1. What is multilevel modelling?
Kelvyn Jones, School of Geographical Sciences, LEMMA, University of Bristol
2nd Oxford Research Methods Festival
July 2006

2. MULTILEVEL MODELS AKA random-effects models, hierarchical models,
variance-components models,
random-coefficient models,
mixed models

3. Two-level hierarchical model
Micro model
Macro models
Combined multilevel model
Level 2 variance
Level 1 variance

4. Three KEY Notions Modelling contextuality: firms as contexts
eg discrimination varies from firm to firm
eg discrimination varies differentially for employees of different ages from firm to firm
Modelling heterogeneity
standard regression models ‘averages’, ie the general relationship
ML models variances
Eg between-firm AND between-employee, within-firm variation
Modelling data with complex structure
- series of structures that ML can handle routinely

5. Structures: UNIT DIAGRAMS
1: Hierarchical structures
a) Pupils nested within schools: modelling progress
NB imbalance More examples follow…...

6. Examples of strict hierarchy
Education
pupils (1) in schools (2)
pupils (1) in classes( 2) in schools (3)
Surveys: 3 stage sampling
respondents (1) in neighbourhoods(2) in regions(3)
Business
individuals(1) within teams(2) within organizations(3)
Psychology
individuals(1) within family(2)
individuals(1) within twin sibling pair(2)
Economics
employees(1) within firms(2)
NB all are structures in the POPULATION (ie exist in reality)

7. 1: Multi-stage samples as hierarchies
Two-level structure imposed by design
Respondents nested within PSU’s
Usually generates dependent data with individuals living within the same PSU can be expected to be more alike than a random sample
If not allowed for, get incorrect estimates of SE’s and therefore Type 1 errors:
Multilevel models model this dependency

8. 1: Hierarchical structures (continued)
b) Repeated measures of voting behaviour at the UK general election

9. 1: Hierarchical structures (continued)
c) Multivariate design for health-related behaviours
Extreme case of rotational designs

10. 2: Non- Hierarchical structures
a) cross-classified structure
b) multiple membership with weights
Can represent reality by COMBINATIONS of different types of structures
But can get complex so….

11. CLASSIFICATION DIAGRAMS
a) 3-level hierarchical structure
b) cross-classified structure

12. CLASSIFICATION DIAGRAMS(cont)
c) multiple membership structure
d) spatial structure

13. ALSPAC All children born in Avon in 1990 followed longitudinally
occasions
Pupil
Teacher
School Cohort
Primary school
Area
ALSPAC
All children born in Avon in 1990 followed longitudinally
Multiple attainment measures on a pupil
Pupils span 3 school-year cohorts (say 1996,1997,1998)
Pupils move between teachers,schools,neighbourhoods
Pupils progress potentially affected by their own changing characteristics, the pupils around them, their current and past teachers, schools and neighbourhoods

14. IS SUCH COMPLEXITY NEEDED?
M. occasions
Pupil
Teacher
School Cohort
Primary school
Area
IS SUCH COMPLEXITY NEEDED?
Complex models are NOT reducible to simpler models
Confounding of variation across levels (eg primary and secondary school variation)

15. Summary
Multilevel models can handle social science research problems with “realistic complexity”
Complexity takes on two forms and two types
As ‘Structure’ ie dependencies
- naturally occurring dependencies
Eg: pupils in schools ; measurements over time
- ‘imposed-by-design’ dependencies
Eg: multistage sample
As ‘Missingness’ ie imbalance
- naturally occurring imbalances
Eg: not answering in a panel study
- ‘imposed-by-design’ imbalances
Eg: rotational questions
Most (all?) social science research problems and designs are a combination of strict hierarchies, cross-classifications and multiple memberships

16. So what? Substantive reasons: richer set of research questions
To what extent are pupils affected by school context in addition to or in interaction with their individual characteristics?
What proportion of the variability in achievement at aged 16 can be accounted for by primary school, secondary school and neighbourhood characteristics?
Technical reasons:
Individuals drawn from a particular ‘groupings’ can be expected to be more alike than a random sample
Incorrect estimates of precision, standard errors, confidence limits and tests; increased risk of finding relationships and differences where none exists

17. Varying relationships: what are random effects?
“There are NO general laws in social science that are constant over time and independent of the context in which they are embedded”
Rein (quoted in King, 1976)

18. VARYING RELATIONS 3 2 1 -1 -2 -3 -4 8 7 6 5 4 Rooms
Multilevel modelling can handle
- multiple outcomes
- categorical & continuous predictors
- categorical and continuous responses
But KISS………
Single response: house price
Single predictor
- size of house, number of rooms
Two level hierarchy
- houses at level 1 nested within
- neighbourhoods at level 2 are the contexts
Set of characteristic plots……………… 3 2 1 -1 -2 -3 -4 8 7 6 5 4
Rooms

