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Multilevel Analysis for Applied Research
It's Just Regression!

Robert Bickel

355 Pages
Size: 7" x 10"
Paperback
March 2007
ISBN 978-1-59385-191-0
Cat. #5191
Price: $52.00 $44.20
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Hardcover
March 2007
ISBN 978-1-59385-429-4
Cat. #5429
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1. Broadening the Scope of Regression Analysis

1.1.Chapter Introduction

1.2. Why Use Multilevel Regression Analysis?

1.3. Limitations of Available Instructional Material

1.4. Multilevel Regression Analysis in Suggestive Historical Context

1.5. It’s Just Regression under Specific Circumstances

1.6. Jumping the Gun to a Multilevel Illustration

1.7. Summing Up

1.8. Useful Resources

2. The Meaning of Nesting

2.1. Chapter Introduction

2.2. Nesting Illustrated: School Achievement and Neighborhood Quality

2.3. Nesting Illustrated: Comparing Public and Private Schools

2.4. Cautionary Comment on Residuals in Multilevel Analysis

2.5. Nesting and Correlated Residuals

2.6. Nesting and Effective Sample Size

2.7. Summing Up

2.8. Useful Resources

3. Contextual Variables

3.1. Chapter Introduction

3.2. Contextual Variables and Analytical Opportunities

3.3. Contextual Variables and Independent Observations

3.4. Contextual Variables and Independent Observations: A Nine-Category Dummy Variable

3.5. Contextual Variables, Intraclass Correlation, and Misspecification

3.6. Contextual Variables and Varying Parameter Estimates

3.7. Contextual Variables and Covariance Structure

3.8. Contextual Variables and Degrees of Freedom

3.9. Summing Up

3.10. Useful Resources

4. From OLS to Random Coefficient to Multilevel Regression

4.1. Chapter Introduction

4.2. Simple Regression Equation

4.3. Simple Regression with an Individual-Level Variable

4.4. Multiple Regression: Adding a Contextual Variable

4.5. Nesting (Again!) with a Contextual Variable

4.6. Is There a Problem with Degrees of Freedom?

4.7. Is There a Problem with Dependent Observations?

4.8. Alternatives to OLS Estimatorspt; FONT-FAMILY: Arial; mso-bidi-font-weight: bold"4.9. The Conceptual Basis of ML Estimators

4.10. Desirable Properties of REML Estimators

4.11. Applying REML Estimators with Random Coefficient Regression Models

4.12. Fixed Components and Random Components

4.13. Interpreting Random Coefficients: Developing a Cautionary Comment

4.14. Subscript Conventions

4.15. Percentage of Variance Explained for Random Coefficient and Multilevel Models

4.16. Grand-Mean Centering

4.17. Grand-Mean Centering, Group-Mean Centering, and Raw Scores Compared

4.18. Summing Up

4.19. Useful Resources

5. Developing the Multilevel Regression Model

5.1. Chapter Introduction

5.2. From Random Coefficient Regression to Multilevel Regression

5.3. Equations for a Random Intercept and Random Slope

5.4. Subscript Conventions for Two-Level Models: Gamma Coefficients

5.5. The Full Equation

5.6. An Implied Cross-Level Interaction Term

5.7. Estimating a Multilevel Model: The Full Equation

5.8. A Multilevel Model with a Random Slope and Fixed Slopes at Level One

5.9. Complexity and Confusion: Too Many Random Components

5.10. Interpreting Multilevel Regression Equations

5.11. Comparing Interpretations of Alternative Specifications

5.12. What Happened to the Error Term?

5.13. Summing Up

5.14. Useful Resources

6. Giving OLS Regression Its Due

6.1. Chapter Introduction

6.2. An Extended Exercise with County-Level Data

6.3. Tentative Specification of an OLS Regression Model

6.4. Preliminary Regression Results

6.5. Surprise Results and Possible Violation of OLS Assumptions

6.6. Curvilinear Relationships: YBUSH by XBLACK, XHISPANIC, XNATIVE

6.7. Quadratic Functional Form

6.8. A Respecified OLS Regression Model

6.9. Interpreting Quadratic Relationships

6.10. Nonadditivity and Interaction Terms

6.11. Further Respecification of the Regression Model

6.12. Clarifying OLS Interaction Effects

6.13. Results for the Respecified OLS Regression Equation for County-Level Data

6.14. Summing Up

6.15. Useful Resources

7. Does Multilevel Regression Have Anything to Contribute?

7.1. Chapter Introduction

7.2. Contextual Effects in OLS Regression

7.3. Respecification and Changing Functional Form

7.4. Addressing the Limitations of OLS

7.5. Counties Nested within States: Intraclass Correlation

7.6. Multilevel Regression Model Specification: Learning from OLS

7.7. Interpreting the Multilevel Regression Equation for County-Level Data

7.8. Knowing When to Stop

7.9. Summing Up

7.10. Useful Resources

8. Multilevel Regression Models with Three Levels

8.1. Chapter Introduction

8.2. Students Nested within Schools and within Districts

8.3. Level One: Students

8.4. Level Two: Schools

8.5. Level Three: Districts

8.6. Notation and Subscript Conventions for Specifying a Three-Level Model

8.7. Estimating a Three-Level Random Coefficient Model

8.8. Adding a Second Level-One Predictor

8.9. Adding a Level-Two Predictor

8.10. Adding a Second Predictor at Level Two and a Predictor at Level Three

8.11. Discretionary Use of Same-Level Interaction Terms

8.12. Ongoing Respecification of a Three-Level Model

8.13. A Level-Two Random Slope at Level Three

8.14. Summing Up

8.15. Useful Resources

9. Familiar Measures Applied to Three-Level Models

9.1. Chapter Introduction

9.2. The Intraclass Correlation Coefficient Revisited

9.3. Percentage of Variance Explained in a Level-One Dependent Variable

9.4. Other Summary Measures Used with Multilevel Regression

9.5. Summing Up

9.6. Useful Resources

10. Determining Sample Sizes for Multilevel Regression

10.1. Chapter Introduction

10.2. Interest in Sample Size in OLS and Multiple Regression

10.3. Sample Size: Rules of Thumb and Data Constraints

10.4. Estimation and Inference for Unstandardized Regression Coefficients

10.5. More Than One Level of Analysis Means More Than One Sample Size

10.6. An Individual-Level OLS Analysis with a Large Sample

10.7. A Group-Level OLS Analysis with a Small Sample

10.8. Standard Errors: Corrected and Uncorrected, Individual and Group Levels

10.9. When Output Is Not Forthcoming!

10.10. Sample Sizes and OLS-Based Commonsense in Multilevel Regression

10.11. Sample Size Generalizations Peculiar to Multilevel Regression

10.12. Level-One Sample Size and Level-Two Statistical Power

10.13. The Importance of Sample Size at Higher Levels

10.14. Summing Up

10.15. Useful Resources

11. Multilevel Regression Growth Models

11.1. Chapter Introduction

11.2. Analyzing Longitudinal Data: Pretest–Posttest

11.3. Nested Measures: Growth in Student Vocabulary Achievement

11.4. Nested Measures: Growth in NCLEX Pass Rates

11.5. Developing Multilevel Regression Growth Models

11.6. Summary Statistics with Growth Models

11.7. Sample Sizes

11.8. The Multilevel Regression Growth Model Respecified

11.9. The Multilevel Regression Growth Model: Further Respecification

11.10. Residual Covariance Structures

11.11. Multilevel Regression Growth Models with Three Levels

11.12. Nonlinear Growth Curves

11.13. NCLEX Pass Rates with a Time-Dependent Predictor

11.14. Summing Up

11.15. Useful Resources

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