1. Introduction Multilevel Analysis
Rens van de Schoot
/ rensvandeschoot.wordpress.com

2. Multilevel Regression Model
Known in literature under a variety of names
Hierarchical linear model (HLM)
Random coefficient model
Variance component model
Multilevel model
Contextual analysis
Mixed Linear Model

3. Hierarchical Data Structure
Three level data structure
Groups at different levels may have different sizes
Response (outcome) variable at lowest level
Explanatory variables at all levels

4. Examples?

5. Traditional Approaches
Disaggregate all variables to the lowest level
Do standard analyses (anova, multiple regression)
Aggregate all variables to the highest level
Ancova with groups as factor
Some improvements:
explanatory variables as deviations from their group mean have both deviation score and disaggregated group mean as predictor (separates individual and group effects)
Why not? What is wrong with this?

6. Problems With Standard Analysis of Hierarchical Data
Multiple Regression assumes
independent observations
independent error terms
equal variances of errors for all observations
(assumption of homoscedastic errors)
normal distribution for errors
With hierarchical data
observations are not independent
errors are not independent
different observations may have errors with different variances (heteroscedastic errors)

7. Problems With Standard Analysis of Hierarchical Data
Observations in the same group are generally not independent
they tend to be more similar than observations from different groups
selection, shared history, contextual group effects
The degree of similarity is indicated by the intraclass correlation rho: r
Standard statistical tests are not at all robust against violation of the independence assumption
That is why we need special multilevel techniques!

8. Sample size?
Hox, J., van de Schoot. R., & Matthijsse, S. (2012). How few countries will do? Comparative survey analysis from a Bayesian perspective. Survey Research Methods, Vol.6, No.2, pp

9. Research questions I/III
Questions with respect to variables at the lowest level
Intelligence (IQ) as predictor of school achievement (SA)

10. Research questions II/III
Questions with respect to the influence of variables at a higher level on the dependent variable on the lowest level
Mean intelligence of a class (MIQ) as predictor of school achievement (SA); (control for individual IQ)

11. Research questions III/III
Questions with respect to the interaction of variables on different levels (moderation effect)
The relation between intelligence and school achievement is not the same in all classes

12. Graphical Picture of Simple Two-level Regression Model
School level
Pupil level
Outcome variable on pupil level
Explanatory variables at both levels: individual & group
Residual error at individual level
Plus residual error at school level

13. Regression analysis
In ordinary regression, with one explanatory variable X:
Yi= b0+ b1Xi+ ei
b0 intercept,
b1 regression slope,
ei residual error term

14. Regression analysis

15. Building the Multilevel Regression Model: Random intercept model
In multilevel regression, at the lowest level:
Yij= b0j+ b1jXij+ eij
b0j intercept,
b1j regression slope,
eij residual error term
subscript i for individuals, j for groups
each group has its own intercept coefficient b0j
and its own slope coefficient b1j

16. Building the Multilevel Regression Model: Intercept only model
In multilevel regression, at the lowest level:
Yij= b0j+ eij
Random intercept model:
b0j= g00+ u0j
g00 is the intercept of b0j u0j is the residual error term in the equation for b0j

17. Building the Multilevel Regression Model: Random intercept model
In multilevel regression, at the lowest level:
Yij= b0j+ b1jXij+ eij
Random intercept model:
b0j= g00+ u0j
g00 is the intercept of b0j u0j is the residual error term in the equation for b0j

18. Building the Multilevel Regression Model: Random intercept model

19. Building the Multilevel Regression Model: Intercept only model
Yij= b0j+ b1jXij+ eij
Random intercept model:
b0j= g00+ u0j
g00 is the intercept of b0j u0j is the residual error term in the equation for b0j
Random slope model:
b1j= g10+ u1j
g10 is the intercept of ß1j
u1j is the residual error term in the equation for b1j

20. Difference with the usual regression model:
Each class has a different intercept coefficient b0j and a different slope coefficient b1j
Since the intercept and the slope coefficients vary across the classes: random coefficients
=> Random intercept model & random slope model

21. Building the Multilevel Regression Model: Random slope model

22. Building the Multilevel Regression Model: the Second (Group) Level
Next step:
explain the variation of the regression coefficients b0j and b1j by introducing explanatory variables at the class level

23. Building the Multilevel Regression Model: the Second (Group) Level
At the lowest (individual) level we have
Yij= b0j+ b1jXij+ eij
b0j= g00+ g01Zj+ u0j
g00 and g01 are the intercept and slope to predict b0j from Zj
u0j is the residual error term in the equation for b0j

