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Series/Number 07–146
LOGISTIC REGRESSION MODELS
FOR ORDINAL RESPONSE VARIABLES
ANN A. O’CONNELL
University of Connecticut
SAGE PUBLICATIONS
International Educational and Professional Publisher
Thousand Oaks London New Delhi
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Copyright © 2006 by Sage Publications, Inc.
All rights reserved. No part of this book may be reproduced or utilized in any form or by any
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For Nathan, and especially for Delaney
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CONTENTS
List of Tables and Figures vii
Series Editor’s Introduction ix
Acknowledgments xi
1. Introduction 1
Purpose of This Book 3
Software and Syntax 4
Organization of the Chapters 5
2. Context: Early Childhood Longitudinal Study 6
Overview of the Early Childhood Longitudinal Study 6
Practical Relevance of Ordinal Outcomes 7
Variables in the Models 8
3. Background: Logistic Regression 10
Overview of Logistic Regression 10
Assessing Model Fit 14
Interpreting the Model 15
Measures of Association 17
EXAMPLE 3.1: Logistic Regression 17
Comparing Results Across Statistical Programs 25
4. The Cumulative (Proportional)
Odds Model for Ordinal Outcomes 27
Overview of the Cumulative Odds Model 27
EXAMPLE 4.1: Cumulative Odds Model
With a Single Explanatory Variable 30
EXAMPLE 4.2: Full-Model
Analysis of Cumulative Odds 41
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Assumption of Proportional Odds and Linearity in the Logit 44
Alternatives to the Cumulative Odds Model 47
EXAMPLE 4.3: Partial Proportional Odds 49
5. The Continuation Ratio Model 54
Overview of the Continuation Ratio Model 54
Link Functions 57
Probabilities of Interest 58
Directionality of Responses and
Formation of the Continuation Ratios 59
EXAMPLE 5.1: Continuation Ratio Model
With Logit Link and Restructuring the Data 60
EXAMPLE 5.2: Continuation Ratio
Model With Complementary Log-Log Link 67
Choice of Link and Equivalence of Two Clog-Log Models 71
Choice of Approach for Continuation Ratio Models 73
EXAMPLE 5.3: Full-Model Continuation
Ratio Analyses for the ECLS-K Data 74
6. The Adjacent Categories Model 76
Overview of the Adjacent Categories Model 76
EXAMPLE 6.1: Gender-Only Model 77
EXAMPLE 6.2: Adjacent Categories Model
With Two Explanatory Variables 82
EXAMPLE 6.3: Full Adjacent Categories
Model Analysis 84
7. Conclusion 85
Considerations for Further Study 87
Notes 89
Appendix A: Chapter 3 91
Appendix B: Chapter 4 92
Appendix C: Chapter 5 94
Appendix D: Chapter 6 98
References 100
Index 104
About the Author 107
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LIST OF TABLES AND FIGURES
Tables
2.1 Proficiency Categories for the ECLS-K
Measures for Early Literacy 7
2.2 Descriptive Statistics at First-Grade Entry, N = 3,365 9
3.1 Cross-Tabulation of Proficiency (0, 1 versus 5)
by Gender, N = 702 18
3.2 Comparison of Results for SPSS, SAS,
and SPSS PLUM for a Dichotomous Outcome:
Proficiency (0, 1 versus 5) by Gender, N = 702 25
4.1 Category Comparisons Associated With
Three Different Ordinal Regression Model Approaches,
Based on a 6-Level Ordinal Outcome (j = 0, 1, 2, 3, 4, 5) 30
4.2 Observed Data Cross-Classification of
Gender by Five Proficiency Categories: Frequency (f),
Proportion (p), Cumulative Proportion (cp),
Cumulative Odds (co), and Odds Ratios (OR) 32
4.3 Predicted Cumulative Logits, Estimated Odds of Being
at or Below Category j for Boys and Girls, Estimated
Cumulative Probabilities (cp), and Estimated Odds
Ratios From the CO Model (SAS With Ascending Option) 35
4.4 Results for Cumulative Odds Model Using
SAS (Ascending), SAS (Descending), SPSS PLUM,
and Multiple Linear Regression on an Ordinal Response
Scale: Proficiency (j = 0, 1, 2, 3, 4, 5) by Gender, N = 3,365 38
4.5 Full-Model Analysis of Cumulative Odds (CO),
SAS (Descending) (Y > cat. j), N = 3,365 42
4.6 Classification Table for Full CO Model, N = 3,365 44
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viii
4.7 Associated Cumulative Binary Models for the CO Analysis
(Descending), Where CUMSP Compares Y < cat. j
j
to Y > cat. j, N = 3,365 46
5.1 Observed ECLS-K Gender Frequency (f),
Category Probability (p), and Conditional Probabilities
