1. By Zach Andersen Jon Durrant Jayson Talakai
Multilevel Analysis By
Zach Andersen
Jon Durrant
Jayson Talakai

2. OUTLINE
Jon – What is Multilevel Regression Jayson – The Model Zach – R code applications / examples

3. WHAT IS MULTILEVEL REGRESSION
Regression models at multiple levels, because of dependencies in nested data Not two stage, this occurs all at once

4. Employees in organizations Firms in various industries
EXAMPLES
Students in schools
Individuals by area
Employees in organizations
Firms in various industries
Repeated observations on a person

5. WHEN TO USE A MULTILEVEL MODEL?
Individual units (often people), with group indicators (e.g. Schools, area).
Dependent variable (level 1)
More than one person per group
Generally we need at least 5 groups, preferably more. (Ugly rule of thumb)

6. WHEN TO USE A MULTILEVEL MODEL?
Use a multilevel model whenever your data is grouped (or nested) into categories (or clusters)
Allows for the study of effects that vary by group
Regular regression ignores the average variation between groups and may lack the ability to generalize

7. DATA STRUCTURE AND DEPENDENCE
Independence makes sense sometimes and keeps statistical theory relatively simple.
Eg; standard error(sample average) = s/n requires that the n observations are independent
But data often have structure, and observations have things in common; same area, same school, repeated observations on the same person
Observations usually cannot be regarded as independent

8. Multilevel Models

9. PROBLEMS CAUSED BY CORRELATION
Imprecise parameter estimates
Incorrect standard errors

10. A SIMPLE 2-LEVEL HIERARCHY
School 1
School 2
Student 1
Student 2
Student 3
Student 1
Student 2
Student 3

11. A SIMPLE 2-LEVEL HIERARCHY
School 1
School 2
Level 2
Student 1
Student 2
Student 3
Student 1
Student 2
Student 3
Level 1

12. The first level of a hierarchy is not necessarily a person
PEOPLE ARE AT LEVEL 1??
The first level of a hierarchy is not necessarily a person

13. A SIMPLE 2-LEVEL HIERARCHY
Industry 1
Industry 2
Level 2
Firm 1
Firm 2
Firm 3
Firm 1
Firm 2
Firm 3
Level 1

14. A SIMPLE 2-LEVEL HIERARCHY
Person 1
Person 2
Level 2
Event 1
Event 2
Event 3
Event 1
Event 2
Event 3
Level 1

15. BRIEF HISTORY
Problems of single level analysis, cross level inferences and ecological fallacy

16. DISCUSSION AS TO WHY A NORMAL REGRESSION CAN BE A POOR MODEL
Because Reality might not conform to the assumptions of linear regression (Independence)
Because in nature observation tend to cluster
A random person in Lubbock is more likely to be a student then a random person in another city (clustering of populations/not independent)
Different clusters react differently

17. Also longitudinal, geographical studies
EXTENSIONS
Focus was initially on hierarchical structures and especially students in schools
Also longitudinal, geographical studies
More recently moved to non hierarchical situations such as cross-classified models. (single level is part of more than one group)

18. INTRACLASS CORRELATION
Level 1 variance explained by the group (level 2)
ICC is the proportion of group-level variance to the total variance
Formula for ICC:
Variance in group
Overall variance

20. Random or Fixed Effects
MULTILEVEL MODELING
Random or Fixed Effects
What are random and fixed effects?
When should you use random and fixed effects?
Types of random effects models
The Model
Assumptions of the model
Building a multilevel model

21. Fixed vs random effects
**Anytime that you see the word “population” substitute it with the word “processes.”

22. INTRODUCING THE MODEL

23. Types of Models: Random Intercepts Model
Intercepts are allowed to vary:
The scores on the dependent variable for each individual observation are predicted by the intercept that varies across groups. 

24. Types of Models: Random Slopes Model
Slopes are different across groups.
This model assumes that intercepts are fixed (the same across different contexts). 

25. Types of Models: Random intercepts and slopes model
Includes both random intercepts and random slopes
Is likely the most realistic type of model, although it is also the most complex.

26. Assumptions for Multilevel Models
Modification of assumptions
Linearity and normality assumptions are retained
Homoscedasticity and independence of observations need to be adjusted.
Observations within a group are more similar to observations in different groups.
Groups are independent from other groups, but observations within a group are not.

27. Multilevel Model: Example

28. Multilevel Model: Level 1 Regression Equation

29. Multilevel Model continued:

30. Multilevel Model continued:

31. Multilevel Model continued:

32. Adding a Random Sample Component

34. EXAMPLES IN R
Example of group effects without Multilevel modeling Example of the Covariance Theorem Example of Random Intercept Model