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QuasiLeast Squares Regression 

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Comment from the Stata technical groupQuasiLeast Squares Regression, by Justine Shults and Joseph M. Hilbe, is a great resource for graduate students and researchers interested in estimating populationaveraged effects from longitudinal and clustered data. The book provides an introduction to quasileastsquares (QLS) estimators and provides many examples using the authorwritten Stata commands. QLS estimators extend GEE estimators to incorporate a wider set of correlation structures. The QLS estimators also allow researchers to determine the correlation structure that best fits their data. 

Table of contentsView table of contents >> 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
1.6 Summary 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 1.7 Exercises 2 Review of Generalized Linear Models
2.1 Background
2.2 Generalized Linear Models
2.2.1 Linear Model
2.3 Generalized Estimating Equations 2.2.2 Generalized Linear Model 2.2.3 Estimation of the Parameters 2.2.4 QuasiLikelihood
2.3.1 Notation for Correlated Data
2.4 Application for Obesity Study Provided in Chapter 1 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 2.5 Exercises II QuasiLeast Squares Theory and Applications
3 History and Theory of QuasiLeast 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
3.3 Stage Two QLS Estimates of the Correlation Parameter for the AR(1) Structure 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
3.3.1 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
4.5 Demonstration of QLS for Analysis of Balanced Familial Data Using Stata Software 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 4.6 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
5.3 The TriDiagonal Correlation Structure 5.2.2 Demonstration of Implementation of the Exchangeable Structure for QLS
5.3.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.4 The AR(1) Structure for Analysis of (Planned) Equally Spaced Longitudinal Data 5.3.2 Demonstration of Implementation of the TriDiagonal Structure for QLS
5.4.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.5 The Markov Structure for Analysis of Unequally Spaced Longitudinal Data 5.4.2 Demonstration of Implementation of the AR(1) Structure for QLS
5.5.1 Solutions to the QLS Stage One and Stage Two Estimating Equations for α
5.6 The Unstructured Matrix for Analysis of Balanced Data 5.5.2 Demonstration of Implementation of the Markov Structure for QLS 5.5.3 Generalized Markov Structure
5.6.1 Obtaining a Solution to the Stage One Estimating Equation for the Unstructured Matrix
5.7 Other Structures 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 5.8 Implementation of QLS for Patterned Correlation Structures
5.8.1 Algorithm for Implementation of QLS Using Software That Allows for Application of a UserSpecified Working Correlation Structure That Is Treated as Fixed and Known in the GEE Estimating Equation for β
5.9 Summary 5.8.2 When Software for GEE Is Not Available, or Is Not Utilized 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 MultiSource Correlated Data That Are Totally Balanced
6.2.1 Example of Multivariate Longitudinal Data That Are Totally Balanced
6.3 MultiSource Correlated Data That Are Balanced within Clusters 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
6.3.1 Example
6.4 MultiSource Correlated Data That Are Unbalanced 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 MultiSource Correlated Data That Are Balanced within Clusters 6.3.5 Implementation of QLS for Analysis
6.4.1 Example
6.5 Asymptotic Relative Efficiency Calculations 6.4.2 Notation 6.4.3 Correlation Structure for Data That Are Unbalanced 6.4.4 Algorithm for Implementation of QLS for MultiSource Correlated Data That Are Unbalanced 6.4.5 Implementation of QLS for Analysis 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
7.2 When Violation Is Likely to Occur 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
7.2.1 When the Model Is Correctly Specified
7.3 Implications of Violation of Constraints for Binary Data 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 7.4 Comparison among GEE, QLS, and MARK1ML
7.4.1 Comparisons with ALR
7.5 PrenticeCorrected 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
8.3 Summary and Recommendations 8.2.2 True Markov Structure 8.2.3 True Decaying Product Structure 8.4 Exercises 9 Sample Size and Demonstration
9.1 TwoGroup Comparisons
9.1.1 TwoGroup Comparisons
9.2 More Complex Situations
9.1.1.1 TimeAveraged Comparison of Group Means
9.1.1.2 TimeAveraged 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 9.3 Worked Example
9.3.1 Sample Size for a Future Study
9.4 Discussion and Summary 9.5 Exercises Bibliography
Index

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