Introduction to Multilevel Modelling
Welcome
About the Authors
Funding
1
Introduction
1.1
Overview
1.2
Goals
1.3
Prerequisites
1.4
Materials
2
Multiple Regression Review
2.1
Learning Objectives
2.2
Data Demonstration
2.2.1
Creating R Projects
2.2.2
Loading Data and Dependencies
2.2.3
Simple Linear Regression
2.2.4
Multiple Regression
2.2.5
Interaction Terms
2.3
Conclusion
3
Approaches to Multilevel Data
3.1
Learning Objectives
3.2
Data Demonstration
3.2.1
Load Data and Dependencies
3.2.2
Dealing with Dependence
3.2.3
Cluster-Robust Standard Errors
3.3
Conclusion
3.4
Further Reading
4
Our First Multilevel Models
4.1
Learning Objectives
4.2
Data Demonstration
4.2.1
Load Data and Dependencies
4.2.2
Why Multilevel Models?
4.2.3
Fixed vs Random Effects
4.2.4
The Null Model
4.2.5
Understanding Variance
4.3
Conclusion
5
Adding Fixed Predictors to MLMs
5.1
Learning Objectives
5.2
Data Demonstration
5.2.1
Load Data and Dependencies
5.2.2
MLM with Level-1 Predictor
5.2.3
Compare Regular and Multilevel Regression
5.2.4
MLM with Level-2 Predictor
5.3
Conclusion
6
Random Effects and Cross-level Interactions
6.1
Learning Objectives
6.2
Data Demonstration
6.2.1
Load Data and Dependencies
6.2.2
MLM with Random Slope Effect
6.2.3
MLM with Crosslevel Effect
6.3
Conclusion
7
Model Estimation Options, Problems, and Troubleshooting
7.1
Learning Objectives
7.2
Data Demonstration
7.2.1
Load Data and Dependencies
7.2.2
Introduction to Estimation Problems
7.2.3
Estimation and Optimizers
7.2.4
Non-Convergence
7.2.5
Singularity
7.2.6
Deviance Testing for Model Comparison
7.3
Conclusion
7.4
Further Reading
8
Centering Options and Interpretations
8.1
Learning Objectives
8.2
Data Demonstration
8.2.1
Load Data and Dependencies
8.2.2
Why Center Variables?
8.2.3
Within, Between, and Contextual Effects
8.2.4
Options for Centering in MLMs
8.2.5
What Kind of Centering Should You Use?
8.3
Conclusion
8.4
Further Reading
9
Multilevel Modelling with Repeated Measures Data
9.1
Learning Objectives
9.2
Data Demonstration
9.2.1
Load Dependencies
9.2.2
Review of Multilevel Modelling Procedure
9.2.3
Multilevel Models for Repeated Measures
9.2.4
Our Data: Reaction Time
9.2.5
Random-Intercept-Only/Null Model
9.2.6
Adding Level-1 Fixed Effects
9.2.7
Adding Random Slopes
9.2.8
Adding Level-2 Fixed Effects
9.2.9
Adding Cross-Level Interactions
9.3
Conclusion
10
Multilevel Modelling with Longitudinal Data
10.1
Learning Objectives
10.2
Data Demonstration
10.2.1
Load Dependencies
10.2.2
Multilevel Models for Longitudinal Data
10.2.3
Visualizing Testosterone Levels Over Time
10.2.4
Random-Intercept-Only/Null Model
10.2.5
Adding Level-1 Fixed and Random Effects
10.2.6
Evidence for Retaining Effects
10.2.7
Adding Level-2 Fixed Effects
10.3
Conclusion
10.4
Further Reading
11
Effect Sizes in Multilevel Models
11.1
Learning Objectives
11.2
Data Demonstration
11.2.1
Load Data and Dependencies
11.2.2
Defining Effect Sizes
11.2.3
R-squared in Multilevel Models
11.2.4
Single Model, Automatic Entry
11.2.5
Single Model, Manual Entry
11.2.6
Model Comparison
11.3
Conclusion
11.4
Additional Reading
12
Assumptions
12.1
Learning Objectives
12.2
Data Demonstration
12.2.1
Load Data and Dependencies
12.2.2
Assumptions of MLMs
12.2.3
Assumption 1: Model Specification
12.2.4
Assumption 2: Functional Form is Correct
12.2.5
An Aside: Extracting Residuals
12.2.6
Assumption 3: Level-1 Residuals are Independent and Normally Distributed
12.2.7
Assumption 4: Level-2 Residuals are Independent and Multivariate Normal
12.2.8
Assumption 5: Residuals at Level-1 and Level-2 are Independent
12.2.9
Assumption 6: Level-1 Residuals Independent of Level-2 Predictors, Level-2 Residuals Independent of Level-1 Predictors
12.3
Conclusion
12.4
Further Reading
Appendix
A
Download Materials
Mairead Shaw
Jessica Kay Flake
Introduction to Multilevel Modelling
A
Download Materials
Chapter
Data
R Script
Worksheet
2
heck2011.csv
02-multiple-regression.R
Linear Regression Review
3