20. Example of varying relations (BJPS 1992)
Stucture: 3 levels strict hierarchy
individuals within constituencies within regions
Response: Voting for labour in 1987
Predictors
1 age, class, tenure, employment status
2 %unemployed, employment change, % in mining in 1981
Expectation: coal mining areas vote for the left
Allow: mining parameters for mining effect(2) to vary over region(3) in a 3-level logistic model

21. Varying relations for Labour voting and % mining

22. Higher-level variables
So far all predictors have been level 1 (Math3, boy/girl); (size,type of property)
Now higher level predictors (contextual,ecological)
- global occurs only at the higher level;
-aggregate based on summarising a level 1 attribute
Example: pupils in classes
progress affected by previous score (L1); class average
score (A:L2); class homogeneity (SD, A:L2); teaching style
(G:L2)
NOW: trying to account for between school differences

23. Propensity for left vote
Main and cross-level relationships: a graphical typology The individual and the ecological - 1
Low SES
Propensity for left vote
High SES
% Working class

24. The individual and the ecological - 2
Low SES
High SES
Propensity for left vote
% Working class

25. The individual and the ecological - 3 consensual
Low SES
High SES
Propensity for left vote
% Working class

26. A graphical typology of cross-level interactions (Jones & Duncan 1993)
Individual
Ecological
Reactive
Consensual
Reactive for W; Consensual for M
Non-linear cross-level interactions

27. STRUCTURE: 2275 voters in 218 constituencies, 1992
RESPONSE: vote Labour not Conservative
PREDICTORS: Level
- individual: age, sex, education, tenure, income 1
: 8-fold classification of class
- constituency:% Local authority renters 2
% Employers and managers;100 - % Unemployed
MODEL: cross-level interactions between INDIVIDUAL&CONSTITUENCY characteristics
Fixed part main effects: 8 fold division of class
Random part at level 2: 2 fold division of class
Working class: unskilled and skilled manual, foreman
Non-working class: public and private-sector salariat, routine non- manual, petty-bourgeoisie, ‘unstated’

28. Cross-level interactions

29. Type of questions tackled by multilevel modelling I
2-level model: current attainment given prior attainment of pupils(1) in schools(2)
NB assuming a random sample of pupils from a random samples of schools
Do Boys make greater progress than Girls (F)
Are boys more or less variable in their progress than girls?(R)
What is the between-school variation in progress? (R)
Is School X different from other schools in the sample in its effect? (F)
continued…….

30. Type of questions tackled by multilevel modelling II
Are schools more variable in their progress for pupils with low prior attainment? (R)
Does the gender gap vary across schools? (R)
Do pupils make more progress in denominational schools?(F)
Are pupils in denominational schools less variable in their progress? (R)
Do girls make greater progress in denominational schools? (F) (cross-level interaction)

31. Fixed and Random classifications
Levels and Variables
Why are schools a level but gender a variable?
Schools = Level = a population of units from which we have taken a random sample
Gender = Variable ≠ a sample out of all possible gender categories
Fixed and Random classifications
Fixed classification
Discrete categories of a variable (eg Gender)
Not sample from a population
Specific categories only contribute to their respective means
Information on Females does contribute to the estimate for Males
Random classification
Generalization of a level (e.g., schools)
Random effects come from a distribution
All schools contribute to between-school variance
Information is exchangeable between schools

32. When levels become variables...
Schools can be treated as a variable and placed in the fixed part; achieved by a set of dummy variables one for each school; target of inference is each specific school; each one treated as an ‘island unto itself’
Schools in the random part, treated as a level, with generalization possible to ALL schools (or ‘population’ of schools), in addition to predicting specific school effects given that they come from an overall distribution

33. Conclusions 3 Substantive advantages
1 Modelling contextuality and heterogeneity
2 Micro AND macro models analysed simultaneously
-avoids ecological fallacy and atomistic fallacy
3 Social contexts maintained in the analysis; permits intensive, qualitative research on ‘interesting’ cases
“The complexity of the world is not ignored in the pursuit of a single universal equation, but the specific of people and places are retained in a model which still has a capacity for generalisation”
And finally

34. LEMMA: http://www.ncrm.ac.uk/nodes/lemma/about.php
Going Further!
Learning
Environment for
Multilevel
Methodology and
Applications
NCRM node based at University of Bristol
LEMMA: