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Quasi-Least Squares Regression [Kõva köide]

(University of Pennsylvania, Philadelphia, USA), (California Institute of Technology, Pasadena, and Arizona State University, Tempe, USA)
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Quasi-Least Squares RegressionSuurem pilt
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Püsilink: https://www.kriso.ee/db/9781420099935.html
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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) regressiona 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 books website.

Arvustused

"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

Preface xiii
I Introduction 1(40)
1 Introduction
3(14)
1.1 GEE and QLS for Analysis of Correlated Data
3(1)
1.2 Why Traditional Approaches for Independent Measurements Are Not Appropriate for Analysis of Longitudinal Weight Loss Study
4(1)
1.3 Attractive Features of Both QLS and GEE
5(2)
1.4 When QLS Might Be Considered as an Alternative to GEE
7(1)
1.5 Motivating Studies
8(7)
1.5.1 Longitudinal Study of Obesity in Children Following Renal Transplant: With Binary and Continuous Measurements That Are Unequally Spaced in Time
8(1)
1.5.2 Longitudinal Study of Sentence Recognition Scores That Stabilize over Time in a Hearing Recognition Study
9(2)
1.5.3 Longitudinal Study for Comparison of Two Treatments for Toenail Infection
11(1)
1.5.4 Multivariate Longitudinal Dataset
12(1)
1.5.5 Familial Dataset
13(2)
1.6 Summary
15(1)
1.7 Exercises
15(2)
2 Review of Generalized Linear Models
17(24)
2.1 Background
17(1)
2.2 Generalized Linear Models
18(4)
2.2.1 Linear Model
18(1)
2.2.2 Generalized Linear Model
18(2)
2.2.3 Estimation of the Parameters
20(1)
2.2.4 Quasi-Likelihood
21(1)
2.3 Generalized Estimating Equations
22(10)
2.3.1 Notation for Correlated Data
22(1)
2.3.2 GEE Estimating Equation for β
22(1)
2.3.3 Working Correlation Structures Available for GEE
23(3)
2.3.4 The Concept of the Working versus the True Correlation Structure
26(1)
2.3.5 Moment Estimates of the Dispersion and the Correlation Parameters
26(2)
2.3.6 Algorithm for Estimation
28(1)
2.3.7 Asymptotic Distribution of the GEE Estimators and Estimates of Covariance
29(3)
2.4 Application for Obesity Study Provided in
Chapter 1
32(7)
2.5 Exercises
39(2)
II Quasi-Least Squares Theory and Applications 41(150)
3 History and Theory of Quasi-Least Squares Regression
43(22)
3.1 Why QLS is a "Quasi"-Least Squares Approach
44(3)
3.2 The Least Squares Approach Employed in Stage One of QLS for Estimation of α
47(7)
3.2.1 Benefits of a Least Squares Approach for Estimation of α
48(3)
3.2.2 QLS Stage One Estimates of α for the AR(1) Structure
51(2)
3.2.3 Limiting Value of the Stage One QLS Estimator of α
53(1)
3.3 Stage Two QLS Estimates of the Correlation Parameter for the AR(1) Structure
54(3)
3.3.1 Elimination of the Asymptotic Bias in the Stage One QLS Estimate of α
54(3)
3.4 Algorithm for QLS
57(2)
3.4.1 Asymptotic Distribution of the Regression Parameter for QLS
59(1)
3.5 Other Approaches Based on GEE
59(1)
3.6 Example
60(2)
3.7 Summary
62(1)
3.8 Exercises
63(2)
4 Mixed Linear Structures and Familial Data
65(18)
4.1 Notation for Data from Nuclear Families
65(1)
4.2 Familial Correlation Structures for Analysis of Data from Nuclear Families
66(3)
4.3 Other Work on Assessment of Familial Correlations with QLS
69(1)
4.4 Justification of Implementation of QLS for Familial Structures via Consideration of the Class of Mixed Linear Correlation Structures
70(3)
4.4.1 Definition of Mixed Linear Correlation Structures
70(1)
4.4.2 Results for General Correlation Structures (for Stage One of QLS) and for Linear Correlation Structures (for Stage Two of QLS)
71(14)
4.4.2.1 Results for Stage One
71(1)
4.4.2.2 Results for Stage Two
72(1)
4.5 Demonstration of QLS for Analysis of Balanced Familial Data Using Stata Software
73(3)
4.6 Demonstration of QLS for Analysis of Unbalanced Familial Data Using R Software
76(1)
4.7 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
77(2)
4.8 Summary and Future Research Directions
79(1)
4.9 Exercises
80(3)
5 Correlation Structures for Clustered and Longitudinal Data
83(30)
5.1 Characteristics of Clustered and Longitudinal Data
84(1)
5.2 The Exchangeable Correlation Structure for Clustered Data
85(4)
5.2.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
85(2)
5.2.2 Demonstration of Implementation of the Exchangeable Structure for QLS
87(2)
5.3 The Tri-Diagonal Correlation Structure
89(2)
5.3.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
89(1)
5.3.2 Demonstration of Implementation of the Tri-Diagonal Structure for QLS
90(1)
