Quasi-Least Squares Regression: 1st Edition (Hardback) book cover

Quasi-Least Squares Regression

1st Edition

By Justine Shults, Joseph M. Hilbe

Chapman and Hall/CRC

221 pages | 8 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9781420099935
pub: 2014-01-28
SAVE ~$20.00
eBook (VitalSource) : 9780429132278
pub: 2014-01-28
from $48.98

FREE Standard Shipping!


Drawing on the authors’ substantial expertise in modeling longitudinal and clustered data, Quasi-Least Squares Regression provides a thorough treatment of quasi-least squares (QLS) regression—a computational approach for the estimation of correlation parameters within the framework of generalized estimating equations (GEEs). The authors present a detailed evaluation of QLS methodology, demonstrating the advantages of QLS in comparison with alternative methods. They describe how QLS can be used to extend the application of the traditional GEE approach to the analysis of unequally spaced longitudinal data, familial data, and data with multiple sources of correlation. In some settings, QLS also allows for improved analysis with an unstructured correlation matrix.

Special focus is given to goodness-of-fit analysis as well as new strategies for selecting the appropriate working correlation structure for QLS and GEE. A chapter on longitudinal binary data tackles recent issues raised in the statistical literature regarding the appropriateness of semi-parametric methods, such as GEE and QLS, for the analysis of binary data; this chapter includes a comparison with the first-order Markov maximum-likelihood (MARK1ML) approach for binary data.

Examples throughout the book demonstrate each topic of discussion. In particular, a fully worked out example leads readers from model building and interpretation to the planning stages for a future study (including sample size calculations). The code provided enables readers to replicate many of the examples in Stata, often with corresponding R, SAS, or MATLAB® code offered in the text or on the book’s website.


"The book does an excellent job of explaining basic concepts and techniques in the analysis of longitudinal and correlated data using QLS and GEE. Well-chosen data examples almost follow all the technical explanations, providing the readers a flavor on what problems QLS solves and how to solve those problems using software. Although the authors mainly use Stata to demonstrate the examples, they also provide web access to R, SAS, and MATLAB code and guidelines to replicate those examples, making the book appealing to a wide audience. The book also successfully incorporates some recent research work without raising its technical level. Therefore, the book will serve as a comprehensible guide to researchers who conduct analysis on correlated data. It would also be a good textbook for graduate students in statistics or biostatistics. Finally, I believe it would be a popular desk reference for methodology-oriented researchers who are interested in longitudinal studies and related fields."

Journal of the American Statistical Association, March 2015

"This book deals with the quasi-least squares (QLS) regression, presenting a computational approach for the estimation of correlation parameters in the context of the generalized estimating equations (GEEs). … The book is provided with illustrative examples for each topic."

Zentralblatt MATH 1306

Table of Contents



When QLS Might Be Considered as an Alternative to GEE

Motivating Studies


Review of Generalized Linear Models


Generalized Linear Models

Generalized Estimating Equations

Application for Obesity Study Provided in Chapter One

Quasi-Least Squares Theory and Applications

History and Theory of QLS Regression

Why QLS Is a "Quasi" Least Squares Approach

The Least-Squares Approach Employed in Stage One of QLS for Estimation of α

Stage-Two QLS Estimates of the Correlation Parameter for the AR(1) Structure

Algorithm for QLS

Other Approaches That Are Based on GEE



Mixed Linear Structures and Familial Data

Notation for Data from Nuclear Families

Familial Correlation Structures for Analysis of Data from Nuclear Families

Other Work on Assessment of Familial Correlations with QLS

Justification of Implementation of QLS for Familial Structures via Consideration of the Class of Mixed Linear Correlation Structures

Demonstration of QLS for Analysis of Balanced Familial Data Using Stata Software

Demonstration of QLS for Analysis of Unbalanced Familial Data Using R Software

Simulations to Compare Implementation of QLS with Correct Specification of the Trio Structure versus Correct Specification with GEE and Incorrect Specification of the Exchangeable Working

Structure with GEE

Summary and Future Research Directions

Correlation Structures for Clustered and Longitudinal Data

Characteristics of Clustered and Longitudinal Data

The Exchangeable Correlation Structure for Clustered Data

The Tri-Diagonal Correlation Structure

The AR(1) Structure for Analysis of (Planned) Equally Spaced Longitudinal Data

The Markov Structure for Analysis of Unequally Spaced Longitudinal Data

The Unstructured Matrix for Analysis of Balanced Data

Other Structures

Implementation of QLS for Patterned Correlation Structures



Analysis of Data with Multiple Sources of Correlation

Characteristics of Data with Multiple Sources of Correlation

Multi-Source Correlated Data That Are Totally Balanced

Multi-Source Correlated Data That Are Balanced within Clusters

Multi-Source Correlated Data That Are Unbalanced

Asymptotic Relative Efficiency Calculations



Correlated Binary Data

Additional Constraints for Binary Data

When Violation of the Prentice Constraints for Binary Data Is Likely to Occur

Implications of Violation of Constraints for Binary Data

Comparison between GEE, QLS, and MARK1ML

Prentice-Corrected QLS and GEE


Assessing Goodness of Fit and Choice of Correlation Structure for QLS and GEE

Simulation Scenarios

Simulation Results

Summary and Recommendations

Sample Size and Demonstration

Two-Group Comparisons

More Complex Situations

Worked Example

Discussion and Summary



Exercises appear at the end of each chapter.

About the Authors

Justine Shults is an associate professor and co-director of the Pediatrics Section in the Division of Biostatistics in the Perelman School of Medicine at the University of Pennsylvania, where she is the principal investigator of the biostatistics training grant in renal and urologic diseases. She is the Statistical Editor of the Journal of the Pediatric Infectious Disease Society and the Statistical Section Editor of Springer Plus. Professor Shults (with N. Rao Chaganty) developed Quasi-Least Squares (QLS) and was funded by the National Science Foundation and the National Institutes of Health to extend QLSand develop user-friendly software for implementing her new methodology. She has authored or co-authored over 100 peer-reviewed publications, including the initial papers on QLS for unbalanced and unequally spaced longitudinal data and on MARK1ML and the choice of working correlation structure for GEE.

Joseph M. Hilbe is a Solar System Ambassador with the Jet Propulsion Laboratory, an adjunct professor of statistics at Arizona State University, and an Emeritus Professor at the University of Hawaii. An elected fellow of the American Statistical Association and an elected member of the International Statistical Institute (ISI), Professor Hilbe is president of the International Astrostatistics Association as well as chair of the ISI Sports Statistics and Astrostatistics committees. He has authored two editions of the bestseller Negative Binomial Regression, Logistic Regression Models, and Astrostatistical Challenges for the New Astronomy. He also co-authored Methods of Statistical Model Estimation (with A. Robinson), Generalized Estimating Equations, Second Edition (with J. Hardin), and R for Stata Users (with R. Muenchen), as well as 17 encyclopedia articles and book chapters in the past five years.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General