What is multilevel modelling?

Kelvyn Jones, School of Geographical Sciences, LEMMA, University of Bristol2nd Oxford Research Methods Festival

July 2006

MULTILEVEL MODELS

AKA

• random-effects models,

• hierarchical models,

• variance-components models,

• random-coefficient models,

• mixed models

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Macro models

Combined multilevel model

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Micro model

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

• Can represent reality by COMBINATIONS of different types of structures

• But can get complex so….

a) cross-classified structure

b) multiple membership with weights

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• 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

occasions

Pupil TeacherSchool 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)

M. occasions

Pupil TeacherSchool Cohort

Primary school

Area

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 dependenciesEg: 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• 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 contextsSet of characteristic plots………………

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87654321Rooms

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

Main and cross-level relationships:a graphical typology

The individual and the ecological - 1

% Working class

Pro

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High SES

Low SES

The individual and the ecological - 2

% Working class

Pro

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High SES

Low SES

The individual and the ecological - 3 consensual

% Working class

Pro

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High SES

Low SES

A graphical typology of cross-level interactions (Jones & Duncan 1993)

Consensual

Individual Ecological

Reactive

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 classRandom part at level 2: 2 fold division of classWorking class: unskilled and skilled manual, foremanNon-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)

Levels and VariablesWhy are schools a level but gender a variable?Schools = Level = a population of units from which we have taken a random sampleGender = Variable ≠ a sample out of all possible gender categories

Fixed and Random classificationsRandom classificationGeneralization of a level (e.g., schools)

Random effects come from a distribution

All schools contribute to between-school variance

Information is exchangeable between schools

Fixed classificationDiscrete 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

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

Conclusions3 Substantive advantages1 Modelling contextuality and heterogeneity2 Micro AND macro models analysed simultaneously

-avoids ecological fallacy and atomistic fallacy3 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

Going Further!

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

Learning

Environment for

MultilevelMethodology and

Applications

NCRM node based at University of Bristol