1. Multilevel Modelling Dr Andrew Bell,
Lecturer in Quantitative Social Sciences

2. A basic linear regression model
Y = B0 + B1*X + e Y e X

3. What’s the problem?
Assume that the residuals (e) are independent from each other.
Ie that the model has accounted for everything systematic
only white noise remains
This is often simply not realistic

4. Multilevel structures
“once you know hierarchies exist, you see them everywhere” (Kreft and De Leeuw, 1998 p1)
Hairs on heads
Students nested in schools
Voters nested in electoral wards
Cows nested in farms
Occasions nested in individuals (panel data)
Occasions nested in students nested in schools

5. Multilevel structures
“once you know hierarchies exist, you see them everywhere” (Kreft and De Leeuw, 1998 p1)
Non-hierarchical structures:
Students in both schools and neighbourhoods
Students in more than one classroom
F1 race results nested in drivers and teams

6. A basic linear regression model
Y = B0 + B1*X + e Y e
But this is real life, points don’t fall on lines. So now the line is our prediction, and Y includes a ‘residual’ X

7. A basic linear regression model
Y = B0 + B1*X + e Y e
But this is real life, points don’t fall on lines. So now the line is our prediction, and Y includes a ‘residual’ X

8. A basic multilevel model
Y = B0 + B1*X + u + e Y u
Separate intercepts by including an extra, higher level residual term e X

9. What are we doing? Modelling complex structures
Modelling heterogeneity
Modelling dependency
Modelling context
Separate intercepts by including an extra, higher level residual term

10. Why use MLMs? Toy example
Sometimes:
single level
models can be
seriously
misleading!
Separate intercepts by including an extra, higher level residual term

11. An example: ‘contextual value added’ school league tables

12. age16attainment = 0.002 + 0.563*age11attainment
What the DoE’s CVA model would look like in MLwiN. Basic line – being 1 pt better at age 11 means your score will be better at age 16. But we are more interested in 0.92 – which is the variance between schools. This allows us to plot the following……
age16attainment = *age11attainment

13. Blue line is a good school, black line is a bad school
Blue line is a good school, black line is a bad school. But not that simple. Multilevel model can be extended to allow slopes to vary…….

14. What is a good school now
What is a good school now? Blue school good for clever kids but bad for bad kids – they’d be better off going to the grey school.
Use as a league table? Becomes too complex for most parents? Also all parents think their kids are clever kids.
But allows complex questions to be answered that couldn’t before – great for social scientists.
Schools make more of a difference to clever kids. Depressingly, if you come into any school at age 11 you may struggle to break into a good school

15. Another example Reinhart and Rogoff – Growth in a time of debt (2010)
“median growth rates for countries with public debt over roughly 90 percent of GDP are several percent lower.”
Conclusions used by Paul Ryan, David Cameron, etc, to justify austerity policies

16. Another example Problems with it (1) Arbitrary exclusion of countries
A weird weighting system
An excel spreadsheet error that excluded a number of observations alphabetically
Herndon, Ash and Pollin (2013) Does high public debt consistently stifle economic growth? A critique of Reinhart and Rogoff. Cambridge Journal of Economics, online.

18. Another example Problems with it (2)
Assumes a single consistent effect, rather than allowing that effect to vary between countries.
But why should the effect of debt on growth be the same in USA and Britain?
Instead, can use multilevel models to allow for differences across countries
Country
Year

19. Another example (and back to 2 levels)
Bell, Jones and Johnston (2015) Stylised fact or situated messiness? The diverse effects of increasing national debt on economic growth. Journal of Economic Geography, 15(2),
Australia
Japan
New Zealand UK US -4 -2 2 4 6 8 70
140
210
Predicted Growth (%GDP)
Debt:GDP ratio
Another example (and back to 2 levels)

20. Variance Functions
Key aim of multilevel models: model heterogeneity – the variance can vary!
As well as estimating separate effects for higher level entities, we can see how much variance varies
Eg schools matter more (are more varied) for clever students

21. Final example: F1 racing
Team
Driver
Team-Year
Observation

22. Final example: F1 racing

23. Final example: F1 racing

24. Overall Why multilevel models are good:
Technically sound: Make realistic assumptions about independent residuals; explicitly model heterogeneity and dependency (rather than ‘correct’ for it)
Substantively interesting: Answer questions you cannot answer with standard modelling techniques
Widely applicable: education, epidemiology, veterinary science, political/electoral science, economics, biology, geography

25. Useful resource CMM LEMMA training courses
Online courses from multilple regression (module 3), up to a range of different MLM structures, outcomes, etc
Practical exercises using MLwiN, Stata, R, SPSS
Free, you just need to register

26. Shameless self promotion
Bell, Andrew, and Kelvyn Jones “Explaining Fixed Effects: Random Effects Modelling of Time-Series Cross-Sectional and Panel Data.” Political Science Research and Methods 3(1): 133–53.
Comparison of MLM (or ‘Random Effects’) to Fixed effects modelling; and argues for the benefits of the former
Bell, Andrew, Ron Johnston, and Kelvyn Jones “Stylised Fact or Situated Messiness? The Diverse Effects of Increasing Debt on National Economic Growth.” Journal of Economic Geography 15(2): 449–72.
Bell, Andrew, James Smith, Clive Sabel, Kelvyn Jones Formula for success: Multilevel modelling of Formula One driver and constructor performance, Journal of Quantitative Analysis in Sports 12(2):