Table of Contents
I Introduction
1 Introduction
1.1 GEE and QLS for Analysis of Correlated Data
1.2 Why Traditional Approaches for Independent Measurements Are Not Appropriate for Analysis of Longitudinal Weight Loss Study
1.3 Attractive Features of Both QLS and GEE
1.4 When QLS Might Be Considered as an Alternative to GEE
1.5 Motivating Studies
1.5.1 Longitudinal Study of Obesity in Children Following Renal Transplant: With Binary and Continuous Measurements That Are Unequally Spaced in Time
1.5.2 Longitudinal Study of Sentence Recognition Scores That Stabilize over Time in a Hearing Recognition Study
1.5.3 Longitudinal Study for Comparison of Two Treatments for Toenail Infection
1.5.4 Multivariate Longitudinal Dataset
1.5.5 Familial Dataset
1.6 Summary
1.7 Exercises
2 Review of Generalized Linear Models
2.1 Background
2.2 Generalized Linear Models
2.2.1 Linear Model
2.2.2 Generalized Linear Model
2.2.3 Estimation of the Parameters
2.2.4 Quasi-Likelihood
2.3 Generalized Estimating Equations
2.3.1 Notation for Correlated Data
2.3.2 GEE Estimating Equation for β
2.3.3 Working Correlation Structures Available for GEE
2.3.4 The Concept of the Working versus the True Correlation Structure
2.3.5 Moment Estimates of the Dispersion and the Correlation Parameters
2.3.6 Algorithm for Estimation
2.3.7 Asymptotic Distribution of the GEE Estimators and Estimates of Covariance
2.4 Application for Obesity Study Provided in Chapter 1
2.5 Exercises
II Quasi-Least Squares Theory and Applications
3 History and Theory of Quasi-Least Squares Regression
3.1 Why QLS is a "Quasi"-Least Squares Approach
3.2 The Least Squares Approach Employed in Stage One of QLS for Estimation of α
3.2.1 Benefits of a Least Squares Approach for Estimation of α
3.2.2 QLS Stage One Estimates of α for the AR(1) Structure
3.2.3 Limiting Value of the Stage One QLS Estimator of α
3.3 Stage Two QLS Estimates of the Correlation Parameter for the AR(1) Structure
3.3.1 Elimination of the Asymptotic Bias in the Stage One QLS Estimate of α
3.4 Algorithm for QLS
3.4.1 Asymptotic Distribution of the Regression Parameter for QLS
3.5 Other Approaches Based on GEE
3.6 Example
3.7 Summary
3.8 Exercises
4 Mixed Linear Structures and Familial Data
4.1 Notation for Data from Nuclear Families
4.2 Familial Correlation Structures for Analysis of Data from Nuclear Families
4.3 Other Work on Assessment of Familial Correlations with QLS
4.4 Justification of Implementation of QLS for Familial Structures via Consideration of the Class of Mixed Linear Correlation Structures
4.4.1 Definition of Mixed Linear Correlation Structures
4.4.2 Results for General Correlation Structures (for Stage One of QLS) and for Linear Correlation Structures (for Stage Two of QLS)
4.4.2.1 Results for Stage One
4.4.2.2 Results for Stage Two
4.5 Demonstration of QLS for Analysis of Balanced Familial Data Using Stata Software
4.6 Demonstration of QLS for Analysis of Unbalanced Familial Data Using R Software
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
4.8 Summary and Future Research Directions
4.9 Exercises
5 Correlation Structures for Clustered and Longitudinal Data
5.1 Characteristics of Clustered and Longitudinal Data
5.2 The Exchangeable Correlation Structure for Clustered Data
5.2.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.2.2 Demonstration of Implementation of the Exchangeable Structure for QLS
5.3 The Tri-Diagonal Correlation Structure
5.3.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.3.2 Demonstration of Implementation of the Tri-Diagonal Structure for QLS
5.4 The AR(1) Structure for Analysis of (Planned) Equally Spaced Longitudinal Data
