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

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

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

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

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

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)

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

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

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

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….

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

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

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

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)

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

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

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)

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

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

Varying relations for Labour voting and % mining

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

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

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

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

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

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’

Cross-level interactions

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…….

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)

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

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

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

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: http://www.ncrm.ac.uk/nodes/lemma/about.php