆ

ž

ⁿ^q^z␣2r2

/

q ²

ž

ⁿ^q^z␣2r2

/

1 n 1 1 z_␣2r2

"z␣r2

)

ⁿ^q^z␣2r2 ␲ˆŽ1y␲ˆ.

^ž

ⁿ^q^z␣2r2

^/

q

^{ž / ž /}

² ²

ž

ⁿ^q^z␣2r2

/

. The midpoint␲˜ of the interval is a weighted average of␲ˆ and , where the12

Ž ² .

weight nr nqz␣r2 given␲ˆ increases as nincreases. Combining terms, this

Ž ² . Ž ² .

midpoint equals␲˜s yqz␣r2r2r nqz␣r2 . This is the sample proportion for an adjusted sample that adds z_␣²r2 observations, half of each type. The square of the coefficient of z_␣r2 in this formula is a weighted average of the variance of a sample proportion when␲s␲ˆ and the variance of a sample

1 2

proportion when ␲s₂, using the adjusted sample size nqz_␣_r2 in place of n. This interval has much better performance than the Wald interval.

The likelihood-ratio-based confidence interval is more complex computa- tionally, but simple in principle. It is the set of␲₀ for which the likelihood- ratio test has a P-value exceeding ␣. Equivalently, it is the set of ␲₀ for

2Ž .

which double the log likelihood drops by less than ␹ ␣₁ from its value at the ML estimate␲ˆsyrn.

1.4.3 Proportion of Vegetarians Example

To collect data in an introductory statistics course, recently I gave the students a questionnaire. One question asked each student whether he or she was a vegetarian. Of ns25 students, ys0 answered ‘‘ yes.’’ They were not a random sample of a particular population, but we use these data to illustrate 95% confidence intervals for a binomial parameter␲.

Since ys0, ␲ˆs0r25s0. Using the Wald approach, the 95% confidence interval for␲ is

'

0"1.96 Ž0.0=1.0.r25 , or Ž0, 0 ..

When the observation falls at the boundary of the sample space, often Wald methods do not provide sensible answers.

Ž .

By contrast, the 95% score interval equals 0.0, 0.133 . This is a more believable inference. For H₀:␲s0.5, for instance, the score test statistic is

Ž .

'

z_Ss 0y0.5r Ž0.5=0.5.r25s y5.0, so 0.5 does not fall in the interval.

Ž .

'

By contrast, forH₀:␲s0.10, z_Ss 0y0.10r Ž0.10=0.90.r25s y1.67, so 0.10 falls in the interval.

(32)

STATISTICAL INFERENCE FOR BINOMIAL PARAMETERS 17 Ž . When ys0 and ns25, the kernel of the likelihood function is

ll

␲ s

0Ž .²⁵ Ž .²⁵ Ž . Ž . Ž .

␲ 1y␲ s 1y␲ . The log likelihood 1.7 is L␲ s25 log 1y␲ .

Ž . Ž .

Note that L␲ˆ sL0 s0. The 95% likelihood-ratio confidence interval is the set of␲₀ for which the likelihood-ratio statistic

y2ŽL0yL1.s y2 LŽ␲0.yLŽ␲ˆ.

s y50 log 1Ž y␲₀.F␹₁²Ž0.05.s3.84.

Ž .

The upper bound is 1yexpy3.84r50 s0.074, and the confidence interval

Ž . w

equals 0.0, 0.074 . In this book, we use the natural logarithm throughout, so

Ž . ^xx

its inverse is the exponential function exp x se . Figure 1.2 shows the likelihood and log-likelihood functions and the corresponding confidence region for␲.

The three large-sample methods yield quite different results. When ␲ is near 0, the sampling distribution of␲ˆ is highly skewed to the right for small n. It is worth considering alternative methods not requiring asymptotic approximations.

FIGURE 1.2 Binomial likelihood and log likelihood when ys0 in ns25 trials, and confidence interval for␲.

Categorical Data Analysis

Categorical Data Analysis

Categorical Data Analysis

Second Edition

䊚

To Jacki

Contents

Preface

C H A P T E R 1

Introduction: Distributions and Inference for Categorical Data

ž /

ž /

'

ž

/

Ž

.

'

Ý

Ž

.

Ł

Ž

. Ž

.

ll

ll

ž

/

ž /

(

Ž

.

Ž

.

Ž .

Ž

.

ll

ll

ll

ll

ll

ll

ll

ll

ll

Ž ll

ll

.

'

'

'

ž

/

ž

/

Ý

(

'

Ž

.

ž

/

ž

/

)

ž

/

ž / ž /

ž

/

'

'

'

ll

^ž

^/

^{ž / ž /}