Introductory Topics

375 Pages·1978·17.113 MB·English

by  Shelby J. Haberman (Auth.)

SHELBY J. HABERMAN
DEPARTMENT OF STATISTICS
UNIVERSITY OF CHICAGO
CHICAGO, ILLINOIS
Analysis of
Qualitative Data
Volume 1 INTRODUCTORY TOPICS
ACADEMIC PRESS New York San Francisco London 1978
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1978, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1 7DX
Library of Congress Cataloging in Publication Data
Haberman, Shelby J
Analysis of qualitative data.
Includes bibliographical references and indexes.
CONTENTS: v. 1. Introductory topics.
1. Log-linear models. 2. Contingency tables.
3. Social sciences—Statistical methods.
I. Title.
HA33.H27 519.5 77-25731
ISBN 0-12-312501-4
PRINTED IN THE UNITED STATES OF AMERICA
Preface
Qualitative data are encountered throughout the social sciences. De­
spite this widespread occurrence, statistical methods for analysis of these
data remained quite primitive until the 1960s. Since the early 1960s,
development of log-linear models has resulted in very rapid progress in
the sophisticated analysis of qualitative and nominal data. Although much
of the development of log-linear models has been brought about by
statisticians such as Leo Goodman and Frederick Mosteller with close
ties to the social sciences, few social scientists have had more than a
limited acquaintance with this class of statistical models.
A social scientist interested in learning about log-linear models cur­
rently has available a welter of journal articles. These articles appear in
the statistics, social science, and biological science literature, and varying
levels of mathematical sophistication are required to read them. Three
recent books, Plackett (1974), Bishop, Fienberg, and Holland (1975), and
Haberman (1974a), are also available. None is specifically oriented to­
ward the needs of social scientists, either in choice of examples and topics
or in mathematical level.
This book attempts to provide an introduction to log-linear models that
is oriented toward social scientists. Mathematical requirements are kept
to a minimum. Some familiarity with matrix algebra sometimes will be
helpful, but no explicit use will be made of calculus. Although the formal
mathematical requirements are limited, the book does contain numerous
algebraic formulas and summation notation is repeatedly used. Without
such expressions it is not possible to present general computational al­
gorithms, and assessment of the variability of parameter estimates is
impossible. The book assumes sufficient prior knowledge of statistics so
vii
viii PREFACE
that concepts such as confidence intervals, hypothesis tests, and point
estimates are familiar. Knowledge of analysis of variance and regression
analysis is very helpful in gaining perspective on the relationships be­
tween the analysis of continuous and of discrete variables; however, this
knowledge is not required for understanding the matérial in this book.
General familiarity with maximum-likelihood estimation is also helpful
but not necessary. These restrictions prevent inclusion of formal proofs of
results; however, such proofs can be found in Haberman (1974a).
Examples used to illustrate the statistical techniques described in this
book are obtained from real data of interest to social scientists. Some
examples will involve basic problems in survey research, such as memory
error. Some examples will consider topics of public interest such as
variations in homicide rates related to variables such as the race and sex
of victim. Still other examples will use the General Social Survey of the
National Opinion Research Center to examine public opinion concerning
abortion.
The first four chapters are ordered by the number of variables under
study. Chapter 1 introduces some basic methods for study of the distribu­
tion of a single polytomous variable. Several simple examples provide an
introduction to the basic estimation and testing procedures associated
with log-linear models.
Chapter 2 considers contingency tables in which two polytomous vari­
ables are cross classified. Tests for independence and methods for de­
scribing departures from independence are discussed.
In Chapter 3 hierarchical log-linear models for three-way tables are
introduced. In these tables, three polytomous variables are cross
classified. Models for independence, conditional independence, and no
three-factor interaction are described. The iterative proportional fitting
algorithm of Deming and Stephen (1940) is introduced.
Chapter 4 extends procedures developed in Chapter 3 to contingency
tables in which four or more variables are cross classified.
Chapter 5 provides an introduction to logit models. In these models
discrete or continuous independent variables are used to predict a
dichotomous dependent variable. The models considered are analogous to
regression models used with continuous dependent variables.
In Chapter 6 multinomial-response models are developed for prediction
of one or more polytomous dependent variables. These models are
generalizations of the logit models of Chapter 5.
Chapters 7 and 8 examine contingency tables with special structures. In
Chapter 7 tables are considered in which certain cells are unusual in their
behavior. The quasi-independence model and other models that ignore
these unusual cells are considered. In Chapter 8 models are considered for
PREFACE ix
contingency tables in which several polytomous variables have the same
categories. Symmetry models, quasi-symmetry models, and distance
models are introduced.
Chapter 9 considers a classical problem of applied statistics, the ad­
justment of data. The methods developed by Deming and Stephan in the
1940s are illustrated, and some newer possibilities are discussed.
Chapter 10 examines the relationship of log-linear models to latent-
