Modeling Correlated Outcomes Using Extensions of Generalized Estimating Equations and Linear Mixed Modeling / by George J. Knafl.

Format:

Book

Author/Creator:

Knafl, George J., author.

Language:

English

Subjects (All):

Statistics.

Biometry.

Statistical Theory and Methods.

Biostatistics.

Local Subjects:

Statistical Theory and Methods.

Biostatistics.

Physical Description:

1 online resource (525 pages)

Edition:

1st ed. 2023.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2023.

Summary:

This book formulates methods for modeling continuous and categorical correlated outcomes that extend the commonly used methods: generalized estimating equations (GEE) and linear mixed modeling. Partially modified GEE adds estimating equations for variance/dispersion parameters to the standard GEE estimating equations for the mean parameters. Fully modified GEE provides alternate estimating equations for mean parameters as well as estimating equations for variance/dispersion parameters. The new estimating equations in these two cases are generated by maximizing a "likelihood" function related to the multivariate normal density function. Partially modified GEE and fully modified GEE use the standard GEE approach to estimate correlation parameters based on the residuals. Extended linear mixed modeling (ELMM) uses the likelihood function to estimate not only mean and variance/dispersion parameters, but also correlation parameters. Formulations are provided for gradient vectors and Hessianmatrices, for a multi-step algorithm for solving estimating equations, and model-based and robust empirical tests for assessing theory-based models. Standard GEE, partially modified GEE, fully modified GEE, and ELMM are demonstrated and compared using a variety of regression analyses of different types of correlated outcomes. Example analyses of correlated outcomes include linear regression for continuous outcomes, Poisson regression for count/rate outcomes, logistic regression for dichotomous outcomes, exponential regression for positive-valued continuous outcome, multinomial regression for general polytomous outcomes, ordinal regression for ordinal polytomous outcomes, and discrete regression for discrete numeric outcomes. These analyses also address nonlinearity in predictors based on adaptive search through alternative fractional polynomial models controlled by likelihood cross-validation (LCV) scores. Larger LCV scores indicate better models but not necessarilydistinctly better models. LCV ratio tests are used to identify distinctly better models. A SAS macro has been developed for analyzing correlated outcomes using standard GEE, partially modified GEE, fully modified GEE, and ELMM within alternative regression contexts. This macro and code for conducting the analyses addressed in the book are available online via the book’s Springer website. Detailed descriptions of how to use this macro and interpret its output are provided in the book.

Contents:

Intro

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

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

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

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

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

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

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

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.

Notes:

Includes bibliographical references and index.

ISBN:

9783031419881

303141988X