24. Building the Multilevel Regression Model: Cross level interaction
At the lowest (individual) level we have
Yij= b0j+ b1jXij+ eij
b0j= g00+ g01Zj+ u0j
g00 and g01 are the intercept and slope to predict b0j from Zj
u0j is the residual error term in the equation for b0j
b1j= g10+ g11Zj+ u1j
g10 and g11 are the intercept and slope to predict ß1j from Zj
u1j is the residual error term in the equation for b1j

25. Building the Multilevel Regression Model: Single Equation Version
At the lowest (individual) level we have
Yij= b0j+ b1jXij+ eij
and at the second (group) level
b0j= g00+ g01Zj+ u0j
b1j= g10+ g11Zj+ u1j
Combining (substitution and rearranging terms) gives
Yij= g00+ g10Xij+ g01Zj+ g11ZjXij+ u1jXij+ u0j+ eij

26. Building the Multilevel Regression Model: Single Equation Version
Yij= [g00+ g10Xij+ g01Zj+ g11ZjXij] + [u1jXij+ u0j+ eij]
This equation has two distinct parts
[g00+ g10Xij+ g01Zj+ g11ZjXij] contains all the fixed coefficients, it is called the fixed part of the model
[u1jXij+ u0j+ eij] contains all the random error terms, it is called the random part of the model

27. Building the Multilevel Regression Model: Interpretation
Yij = [g00+ g10Xij+ g01Zj+ g11ZjXij] + [u1jXij+ u0j+ eij]
Several error variances
e2 variance of the lowest level errors eij
s2u0 variance of the highest level errors u0j
s2u1 variance of the highest level errors u1j
su01 covariance of u0j and u1j

28. Full Multilevel Regression Model
Explanatory variables at all levels
Higher level variables predict variation of lowest level intercept and slopes
Predicting the intercept implies a direct effect
Predicting slopes implies cross-level interactions

29. Model Exploration 1 Intercept-only model
calculate intraclass correlation
2 Fixed model, 1st level predictor variables
test individual slopes for significance
3 Model intercept by 2nd level predictor variables
test for significance, how much intercept variance explained?
4 Random coefficient model
test if any 1st level slope has a significant variance component (this is best done one-by-one)
5 Model random slopes by higher level variables: cross level interactions
test for significance, how much slope variance is explained?
Note: sometimes it is interesting to calculate the intraclass correlation for all variables!

30. Example: Popularity in Schools
Outcome: popularity rating
100 classes, 2000 pupils
Explanatory variables
Pupil level: sex (0=boy, 1=girl)
Class level: teacher experience (in years)

31. Graphical Picture of Simple Two-level Regression Model

32. Popularity Example: Intercept-only Model
Popularityij = g00+ u0j+ eij
Estimates (st. err.)
g00 = 5.31 (.10) (This is just the overall average popularity)
se2 = 0.64 (.02)
s2u0 = 0.88 (.13)

33. Popularity Example: Fixed Model
Popularityij = g00 + g10sexij + u0j + eij
Estimates (st. err.)
g00 = 4.89 (.10),
g10 = 0.84 (.03)
se2 = 0.46 (.02)
s2u0 = 0.85 (.12)

34. Popularity Example: Fixed Model + Higher Level Variable
Popularityij = g00 + g10 sexij + g01 t.exp.j + u0j + eij
Estimates (st. err.)
g00 = 3.56 (.17),
g10 = 0.84 (.03),
g01 = 0.09 (.01)
se2 = 0.46 (.02)
s2u0 = 0.48 (.07)

35. Popularity Example: Random Coefficient Model
Popularityij =
g00 + g10 sexij + g01 t.exp.j + u0j + u1j sexij + eij
Estimates (st. err.)
g00 = 3.34 (.16), g10 = 0.84 (.06), g01 = 0.11 (.01)
se2 = 0.39 (.01)
s2u0 = 0.41 (.06)
su01 = 0.02 (.04) (covariance between intercept and slope)
s2u1 = 0.27 (.05)
Slope variation for sex

36. Popularity Example: Random Coefficient Model + Interaction
Popularityij = g00 + g10 sexij + g01 t.exp.j
+ g11 sexij t.exp.j + u0j + u1j sexij + eij
Estimates (st. err.)
g00 = 3.31 (.16), g10 = 1.33 (.13), g01 = 0.11 (.01),
g11 = (.01)
se2 = 0.39 (.01)
s2u0 = 0.40 (.06)
su01 = 0.02 (.04)
s2u1 = 0.22 (.04)
Smaller, but still significant slope variation for sex

37. 5-day course Multilevel Analyses in Mplus 21-25 jan. 2013
The 9th International  Multilevel Conference is on March (2013).
Prior to the conference (26th of March) a one-day course is taught by prof. Stef van Buuren on Mutiple Imputation of Multilevel missing data in MICE.
5th Mplus users meeting will be organized, 25th of March