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Ordered Regression Models: Parallel, Partial, and Non-Parallel Alternatives

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Authors:	Andrew S. Fullerton and Jun Xu
Publisher:	CRC Press
Copyright:	2016
ISBN-13:	978-1-4665-6973-7
Pages:	171; hardcover

Authors:	Andrew S. Fullerton and Jun Xu
Publisher:	CRC Press
Copyright:	2016
ISBN-13:
Pages:	171; eBook
Price:	$0.00

Authors:	Andrew S. Fullerton and Jun Xu
Publisher:	CRC Press
Copyright:	2016
ISBN-13:
Pages:	171; Kindle
Price:	$

Comment from the Stata technical group

In Ordered Regression Models: Parallel, Partial, and Non-Parallel Alternatives, Fullerton and Xu provide a thorough treatment of models for ordinal data. This book will appeal to researchers from any discipline who wish to build on their knowledge of linear, logistic, and probit regression and learn both theoretical and practical concepts related to a variety of models for ordinal outcomes.

As the title indicates, the models presented are partitioned into three groups based on whether a parallel regression assumption is made for all covariates, for a subset of the covariates, or for none of the covariates. Under each of these assumptions, the authors describe three models—cumulative, continuation ratio, and adjacent category—from which a researcher can choose, depending on the probability of interest. They also include worked examples with real data and provide advice regarding interpretation, presentation of results, choice of model, and common problems that arise. Example Stata commands for fitting these models are shown at the end of the chapters.

View table of contents >>

Preface

Authors

1. Introduction

1.1 Ordinal Variables versus Ordinal Models
1.2 Brief History of Binary and Ordered Regression Models
1.3 Three Approaches to Ordered Regression Models

1.3.1 Cumulative Models
1.3.2 Stage Models
1.3.3 Adjacent Models

1.4 The Parallel Regression Assumption
1.5 A Typology of Ordered Regression Models
1.6 Link Functions
1.7 Asymmetrical Relationships in Partial and Nonparallel Models
1.8 Hypothesis Testing and Model Fit in Ordered Regression Models
1.9 Datasets Used in the Empirical Examples

1.9.1 Cumulative Models of Self-Rated Health
1.9.2 Stage Models of Educational Attainment
1.9.3 Adjacent Models of Welfare Attitudes

1.10 Example: Education and Welfare Attitudes
1.11 Organization of the Book

2. Parallel Models

2.1 Parallel Cumulative Model

2.1.1 A Latent Variable Model
2.1.2 A Nonlinear Cumulative Probability Model
2.1.3 Interpreting the Results from Ordered Regression Models
2.1.4 Example: Parallel Cumulative Models of General Self-Rated Health

2.2 Parallel Continuation Ratio Model

2.2.1 A Latent Variable Model
2.2.2 A Nonlinear Conditional Probability Model
2.2.3 Potential Sample Selection Bias and Scaling Effects in Stage Models
2.2.4 Example: Parallel Continuation Ratio Models of Educational Attainment

2.3 Parallel Adjacent Category Model

2.3.1 A Nonlinear Adjacent Probability Model
2.3.2 Example: Parallel Adjacent Category Models of Welfare Spending Attitudes

2.4 Estimation

2.4.1 Basics of Maximum Likelihood Estimation
2.4.2 Common Problems in the Use of MLE for Ordered Regression Models
2.4.3 Person-Level and Person/Threshold-Level Estimation

2.5 Conclusions
2.6 Appendix

2.6.1 Equivalence of Two Parallel Complementary Log Log Models
2.6.2 Stata Codes for Parallel Ordered Logit Models
2.6.3 R Codes for Parallel Ordered Logit Models

3. Partial Models

3.1 Unconstrained versus Constrained Partial Models
3.2 Partial Cumulative Models

3.2.1 Unconstrained Partial Cumulative Model
3.2.2 Constrained Partial Cumulative Model
3.2.3 Example: Partial Cumulative Models of Self-Rated Health

3.3 Partial Continuation Ratio Models

3.3.1 Unconstrained Partial Continuation Ratio Models
3.3.2 Constrained Partial Continuation Ratio Models
3.3.3 Example: Partial Continuation Ratio Models of Educational Attainment

