Preface
Acknowledgments
About This Book
Contents
About the Author
Abbreviations
Chapter 1: Introduction
1.1 Background
1.2 Overview of Part I
1.3 Overview of Part II
1.4 Overview of Part III
References
Part I: Continuous, Count, and Dichotomous Outcomes
Chapter 2: Standard GEE Modeling of Correlated Univariate Outcomes
2.1 Correlated Univariate Outcomes
2.2 Generalized Linear Modeling
2.2.1 Linear Regression with Identity Link Function
2.2.2 Poisson Regression with Natural Log Link Function
2.2.3 Logistic Regression with Logit Link Function
2.2.4 Exponential Regression with Natural Log Link Function
2.3 Modeling Correlations
2.3.1 Independent Correlations
2.3.2 Exchangeable Correlations
2.3.3 Autoregressive Order 1 Correlations
2.3.4 Unstructured Correlations
2.4 Standard GEE Modeling
2.4.1 Estimating the Correlation Structure
2.4.2 Estimating the Covariance Matrix for Mean Parameter Estimates
2.4.3 Parameter Estimation Problems
2.5 The Likelihood Function
2.6 Likelihood Cross-Validation
2.6.1 Choosing the Number of Folds
2.6.2 LCV Ratio Tests
2.6.3 Penalized Likelihood Criteria
2.7 Adaptive Regression Modeling of Means
2.8 Example Data Sets
2.8.1 The Dental Measurement Data
2.8.2 The Epilepsy Seizure Rate Data
2.8.3 The Dichotomous Respiratory Status Data
2.8.4 The Blood Lead Level Data
References
Chapter 3: Partially Modified GEE Modeling of Correlated Univariate Outcomes
3.1 Including Non-constant Dispersions
3.2 Adding Estimating Equations for the Dispersions Based on the Likelihood
3.3 Estimating the Correlation Structure
3.4 Estimating the Covariance Matrix for Coefficient Parameter Estimates
3.5 The Constant Dispersion Model
3.6 Degeneracy in Correlation Parameter Estimation
3.7 The Estimation Process
3.7.1 Step 1 Adjustment
3.7.2 Step 2 Adjustment
3.7.3 Stopping the Estimation Process
3.7.4 Initial Estimates
3.7.5 Other Computational Issues
3.7.6 Recommended Tolerance Settings
3.8 Variation in Measurement Conditions
References
Chapter 4: Fully Modified GEE Modeling of Correlated Univariate Outcomes
4.1 Estimating Equations for Means and Dispersions Based on the Likelihood
4.2 Alternate Regression Types
4.2.1 Linear Regression with Identity Link Function
4.2.2 Poisson Regression with Natural Log Link Function
4.2.3 Logistic Regression with Logit Link Function
4.2.4 Exponential Regression with Natural Log Link Function
4.2.5 Inverse Gaussian Regression with Natural Log Link Function
4.3 The Parameter Estimation Process
4.3.1 Revised Stopping Criteria
4.3.2 Initial Estimates
4.4 Singleton Univariate Outcomes
References
Chapter 5: Extended Linear Mixed Modeling of Correlated Univariate Outcomes
5.1 Estimating Equations for Means, Dispersions, and Correlations Based on the Likelihood
5.2 Adjustments to the Estimation Process
5.3 Exchangeable Correlation Structure Computations
5.3.1 A General Class of Symmetric Matrices
5.3.2 Eigenvalues of the EC Correlation Matrix
5.3.3 Inverse of the EC Correlation Matrix
5.3.4 Square Root of the EC Correlation Matrix
5.3.5 Inverse of the Square Root of the EC Correlation Matrix
5.3.6 Derivatives with Respect to the Constant EC Correlation
5.4 Spatial Autoregressive Order 1 Correlation Structure Computations
5.4.1 Square Root and Determinant of the Spatial AR1 Correlation Matrix
5.4.2 Inverse of the Square Root of the Spatial AR1 Correlation Matrix
5.4.3 Derivatives with Respect to the Spatial Autocorrelation
5.5 Unstructured Correlation Structure Computations
5.6 Verifying Gradient and Hessian Computations
5.7 Direct Variance Modeling
References
Chapter 6: Example Analyses of the Dental Measurement Data
6.1 Choosing the Number of Folds and the Correlation Structure
6.2 Assessing Linearity of Means in Child Age
