Multilevel Analysis
By
Zach Andersen
Jon Durrant
Jayson Talakai
OUTLINE
Jon – What is Multilevel Regression
Jayson – The Model
Zach – R code applications / examples
WHAT IS MULTILEVEL REGRESSION
Regression models at multiple levels, because of
dependencies in nested data
Not two stage, this occurs all at once
EXAMPLES
•Students in schools
•Individuals by area
•Employees in organizations
•Firms in various industries
•Repeated observations on a person
https://www.youtube.com/watch?v=wom6uPdI-P4
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)
https://www.youtube.com/watch?v=wom6uPdI-P4
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
http://www.princeton.edu/~otorres/Multilevel101.pdf
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
https://www.youtube.com/watch?v=wom6uPdI-P4
Multilevel Models
https://www.youtube.com/watch?v=wrTiCfgGdro
PROBLEMS CAUSED BY CORRELATION
•Imprecise parameter estimates
•Incorrect standard errors
A SIMPLE 2-LEVEL HIERARCHY
School 1
Student 1
Student 2
School 2
Student 3
https://www.youtube.com/watch?v=wom6uPdI-P4
Student 1
Student 2
Student 3
A SIMPLE 2-LEVEL HIERARCHY
School 1
Student 1
Student 2
Level 2
Student 3
Student 1
Level 1
https://www.youtube.com/watch?v=wom6uPdI-P4
School 2
Student 2
Student 3
PEOPLE ARE AT LEVEL 1??
The first level of a hierarchy is not necessarily a
person
https://www.youtube.com/watch?v=wom6uPdI-P4
A SIMPLE 2-LEVEL HIERARCHY
Level 2
Industry 1
Firm 1
Firm 2
Firm 3
Firm 1
Level 1
https://www.youtube.com/watch?v=wom6uPdI-P4
Industry 2
Firm 2
Firm 3
A SIMPLE 2-LEVEL HIERARCHY
Person 1
Event 1
Event 2
Level 2
Event 3
Event 1
Level 1
https://www.youtube.com/watch?v=wom6uPdI-P4
Person 2
Event 2
Event 3
BRIEF HISTORY
•Problems of single level analysis, cross level
inferences and ecological fallacy
https://www.youtube.com/watch?v=wom6uPdI-P4
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
https://www.youtube.com/watch?v=wom6uPdI-P4
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)
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
http://en.wikipedia.org/wiki/Intraclass_correlation
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
FIXED VS RANDOM EFFECTS
**Anytime that you see the word “population” substitute it with the word
“processes.”
http://www2.sas.com/proceedings/forum2008/374-2008.pdf
INTRODUCING THE MODEL
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.
http://en.wikipedia.org/wiki/Multilevel_model
Types of Models: Random Slopes Model
• Slopes are different across groups.
• This model assumes that intercepts
are fixed (the same across different
contexts).
http://en.wikipedia.org/wiki/Multilevel_model
http://www.strath.ac.uk/aer/materials/5furtherquantitativeresearchdesignanda
nalysis/unit4/randomslopemodelling/
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.
http://en.wikipedia.org/wiki/Multilevel_model
Assumptions for Multilevel Models
Modification of assumptions
Linearity and normality assumptions are retained
Homoscedasticity and independence of observations need to be
adjusted.
1. Observations within a group are more similar to observations in
different groups.
2. Groups are independent from other groups, but observations within
a group are not.
http://en.wikipedia.org/wiki/Multilevel_model
Multilevel Model: Example
http://faculty.smu.edu/kyler/training/AERA_overheads.pdf
Multilevel Model: Level 1 Regression Equation
http://faculty.smu.edu/kyler/training/AERA_overheads.pdf
Multilevel Model continued:
http://faculty.smu.edu/kyler/training/AERA_overheads.pdf
Multilevel Model continued:
http://faculty.smu.edu/kyler/training/AERA_overheads.pdf
Multilevel Model continued:
http://faculty.smu.edu/kyler/training/AERA_overheads.pdf
Adding a Random Sample Component
http://faculty.smu.edu/kyler/training/AERA_overheads.pdf
EXAMPLES IN R
Example of group effects without Multilevel
modeling
Example of the Covariance Theorem
Example of Random Intercept Model