5.4.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.4.2 Demonstration of Implementation of the AR(1) Structure for QLS
5.5 The Markov Structure for Analysis of Unequally Spaced Longitudinal Data
5.5.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.5.2 Demonstration of Implementation of the Markov Structure for QLS
5.5.3 Generalized Markov Structure
5.6 The Unstructured Matrix for Analysis of Balanced Data
5.6.1 Obtaining a Solution to the Stage One Estimating Equation for the Unstructured Matrix
5.6.2 Obtaining a Solution to the Stage Two Estimating Equation for the Unstructured Matrix
5.6.3 Demonstration of Implementation of the Unstructured Matrix for QLS
5.7 Other Structures
5.8 Implementation of QLS for Patterned Correlation Structures
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
5.8.2 When Software for GEE Is Not Available, or Is Not Utilized
5.9 Summary
5.10 Exercises
5.11 Appendix
6 Analysis of Data with Multiple Sources of Correlation
6.1 Characteristics of Data with Multiple Sources of Correlation
6.2 Multi-Source Correlated Data That Are Totally Balanced
6.2.1 Example of Multivariate Longitudinal Data That Are Totally Balanced
6.2.2 Notation
6.2.3 Working Correlation Structure for Balanced Data
6.2.4 Prior Implementation of the Kronecker Product Structure
6.2.5 Implementation of QLS for Analysis
6.3 Multi-Source Correlated Data That Are Balanced within Clusters
6.3.1 Example
6.3.2 Notation
6.3.3 Correlation Structure for Data That Are Balanced within Clusters
6.3.4 Algorithm for Implementation of QLS for Multi-Source Correlated Data That Are Balanced within Clusters
6.3.5 Implementation of QLS for Analysis
6.4 Multi-Source Correlated Data That Are Unbalanced
6.4.1 Example
6.4.2 Notation
6.4.3 Correlation Structure for Data That Are Unbalanced
6.4.4 Algorithm for Implementation of QLS for Multi-Source Correlated Data That Are Unbalanced
6.4.5 Implementation of QLS for Analysis
6.5 Asymptotic Relative Efficiency Calculations
6.6 Summary
6.7 Exercises
6.8 Appendix: The Limiting Value of the QLS Estimate of the Association Parameter When the True Correlation Structure Is Misspecified as Exchangeable
7 Correlated Binary Data
7.0.1 Notation for Correlated Binary Data
7.1 Additional Constraints for Binary Data
7.1.1 Negative Estimated Bivariate Probabilities for the Toenail Data
7.1.2 Prentice Constraints to Ensure Valid Induced Bivariate Distributions
7.1.3 Simplification of the Prentice Constraints for Decaying Product Correlation Structures
7.1.4 Conditions to Ensure the Existence of an Underlying Multivariate Distribution
7.2 When Violation Is Likely to Occur
7.2.1 When the Model Is Correctly Specified
7.2.2 When the Working Structure Is Incorrectly Specified
7.2.3 When the Model for the Mean Is Incorrect
7.2.4 When the Assumption of Missing Completely at Random Is Violated
7.3 Implications of Violation of Constraints for Binary Data
7.4 Comparison among GEE, QLS, and MARK1ML
7.4.1 Comparisons with ALR
7.5 Prentice-Corrected QLS and GEE
7.6 Summary
7.7 Exercises
8 Assessing Goodness of Fit and Choice of Correlation Structure for QLS and GEE
8.1 Simulation Scenarios
8.2 Simulation Results
8.2.1 True AR(1) Structure
8.2.2 True Markov Structure
8.2.3 True Decaying Product Structure
8.3 Summary and Recommendations
8.4 Exercises
9 Sample Size and Demonstration
9.1 Two-Group Comparisons
9.1.1 Two-Group Comparisons
9.1.1.1 Time-Averaged Comparison of Group Means
9.1.1.2 Time-Averaged Comparison of Proportions
9.1.1.3 Comparison of Change over Time for Continuous Outcomes
9.1.1.4 Comparison of Change over Time for Binary Outcomes
9.2 More Complex Situations
9.3 Worked Example
9.3.1 Sample Size for a Future Study
9.4 Discussion and Summary
9.5 Exercises
Bibliography
Index