6.3 Comparison to Standard GEE Modeling
6.4 Modeling Means and Variances in Child Age
6.5 Adaptive Additive Models in Child Age and Child Gender
6.6 Adaptive Moderation of the Effect of Child Age by Child Gender
6.7 Comparison to Standard Linear Moderation
6.8 Analysis Summary
6.9 Example SAS Code for Analyzing the Dental Measurement Data
6.9.1 Modeling Means in Child Age Assuming Constant Variances
6.9.2 Modeling Means and Variances in Child Age
6.9.3 Additive Models in Child Age and Child Gender
6.9.4 Moderation Models in Child Age and Child Gender
6.9.5 Example Output
Reference
Chapter 7: Example Analyses of the Epilepsy Seizure Rate Data
7.1 Choosing the Number of Folds and the Correlation Structure
7.2 Assessing Linearity of the Log of the Means in Visit
7.3 Comparison to Standard GEE Modeling
7.4 Modeling Means and Dispersions in Visit
7.5 Additive Models in Visit and Being in the Intervention Group
7.6 Adaptive Moderation of the Effect of Visit by Being in the Intervention Group
7.7 Comparison of Linear Additive and Moderation Models with Constant Dispersions
7.8 Direct Variance Modeling of Epilepsy Seizure Rates
7.9 Analysis Summary
7.10 Example SAS Code for Analyzing the Epilepsy Seizure Rate Data
7.10.1 Modeling Means in Visit Assuming Constant Dispersions
7.10.2 Modeling Means and Dispersions in Visit
7.10.3 Additive Models in Visit and Being in the Intervention Group
7.10.4 Moderation Models in Visit and Being in the Intervention Group
7.10.5 Direct Variance Modeling
7.10.6 Example Output
Reference
Chapter 8: Example Analyses of the Dichotomous Respiratory Status Data
8.1 Choosing the Number of Folds and the Correlation Structure
8.2 Assessing Linearity of the Logits of the Means in Visit
8.3 Assessing Unit Versus Constant Dispersions
8.4 Comparison to Standard GEE Modeling
8.5 Modeling Means and Dispersions in Visit
8.6 Additive Models in Visit and Being on Active Treatment
8.7 Adaptive Moderation of the Effect of Visit by Being on Active Treatment
8.8 Comparison to Standard Linear Moderation
8.9 Direct Variance Modeling of Dichotomous Respiratory Status
8.10 Analysis Summary
8.11 Example SAS Code for Analyzing the Dichotomous Respiratory Status Data
8.11.1 Modeling Means in Visit Assuming Constant Dispersions
8.11.2 Modeling Means and Dispersions in Visit
8.11.3 Additive Models in Visit and Being on Active Treatment
8.11.4 Moderation Models in Visit and Being on Active Treatment
8.11.5 Direct Variance Modeling
8.11.6 Example Output
Reference
Chapter 9: Example Analyses of the Blood Lead Level Data
9.1 Choosing the Number of Folds and the Correlation Structure
9.2 Assessing Linearity of the Log of the Means in Week
9.3 Comparison to Standard GEE Modeling
9.4 Modeling Means and Dispersions in Week
9.5 Additive Models in Week and Being on Succimer
9.6 Adaptive Moderation of the Effect of Week by Being on Succimer
9.7 Direct Variance Modeling of Blood Lead Level Data
9.8 Analysis Summary
9.9 Example SAS Code for Analyzing the Blood Lead Level Data
9.9.1 Modeling Means in Week Assuming Constant Dispersions
9.9.2 Modeling Means and Dispersions in Week
9.9.3 Additive Models in Week and Being on Succimer
9.9.4 Moderation Models in Week and Being on Succimer
9.9.5 Direct Variance Modeling
9.9.6 Example Output
Reference
Part II: Polytomous Outcomes
Chapter 10: Multinomial Regression
10.1 Standard GEE Modeling
10.2 Partially and Fully Modified GEE Modeling
10.3 Alternate Correlation Structures
10.3.1 Independent Correlations
10.3.2 Exchangeable Correlations
10.3.3 Spatial Autoregressive Order 1 Correlations
10.3.4 Unstructured Correlations
10.3.5 Degeneracy in Correlation Estimates
10.4 Extended Linear Mixed Modeling