5.4 The AR(1) Structure for Analysis of (Planned) Equally Spaced Longitudinal Data
91(3)
5.4.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
91(1)
5.4.2 Demonstration of Implementation of the AR(1) Structure for QLS
92(2)
5.5 The Markov Structure for Analysis of Unequally Spaced Longitudinal Data
94(4)
5.5.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
94(2)
5.5.2 Demonstration of Implementation of the Markov Structure for QLS
96(1)
5.5.3 Generalized Markov Structure
97(1)
5.6 The Unstructured Matrix for Analysis of Balanced Data
98(8)
5.6.1 Obtaining a Solution to the Stage One Estimating Equation for the Unstructured Matrix
99(2)
5.6.2 Obtaining a Solution to the Stage Two Estimating Equation for the Unstructured Matrix
101(1)
5.6.3 Demonstration of Implementation of the Unstructured Matrix for QLS
102(4)
5.7 Other Structures
106(1)
5.8 Implementation of QLS for Patterned Correlation Structures
107(2)
5.8.1 Algorithm for Implementation of QLS Using Software That Allows for Application of a User-Specified Working Correlation Structure That Is Treated as Fixed and Known in the GEE Estimating Equation for β
107(1)
5.8.2 When Software for GEE Is Not Available, or Is Not Utilized
108(1)
5.9 Summary
109(1)
5.10 Exercises
109(1)
5.11 Appendix
110(3)
6 Analysis of Data with Multiple Sources of Correlation
113(28)
6.1 Characteristics of Data with Multiple Sources of Correlation
113(1)
6.2 Multi-Source Correlated Data That Are Totally Balanced
113(10)
6.2.1 Example of Multivariate Longitudinal Data That Are Totally Balanced
113(1)
6.2.2 Notation
114(1)
6.2.3 Working Correlation Structure for Balanced Data
115(2)
6.2.4 Prior Implementation of the Kronecker Product Structure
117(1)
6.2.5 Implementation of QLS for Analysis
118(5)
6.3 Multi-Source Correlated Data That Are Balanced within Clusters
123(6)
6.3.1 Example
123(1)
6.3.2 Notation
123(1)
6.3.3 Correlation Structure for Data That Are Balanced within Clusters
124(1)
6.3.4 Algorithm for Implementation of QLS for Multi-Source Correlated Data That Are Balanced within Clusters
124(2)
6.3.5 Implementation of QLS for Analysis
126(3)
6.4 Multi-Source Correlated Data That Are Unbalanced
129(5)
6.4.1 Example
129(1)
6.4.2 Notation
130(1)
6.4.3 Correlation Structure for Data That Are Unbalanced
130(1)
6.4.4 Algorithm for Implementation of QLS for Multi-Source Correlated Data That Are Unbalanced
131(2)
6.4.5 Implementation of QLS for Analysis
133(1)
6.5 Asymptotic Relative Efficiency Calculations
134(2)
6.6 Summary
136(2)
6.7 Exercises
138(1)
6.8 Appendix: The Limiting Value of the QLS Estimate of the Association Parameter When the True Correlation Structure Is Misspecified as Exchangeable
139(2)
7 Correlated Binary Data
141(20)
7.0.1 Notation for Correlated Binary Data
142(1)
7.1 Additional Constraints for Binary Data
142(7)
7.1.1 Negative Estimated Bivariate Probabilities for the Toenail Data
143(1)
7.1.2 Prentice Constraints to Ensure Valid Induced Bivariate Distributions
144(2)
7.1.3 Simplification of the Prentice Constraints for Decaying Product Correlation Structures
146(3)
7.1.4 Conditions to Ensure the Existence of an Underlying Multivariate Distribution
149(1)
7.2 When Violation Is Likely to Occur
149(5)
7.2.1 When the Model Is Correctly Specified
150(1)
7.2.2 When the Working Structure Is Incorrectly Specified
150(4)
7.2.3 When the Model for the Mean Is Incorrect
154(1)
7.2.4 When the Assumption of Missing Completely at Random Is Violated
154(1)
7.3 Implications of Violation of Constraints for Binary Data
154(1)
7.4 Comparison among GEE, QLS, and MARK1ML
155(2)
7.4.1 Comparisons with ALR
156(1)
7.5 Prentice-Corrected QLS and GEE
157(2)
7.6 Summary
159(1)
7.7 Exercises
160(1)
8 Assessing Goodness of Fit and Choice of Correlation Structure for QLS and GEE
161(14)
8.1 Simulation Scenarios
166(2)
8.2 Simulation Results
168(3)
8.2.1 True AR(1) Structure
168(1)
8.2.2 True Markov Structure
168(1)
8.2.3 True Decaying Product Structure
169(2)
8.3 Summary and Recommendations
171(1)
8.4 Exercises
172(3)
9 Sample Size and Demonstration
175(16)
9.1 Two-Group Comparisons
177(5)
9.1.1 Two-Group Comparisons
177(6)
9.1.1.1 Time-Averaged Comparison of Group Means
177(3)
9.1.1.2 Time-Averaged Comparison of Proportions
180(1)
9.1.1.3 Comparison of Change over Time for Continuous Outcomes
181(1)
9.1.1.4 Comparison of Change over Time for Binary Outcomes
182(1)
9.2 More Complex Situations
182(1)
9.3 Worked Example
183(5)
9.3.1 Sample Size for a Future Study
187(1)
9.4 Discussion and Summary
188(2)
9.5 Exercises
190(1)
Bibliography 191(10)
Index 201
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 QLS and 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.