Modeling Correlated Outcomes Using Extensions of Generalized Estimating Equations and Linear Mixed Modeling

This book formulates methods for modeling continuous and categorical correlated outcomes extending the commonly used methods: generalized estimating equations (GEE) and linear mixed modeling. Partially modified GEE adds estimating equations for variance/dispersion parameters to the standard GEE estimating equations for the mean parameters. Fully modified GEE uses alternate estimating equations for the mean parameters. The new estimating equations in these two cases are generated by maximizing a "likelihood" function related to the multivariate normal density function. Partially modified GEE and fully modified GEE use the standard GEE approach to estimate correlation parameters based on the residuals. Extended linear mixed modeling (ELMM) uses the likelihood function to estimate not only mean and variance/dispersion parameters, but also correlation parameters. Formulations are provided for gradient vectors and Hessian matrices, for a multi-step algorithm for solving estimating equations, and model-based and robust empirical tests for assessing theory-based models. Directly specified correlation structures are considered as well as covariance structures based on random effect/coefficients.
 
Standard GEE, partially modified GEE, fully modified GEE, and ELMM are demonstrated and compared using a variety of regression analyses of different types of correlated outcomes. Example analyses of correlated outcomes include linear regression for continuous outcomes, Poisson regression for count/rate outcomes, logistic regression for dichotomous outcomes, exponential regression for positive-valued continuous outcome, multinomial regression for general polytomous outcomes, ordinal regression for ordinal polytomous outcomes, and discrete regression for discrete numeric outcomes. These analyses also address nonlinearity in predictors based on adaptive search through alternative fractional polynomial models controlled by likelihood cross-validation (LCV) scores. Larger LCV scores indicate better models but not necessarily distinctly better models. LCV ratio tests are used to identify distinctly better models.
 
A SAS® macro has been developed for analyzing correlated outcomes using standard GEE, partially modified GEE, fully modified GEE, and ELMM within alternative regression contexts. This macro and code for conducting the analyses addressed in the book are available as supplementary materials upon request from the author. Detailed descriptions of how to use this macro and interpret its output are provided in the book.

List of contents

Introduction.- Part I Continuous, Count, and Dichotomous Outcomes.- 2. Standard GEE Modeling of Correlated Univariate Outcomes.- 3. Partially Modified GEE Modeling of Correlated Univariate Outcomes.- 4. Fully Modified GEE Modeling of Correlated Univariate Outcomes.- 5. Extended Linear Mixed Modeling of Correlated Univariate Outcomes.- 6. Example Analyses of the Dental Measurement Data.- 7. Example Analyses of the Epilepsy Seizure Rate Data.- 8. Example Analyses of the Dichotomous Respiratory Status Data.- 9. Example Analyses of the Blood Lead Level Data.- Part II Polytomous Outcomes.- 10. Multinomial Regression.- 11. Ordinal Regression.- 12. Discrete Regression.- 13. Example Multinomial and Ordinal Regression Analyses.- 14. Example Discrete Regression Analyses.- Part III Adaptive Analysis Strategies.- 15. Alternative Analyses.- 16. Additional Example Analyses.

About the author

George J. Knafl is Biostatistician and Professor Emeritus in the School of Nursing of the University of North Carolina at Chapel Hill where he taught statistics courses for doctoral nursing students, consulted with doctoral students and faculty on their research, and conducted his own research. He has over 45
years of experience in teaching, consulting, and research in statistics. He has continued to conduct research involving development of methods for searching through alternative models for different types of statistical data and application of those methods to the analysis of a variety of health science data sets. He is also Professor Emeritus in the College of Computing and Digital Media at DePaul University and has served on the faculties of the Schools of Nursing at Yale University and at the Oregon Health and Science University.

Summary

This book formulates methods for modeling continuous and categorical correlated outcomes extending the commonly used methods: generalized estimating equations (GEE) and linear mixed modeling. Partially modified GEE adds estimating equations for variance/dispersion parameters to the standard GEE estimating equations for the mean parameters. Fully modified GEE uses alternate estimating equations for the mean parameters. The new estimating equations in these two cases are generated by maximizing a "likelihood" function related to the multivariate normal density function. Partially modified GEE and fully modified GEE use the standard GEE approach to estimate correlation parameters based on the residuals. Extended linear mixed modeling (ELMM) uses the likelihood function to estimate not only mean and variance/dispersion parameters, but also correlation parameters. Formulations are provided for gradient vectors and Hessian matrices, for a multi-step algorithm for solving estimating equations, and model-based and robust empirical tests for assessing theory-based models. Directly specified correlation structures are considered as well as covariance structures based on random effect/coefficients.
 
Standard GEE, partially modified GEE, fully modified GEE, and ELMM are demonstrated and compared using a variety of regression analyses of different types of correlated outcomes. Example analyses of correlated outcomes include linear regression for continuous outcomes, Poisson regression for count/rate outcomes, logistic regression for dichotomous outcomes, exponential regression for positive-valued continuous outcome, multinomial regression for general polytomous outcomes, ordinal regression for ordinal polytomous outcomes, and discrete regression for discrete numeric outcomes. These analyses also address nonlinearity in predictors based on adaptive search through alternative fractional polynomial models controlled by likelihood cross-validation (LCV) scores. Larger LCV scores indicate better models but not necessarily distinctly better models. LCV ratio tests are used to identify distinctly better models.
 
A SAS® macro has been developed for analyzing correlated outcomes using standard GEE, partially modified GEE, fully modified GEE, and ELMM within alternative regression contexts. This macro and code for conducting the analyses addressed in the book are available as supplementary materials upon request from the author. Detailed descriptions of how to use this macro and interpret its output are provided in the book.

Product details

• Authors: George J Knafl, George J. Knafl
• Publisher: Springer International Publishing
• Languages: English
• Product format: Hardback
• Release: 09.10.2025
• EAN: 9783032009883
• ISBN: 978-3-032-00988-3
• Illustrations: Approx. 565 p. With online files/update.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; Moderation, REGRESSION, Biostatistics, Linear Models and Regression, Parametric Inference, Nonlinearity, Newton's method, Extended Linear Mixed Modeling, Adaptive Modeling, Generalized Estimating Equations, Likelihood Cross-Validation, Correlated Outcomes, Random Effects/Coefficients