10.4.1 Estimating Equations for Means, Dispersions, and Correlations Based on the Likelihood
10.4.2 First Partial Derivatives with Respect to Mean Parameters
10.4.3 First Partial Derivatives with Respect to Correlation Parameters
10.4.4 Second Partial Derivatives with Respect to Mean Parameters
10.4.5 Second Partial Derivatives with Respect to Correlation Parameters
10.4.6 Second Partial Derivatives with Respect to Mean and Dispersion Parameters
10.4.7 Second Partial Derivatives with Respect to Mean and Correlation Parameters
10.4.8 Second Partial Derivatives with Respect to Dispersion and Correlation Parameters
References
Chapter 11: Ordinal Regression
11.1 Ordinal Regression Based on Individual Outcomes
11.1.1 Standard GEE Modeling
11.1.2 Partially and Fully Modified GEE Modeling
11.1.3 Alternate Correlation Structures
11.1.3.1 Independent Correlations
11.1.3.2 Exchangeable Correlations
11.1.3.3 Autoregressive Correlations
11.1.3.4 Unstructured Correlations
11.1.3.5 Degeneracy in Correlation Estimates
11.1.4 Extended Linear Mixed Modeling
11.1.4.1 Estimating Equations for Means, Dispersions, and Correlations Based on the Likelihood
11.1.4.2 First Partial Derivatives with Respect to Mean Parameters
11.1.4.3 First Partial Derivatives with Respect to Correlation Parameters
11.1.4.4 Second Partial Derivatives with Respect to Mean Parameters
11.1.4.5 Second Partial Derivatives with Respect to Correlation Parameters
11.1.4.6 Second Partial Derivatives with Respect to Mean and Dispersion Parameters
11.1.4.7 Second Partial Derivatives with Respect to Mean and Correlation Parameters
11.1.4.8 Second Partial Derivatives with Respect to Dispersion and Correlation Parameters
11.2 Ordinal Regression Based on Cumulative Outcomes
11.2.1 Standard GEE Modeling
11.2.2 Partially and Fully Modified GEE Modeling
11.2.3 Alternate Correlation Structures
11.2.4 Extended Linear Mixed Modeling
11.2.4.1 Estimating Equations for Means, Dispersions, and Correlations Based on the Likelihood
11.2.4.2 First Partial Derivatives with Respect to Mean Parameters
11.2.4.3 First Partial Derivatives with Respect to Correlation Parameters
11.2.4.4 Second Partial Derivatives with Respect to Mean Parameters
11.2.4.5 Second Partial Derivatives with Respect to Correlation Parameters
11.2.4.6 Second Partial Derivatives with Respect to Mean and Dispersion Parameters
11.2.4.7 Second Partial Derivatives with Respect to Mean and Correlation Parameters
11.2.4.8 Second Partial Derivatives with Respect to Dispersion and Correlation Parameters
References
Chapter 12: Discrete Regression
12.1 Singleton Univariate Discrete Outcomes
12.1.1 Multinomial Probabilities
12.1.2 Ordinal Probabilities
12.1.3 Censored Poisson Probabilities
12.1.4 Direct Variance Modeling
12.1.4.1 Multinomial Probabilities
12.1.4.2 Ordinal Probabilities
12.1.4.3 Censored Poisson Probabilities
12.2 Correlated Univariate Discrete Outcomes
12.2.1 Multinomial Probabilities
12.2.1.1 Standard GEE Modeling
12.2.1.2 Partially Modified GEE Modeling
12.2.1.3 Fully Modified GEE Modeling
12.2.1.4 Extended Linear Mixed Modeling
12.2.2 Ordinal Probabilities
12.2.2.1 Standard GEE Modeling
12.2.2.2 Partially Modified GEE Modeling
12.2.2.3 Fully Modified GEE Modeling
12.2.2.4 Extended Linear Mixed Modeling
12.2.3 Censored Poisson Probabilities
12.2.3.1 Standard GEE Modeling
12.2.3.2 Partially Modified GEE Modeling
12.2.3.3 Fully Modified GEE Modeling
12.2.3.4 Extended Linear Mixed Modeling
12.2.4 Direct Variance Modeling
12.2.4.1 Multinomial Probabilities
12.2.4.2 Ordinal Probabilities
12.2.4.3 Censored Poisson Probabilities
Chapter 13: Example Multinomial and Ordinal Regression Analyses
13.1 The Polytomous Respiratory Status Data
13.2 Multinomial Regression Analyses