structure models. The material in this chapter is quite new. It is intended
to provide an indication of possible future developments.
When used as a textbook in a one-quarter or one-semester course, the
first five chapters, which constitute Volume 1, can be regarded as a
minimal goal.
A cknowledgments
Work on this book has been supported in part by National Science
Foundation Grant No. SOC72-05228 A04, National Science Foundation
Grant No. MCS72-04364 A04, and National Institutes of Health Grant
No. GM22648.
Discussions with Professors Leo Goodman, William Mason, and Clif­
ford Clogg have been particularly helpful in preparation of the manu­
script. Much of the data used in this book has been provided by the
National Opinion Research Center of the University of Chicago. Thanks
are also due to the editors of Sociological Analysis for permission to use
Table 2.2, to John Wiley & Sons for permission to use Tables 2.4 and
2.14, and to Harcourt Brace Jovanovich for permission to use Tables
2.1 and 2.12.
Contents of Volume 2
Ό MULTINOMIAL-RESPONSE MODELS
7 INCOMPLETE TABLES
O SYMMETRICAL TABLES
\)ADJUSTMENT OF DATA
10 LATENT-CLASS MODELS
APPENDIX: COMPUTER PROGRAMS FOR COMPUTATION
OF MAXIMUM-LIKELIHOOD ESTIMATES
1
Po ly torno us Responses
The basic tools used to analyze frequency tables by means of log-linear
models may be introduced by an examination of some one-way frequency
tables which summarize observations on a single polytomous variable,
examples for which are provided in Sections 1.1 to 1.4. In Section 1.5, some
general observations are made concerning use of log-linear models with
one-way tables; chapter references are provided here.
Sections 1.1 and 1.2 treat multinomial data. The example in Section 1.1
uses memory data to illustrate use of Pearson and likelihood-ratio chi-
square statistics, adjusted residuals, and linear combinations of frequency
counts to examine departures from a model of equal probabilities for each
class. Departures from this model, which has many formal analogies to those
of simple linear regression, are then described by a log-linear model. The
analogies are used to describe the Newton-Raphson algorithm for com­
putation of maximum-likelihood estimates in terms of a series of weighted
regression analyses. Formulas used in weighted regression are also used to
obtain normal approximations for the distribution of estimated parameters.
Once maximum-likelihood estimates are obtained, chi-square tests and
adjusted residuals are used to explore the adequacy of the proposed model.
In Section 1.2, which has a similar structure to Section 1.1, self-classi­
fication by social class is considered. Chi-square statistics and adjusted
residuals are now used to determine whether the distribution of self-classi­
fications is symmetrical. Departures from symmetry are examined by a
log-linear model which is formally analogous to a multiple regression model,
and shows that computational procedures for maximum-likelihood estimates
correspond to weighted multiple regressions.
In Sections 1.3 and 1.4 attention shifts to Poisson data. Section 1.3 con­
siders whether daily suicide rates are seasonally dependent and models of
1
2 1. POL TTOMO US RESPONSES
constant daily rates and of constant daily rates for each season are tested with
chi-square statistics and adjusted residuals. Regional variations in suicide
rates are considered in Section 1.4 and simultaneous confidence intervals
are introduced to determine which regional differences are clearly established.
Section 1.5 provides the general results on which the chapter is based.
Maximum-likelihood equations and the Newton-Raphson algorithm are
introduced for general log-linear models for both multinominal and Poisson
data. General analogies with weighted regression analysis are used to de­
scribe the Newton-Raphson algorithm and to develop large-sample ap­
proximations to the distributions of parameter estimates, residuals and
chi-square statistics.
1.1 Effects of Memory—An introduction
to Log-Linear Models
Analysis of data by log-linear models involves several distinct stages.
First, a plausible model is proposed for the data under study. Second,
unknown parameters in the model are estimated from the data, generally by
the method of maximum likelihood. This methods yields estimates available
in closed form, as in the equiprobability model, or estimates computed
by a version of the Newton-Raphson algorithm with analogies to weighted
regression analysis, as in the log-linear time-trend model. Third, these param­
eter estimates are used in statistical tests of the model's adequacy. Pearson
and likelihood-ratio chi-square tests provide overall measures of the com­
patibility of the model and the data. More specific insight into deviations
between model and data are provided by analysis of adjusted residuals and
of selected linear combinations of frequencies. Two possibilities exist at
the fourth step. If the model appears adequate, then the parameter estimates
are used to obtain quantitative implications concerning the data. Asymptotic
standard deviations and approximate confidence intervals are major tools
at this point. If the model appears inadequate, then the residual analyses
and analyses of linear combinations of frequencies of the third stage are
employed to suggest new models that are more consistent with the data,
to which the new model are then applied. Thus, analysis will often be an
iterative process in which the four-stage analysis is applied to several distinct
models, many of which are suggested by previous exploration of the data.
As an introduction to the use of log-linear models, consider the data
shown in Table 1.1. These data were compiled to study a problem observed
during a study conducted to explore the relationship between life stresses
and illnesses. The study, which is described in Uhlenhuth, Lipman, Baiter,