What is multilevel modelling?

1. What ismultilevel 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. occasions Primary school School Cohort Teacher Area Pupil 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. M. occasions Primary school School Cohort Teacher Area Pupil 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. Rooms 1 2 3 4 5 6 7 8 -4 -3 -2 -1 0 1 2 3 VARYING RELATIONS • 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………………

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. Main and cross-level relationships:a graphical typologyThe 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. 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. Going Further! Learning Environment for Multilevel Methodology and Applications NCRM node based at University of Bristol LEMMA: http://www.ncrm.ac.uk/nodes/lemma/about.php