13.2.1 Alternative Correlation Structures
13.2.2 Adaptive Modeling of Means in Visit with Constant Dispersions
13.2.3 Assessing Linearity of Generalized Logits of the Means in Visit
13.2.4 Assessing Constant Versus Unit Dispersions
13.2.5 Estimated Probabilities for Trichotomous Respiratory Status Levels
13.3 Ordinal Regression Analyses Using Individual Outcomes
13.3.1 Alternative Correlation Structures
13.3.2 Adaptive Modeling of Means in Visit with Constant Dispersions
13.3.3 Assessing Linearity of Cumulative Logits of the Means in Visit
13.3.4 Assessing Constant Versus Unit Dispersions
13.3.5 Adaptive Models for Means in Visit and Active
13.3.6 Estimated Probabilities for Trichotomous Respiratory Status Levels
13.4 Ordinal Regression Analyses Using Cumulative Outcomes
13.4.1 Alternative Correlation Structures
13.4.2 Adaptive Modeling of Means in Visit with Constant Dispersions
13.4.3 Assessing Linearity of Cumulative Logits of the Means in Visit
13.4.4 Assessing Constant Versus Unit Dispersions
13.4.5 Adaptive Models for Means in Visit and Active
13.4.6 Estimated Probabilities for Trichotomous Respiratory Status Levels
13.5 Analysis Summary
13.5.1 Multinomial Regression Analyses
13.5.2 Ordinal Regression Analyses Based on Individual Outcomes
13.5.3 Ordinal Regression Analyses Based on Cumulative Outcomes
13.5.4 All Models for Trichotomous Respiratory Status
13.5.5 Selected Model for Trichotomous Respiratory Status in Visit and Being on Active Treatment
13.6 Example SAS Code for Analyzing the Trichotomous Respiratory Status Data
13.6.1 Modeling Means in Visit Assuming Constant Dispersions
13.6.2 Modeling Means and Dispersions in Visit
13.6.3 Additive Models in Visit and Being on Active Treatment
13.6.4 Moderation Models in Visit and Being on Active Treatment
13.6.5 Example Output
References
Chapter 14: Example Discrete Regression Analyses
14.1 Multinomial Probabilities
14.1.1 Choosing the Number of Folds and the Correlation Structure
14.1.2 Assessing Linearity of the Generalized Logits for the Probabilities in Visit
14.1.3 Modeling Probabilities and Dispersions in Visit
14.1.4 Adaptive Models in Visit and Active Treatment Assuming Constant Dispersions
14.2 Ordinal Probabilities
14.2.1 Choosing the Number of Folds and the Correlation Structure
14.2.2 Assessing Linearity of the Cumulative Logits of the Probabilities in Visit
14.2.3 Comparison to Standard GEE Modeling
14.2.4 Modeling Probabilities and Dispersions in Visit
14.2.5 Adaptive Models in Visit and Active Treatment Assuming Constant Dispersions
14.2.6 Comparison to Linear Additive and Moderation Models with Constant Dispersions
14.2.7 Adaptive Models in Visit and Active Treatment for Probabilities and Dispersions
14.2.8 Direct Variance Discrete Regression Modeling of Trichotomous Respiratory Status
14.2.9 Unit Dispersion Modeling
14.3 Censored Poisson Probabilities
14.3.1 Choosing the Number of Folds and the Correlation Structure
14.3.2 Assessing Linearity in Visit
14.3.3 Comparison to Standard GEE Modeling
14.3.4 Modeling Probabilities and Dispersions in Visit
14.3.5 Adaptive Models in Visit and Active Treatment Assuming Constant Dispersions
14.3.6 Comparison to Linear Additive and Moderation Models with Constant Dispersions
14.3.7 Adaptive Models in Visit and Active Treatment for Probabilities and Dispersions
14.3.8 Direct Variance Discrete Regression Modeling of Trichotomous Respiratory Status
14.3.9 Unit Dispersion Modeling
14.3.10 Comparison to Ordinal Probability Modeling
14.4 Analysis Summary
14.4.1 Multinomial Probability Analyses
14.4.2 Ordinal Probability Analyses
14.4.3 Censored Poisson Probability Analyses
14.4.4 All Models for Trichotomous Respiratory Status
14.4.5 Selected Model for Trichotomous Respiratory Status in Visit and Being on Active Treatment
14.5 Example SAS Code for Analyzing the Trichotomous Respiratory Status Data
14.5.1 Modeling Probabilities in Visit Assuming Constant Dispersions
14.5.2 Modeling Probabilities and Dispersions in Visit
14.5.3 Additive Models in Visit and Being on Active Treatment
14.5.4 Moderation Models in Visit and Being on Active Treatment
14.5.5 Direct Variance Modeling
14.5.6 Unit Variance Modeling
14.5.7 Example Output
Reference
Part III: Adaptive Analysis Strategies
Chapter 15: Alternative Analyses
15.1 Analyzing the Dental Measurement Data
15.1.1 Alternative Correlation Structures
15.1.2 Models Based on Child Age
15.1.3 Models Based on Child Age and Child Gender
15.1.4 Clock Times for Analyses
15.2 Analyzing the Epilepsy Seizure Rate Data
15.2.1 Alternative Correlation Structures
15.2.2 Models Based on Visit
15.2.3 Models Based on Visit and Treatment Group
15.2.4 Clock Times for Analyses
15.3 Analyzing the Dichotomous Respiratory Status Data
15.3.1 Alternative Correlation Structures
15.3.2 Models Based on Visit
15.3.3 Models Based on Visit and Treatment Group
15.3.4 Clock Times for Analyses
15.4 Analyzing the Blood Lead Level Data
15.4.1 Alternative Correlation Structures
15.4.2 Models Based on Week
15.4.3 Additive Models Based on Week and Being on Succimer
15.4.4 Moderation Model Based on Week and Being on Succimer
15.4.5 Clock Times for Analyses
15.5 Analyzing the Trichotomous Respiratory Status Data Using Multinomial/Ordinal Regression
15.5.1 Alternative Correlation Structures
15.5.2 Models Based on Visit
15.5.3 Models Based on Visit and Being on Active Treatment
15.5.4 Clock Times for Analyses
15.6 Analyzing the Trichotomous Respiratory Status Data Using Discrete Regression
15.6.1 Alternative Correlation Structures
15.6.2 Models Based on Visit
15.6.3 Models Based on Visit and Treatment Group
15.6.4 Clock Times for Analyses
15.6.5 Comparison of Discrete Regression to Multinomial/Ordinal Regression
15.7 Overview of Analysis Results
15.8 Strategies for Analyzing Correlated Outcomes
15.9 Evaluation of ELMM for Theory-Based Models of Correlated Outcomes
15.10 Future Work
References
Chapter 16: Additional Example Analyses
16.1 Analyses of Data for Single Mothers on Managing a ChildΒ΄s Chronic Condition
16.1.1 The Single Mothers Data
16.1.2 Alternate Correlation Structures
16.1.3 Models Based on Scale
16.1.4 Models Based on Scale and Family Functioning
16.1.5 Clock Times for Analyses
16.2 Analyses of the Intensity of Conduct-Disordered Behaviors
16.2.1 The Partnered Parents Data
16.2.2 Alternate Numbers of Folds
16.2.3 Models Based on Parent, Family, and Family Management Types
16.2.4 Models Based on Parent, Family, and Family Management Types and on Family Functioning
16.2.5 Clock Times for Analyses
16.3 Analyses of the Number of Problematic Conduct-Disordered Behaviors
16.3.1 Alternate Numbers of Folds
16.3.2 Models Based on Parent, Family, and Family Management Types
16.3.3 Models Based on Parent, Family, and Family Management Types and on Family Functioning
16.3.4 Clock Times for Analyses
16.4 Analyses of a High Level of Intensity of Conduct-Disordered Behaviors
16.4.1 Alternate Numbers of Folds
16.4.2 Models Based on Parent, Family, and Family Management Types
16.4.3 Models Based on Parent, Family, and Family Management Types and on Family Functioning
16.4.4 Clock Times for Analyses
16.5 Analysis Summary
16.5.1 Analyses of the Single Mothers Data
16.5.2 Analyses of the Intensity of Conduct-Disordered Behaviors
16.5.3 Analyses of the Number of Problematic Conduct-Disordered Behaviors
16.5.4 Analyses of a High Level of Intensity of Conduct-Disordered Behaviors
References
Index