P(Beyond Category j Given at Least Category j) (δ ) 61
j
5.2 CR Model (Logit Link) Using Restructured
Data Set, N = 13,053; and Logistic Regression
Results for Each of the Five Conditional Binary Logistic
Models (P(Beyond Category j|Response in at Least Category j)) 63
5.3 Observed Proportions (δ ) for P(Y > j|Y > j), Predictions,
j
and Observed and Estimated ORs for Gender Model,
CR Analysis With Logit Link 65
5.4 Parameter Estimates for CR Models With
Clog-Log Link on Restructured Data Set,
N =13,053; and on Original Data Set, N = 3,365 68
5.5a Observed Proportions (δ ) for P(Y > j|Y > j), Predictions,
j
Estimated Hazards and Complements, and Estimated HRs
for Gender Models, CR Analyses With Clog-Log Link
(Using Restructured Data Set) 69
5.5b Observed Proportions (δ ) for P(Y > j|Y > j),
j
Predictions, Estimated Probabilities and Complements,
and Estimated HRs for Gender Models, CR Analyses
With Clog-Log Link (Using Original Data Set) 70
5.6 CR-Full-Model (Logit Link) Using
Restructured Data Set, N = 13,053; Binary CR
j
(Logit Link) Analyses for P(Y > cat. j|Y > cat. j);
and SPSS Clog-Log PLUM Analysis 75
6.1 Intercepts for the J – 1 = 5 AC Response Functions 80
* ^*
6.2 Observed (π ) and Predicted ( p )
j j
Conditional AC Probabilities 81
6.3 Adjacent Category Binary Logits for the Full Models 84
Figures
3.1 Selected Output: SPSS Logistic Regression Example 20
4.1 SAS Cumulative Odds Model Example: Gender 33
4.2 Partial Proportional Odds for Minority: GEE Analysis 51
6.1 PROC CATMOD Results: Simple Gender Model 79
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SERIES EDITOR’S INTRODUCTION
Over the past three decades, logit type models have become the most
popular statistical methods in the social sciences. In response to the need
for understanding such models and showing how to correctly use them in
various contexts, the Sage QASS (Quantitative Applications in the Social
Sciences) series has given considerable attention to their exposition: The
coverage includes No. 45 in the series, Linear Probability, Logit, and Probit
Models, by Aldrich and Nelson; No. 86, Logit Modeling, by DeMaris;
No. 101, Interpreting Probability Models: Logit, Probit, and Other
Generalized Linear Models, by Liao; No. 106, Applied Logistic Regression,
by Menard; No. 132, Logistic Regression: A Primer, by Pampel; No. 134,
Generalized Linear Models: A Unified Approach, by Gill; No. 135,
Interaction Effects in Logistic Regression, by Jaccard; and No. 138, Logit
and Probit: Ordered and Multinomial Models, by Borooah. Why did my
predecessor, Michael Lewis-Beck, who reviewed the prospectus and earlier
drafts, put in the good work of editing another book on logit models for the
series?
Since Rensis Likert’s 1932 publication of A Technique for the
Measurement of Attitudes, surveying human attitudes has never been the
same. Indeed, any social surveys today will include the Likert-type scale as
a staple means for asking questions. A typical Likert-type scale has five cat-
egories (e.g., strongly disagree, disagree, undecided, agree, strongly agree)
to gauge one’s response to a question, though it may have anywhere
between three and seven or more response categories. If we code the five
categories 1 to 5, we could estimate a linear regression model of a Likert-
type scale, and that was the choice of method in the early days for analyz-
ing such data. There are, however, some obvious problems. First and
foremost, classical linear regression assumes a continuous dependent vari-
able with equally spaced, ordered response categories. A Likert-type scale,
or any other ordinal scale, is, albeit ordered, not necessarily equally spaced
between categories. Second, and perhaps more important, such a scale
would not give the normal distribution that the classical linear regression
assumes the data to display.
To analyze ordinal data of this nature, there are other methods available,
most often in the form of contingency tables and log-linear models. The
Sage QASS series has also given attention to the topic, with the titles
related to the topic including: No. 8, Analysis of Ordinal Data, by
Hildebrand, Laing, and Rosenthal; No. 20, Log-Linear Models, by Knoke
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