3.4 Partial Adjacent Category Models

3.4.1 Unconstrained Partial Adjacent Category Models
3.4.2 Constrained Partial Adjacent Category Models
3.4.3 Example: Partial Adjacent Category Models of Welfare Spending Attitudes

3.5 Dimensionality in Partial Models
3.6 Conclusions

3.6.1 Guidelines for Choosing a Partial or Parallel Ordered Regression Model

3.7 Appendix

3.7.1 Stata Codes for Partial Ordered Logit Models
3.7.2 R Codes for Partial Ordered Logit Models

4. Nonparallel Models

4.1 The Nonparallel Cumulative Model

4.1.1 Example: Nonparallel Cumulative Models of Self-Rated Health

4.2 The Nonparallel Continuation Ratio Model

4.2.1 Example: Nonparallel Continuation Ratio Models of Educational Attainment

4.3 The Nonparallel Adjacent Category Model

4.3.1 Example: Nonparallel Adjacent Category Models of Welfare Spending Attitudes

4.4 Practical Issues in the Estimation of Nonparallel Models
4.5 Conclusions

4.5.1 Guidelines for Choosing a Parallel, Partial, or Nonparallel Ordered Regression Model

4.6 Appendix

4.6.1 Stata Codes for Nonparallel Ordered Logit Models
4.6.2 R Codes for Nonparallel Ordered Logit Models

5. Testing the Parallel Regression Assumption

5.1 Wald and LR Tests
5.2 The Score Test

5.2.1 Example: Cumulative Logit Model of General Self-Rated Health

5.3 The Brant Test

5.3.1 Example: Cumulative Logit Model of General Self-Rated Health

5.4 Additional Wald and LR Tests

5.4.1 Wald and LR Tests Based on a Single Reference Model
5.4.2 Example #1: Cumulative Logit Model of General Self-Rated Health
5.4.3 Example #2: Continuation Ratio Logit Model of Educational Attainment
5.4.4 Example #3: Adjacent Category Logit Model of Welfare Attitudes
5.4.5 Iterative Wald and LR Tests

5.5 Limitations of Formal Tests of the Parallel Assumption
5.6 Model Comparisons Using the AIC and the BIC

5.6.1 Example #1: Cumulative Logit Model of General Self-Rated Health
5.6.2 Example #2: Continuation Ratio Logit Model of Educational Attainment
5.6.3 Example #3: Adjacent Category Logit Model of Welfare Attitudes

5.7 Comparing Coefficients across Cutpoint Equations
5.8 Comparing AMEs and Predicted Probabilities across Models

5.8.1 Example #1: Cumulative Logit Model of General Self-Rated Health
5.8.2 Example #2: Continuation Ratio Logit Model of Educational Attainment
5.8.3 Example #3: Adjacent Category Logit Model of Welfare Attitudes

5.9 Conclusions
5.10 Appendix

5.10.1 Comparisons among the Wald, LR, and Score Tests

6. Extensions

6.1 Heterogeneous Choice Models
6.2 Empirical Examples of Heterogeneous Choice Models
6.3 Group Comparisons Using Heterogeneous Choice Models
6.4 Introduction to Multilevel Ordered Response Regression

6.4.1 Varying Intercepts
6.4.2 Varying Slopes
6.4.3 Intercepts and Slopes as Outcomes
6.4.4 Multilevel Nonparallel Ordered Regression Models
6.4.5 Multilevel Ordered Regression Models Using Alternative Link Functions

6.5 Bayesian Analysis of Ordered Response Regression
6.6 Empirical Examples of Bayesian Ordered Regression Models

6.6.1 Bayesian Estimation of the Parallel Cumulative Model
6.6.2 Bayesian Estimation of the Nonparallel Cumulative Model
6.6.3 Bayesian Estimation of the Random Cutpoint Multilevel Stereotype Logit Model

6.7 Conclusion

References

Index

Ordered Regression Models: Parallel, Partial, and Non-Parallel Alternatives

Comment from the Stata technical group

Table of contents

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Ordered Regression Models: Parallel, Partial, and Non-Parallel Alternatives

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