THREE-LEVEL MODEL Two views The intractable statistical complexity that is occasioned by unduly ambitious three-level models (Bickel, 2007, 246) AND higher.

THREE-LEVEL MODELTwo views

• “The intractable statistical complexity that is occasioned by unduly ambitious three-level models” (Bickel, 2007, 246) AND

• “higher levels may have substantial effects, but without the guidance of well-developed theory or rich substantive literature, unproductive guesswork, data dredging and intractable statistical complications come to the fore” (Bickel, 2007, 219)

•But technically, a three-level model is a straightforward development of 2-level model; substantively research problems are not confined to 2 levels!

THREE-LEVEL MODEL•|Unit and classification diagrams, dataframes• Some example of applied research• Algebraic specification of 3 level random-intercepts model• Various forms of the VPC • Specifying models in MLwiN • Residuals

• Applying the model- the repeated cross-sectional model; changing school

performance

• Further levels:- as structures etc- in MLwin

Student

School

Class

Student St1 St2 St3 St1 St2 St1 St2 St3 St1 St2 St3 St4

School Sc1 Sc2 Sc3

Class C1 C2 C1 C2

• Student achievement affected by student characteristics, class characteristics and school characteristics

•Need more than 1 class per school; imbalance allowed

•Need lots of pupils in lots of classes in lots of schools!

Three-level modelsUnit and classification diagrams

Data Frame for 3 level modelClassifications or levels

Response Explanatory variables

NB categorical and continuous variables can be included at any level

Student

iClass j

School k

Current Exam scoreijk

Student previousExamination scoreijk

Student genderijk

Class teaching stylejk

School typek

1 1 1 75 56 M Formal State

2 1 1 71 45 M Formal State

3 1 1 91 72 F Formal State

1 2 1 68 49 F Informal State

2 2 1 37 36 M Informal State

1 1 2 67 56 M Formal Private

2 1 2 82 76 F Formal Private

3 1 2 85 50 F Formal Private

1 1 3 54 39 M Informal State

NB must be sorted correctly for MLwiN, recognises units by change in higher-level indices

Some examples (with references)• West, B T et al (2007) Linear mixed models, Chapman

and Hall, Boca Raton• Dependent variable: student’s gain in Maths score,

kindergarten to first grade• Explanatory variables- 1: Student (1190): Maths score in kindergarten, Sex,

Minority, SES- 2: Classroom (312) Teacher’s years of teaching

experience, Teacher’s maths experience, teacher’s maths knowledge

- 3: School (107) % households in n’hood of school in poverty

NB lacks power to infer to specific classes/schools?

Some examples continued• Bickel, R (2007) Multilevel analysis for

applied research, Guildford Press, New York

• Dependent variable: Maths score for 8th graders in Kentucky

• Explanatory variables- 1: Student (50,000): Gender, Ethnicity,- 2: Schools (347) School size, % of school

students receiving free/reduced cost lunch - 3: Districts (107) District school size

Some examples continued• Ramano, E et al (2005) Multilevel correlates of

childhood physical aggression and prosocial behaviour Journal of Abnormal Child Psychology, 33, 565-578

- individual, family and neighbourhood

• Wiggins, R et al (2002) Place and personal circumstances in a multilevel account of women’s long-term illness Social Science & Medicine, 54, 827-838

- Large scale study, 75k+ women in 9539 wards in 401 districts; used PCA to construct level-2 variables from census data

Algebraic specification of random intercepts model

ijkjkkjk

ijkijkijk

euv

xy

00000

10

i level 1 (e.g. pupil), j level 2 (e.g. class) , k level 3 (e.g. school)

kv0 is the random effect at the school level, an allowed-to-vary departure from the grand mean;

jku0 is the random effect at the class level, a departure from the school effect;

ijke is the random effect at the pupil level, a departure from the class effect within a school Variance between schools = Var (v0k) = 2

0v Variance between classes within schools = Var (u0jk) = 2

0u Variance between pupils within classes within schools = Var (eijk) = 2

e

Variance between classes = 20v + 2

0u Random effects at different levels assumed to be uncorrelated

Various forms of the VPC for random intercepts model

Proportion variance due to differences between schools

= intra-school correlation = 220

20

20

euv

v

Proportion variance due to differences between classes

= intra-class correlation = 220

20

20

20

euv

uv

Correlation structure of

3 level modelS 1 1 1 2 2 2 2 3 3 3

C 1 1 2 1 1 2 2 1 1 2

P 1 2 3 1 2 3 4 1 2 3

1 1 1 1 0 0 0 0 0 0 0

1 1 2 1 0 0 0 0 0 0 0

1 2 3 1 0 0 0 0 0 0 0

2 1 1 0 0 0 1 0 0 0

2 1 2 0 0 0 1 0 0 0

2 2 3 0 0 0 1 0 0 0

2 2 4 0 0 0 1 0 0 0

3 1 1 0 0 0 0 0 0 0 1

3 1 2 0 0 0 0 0 0 0 1

3 2 3 0 0 0 0 0 0 0 1

Intra-class correlation (within same school & same class)

Intra-school correlation (within same school, different class)

Example: pupils within classes within schools (Snijder & Bosker data)

Response is score on maths test at age 14

Estimate S.E. Fixed β0 7.96 0.23 Random

20v (school) 2.124 0.546

20u (class) 1.746 0.226

2e (pupil) 7.816 0.186

Total variance is 2.124 + 1.746 + 7.816 = 11.686

Intra-school correlation = 2.124/11.686 = 0.18 (similarity of pupils in same school).

Intra-class correlation = (2.124+1.746)/11.686 = 0.33 (similarity of pupils in same class, in same school).

The similarity of classes within the same school (correlation between mean maths score for 2 randomly selected classes in a randomly selected school) is 2.124/(2.124+1.746) = 0.55.

Variance Partition Coefficients:pupils within classes within schools (Snijder & Bosker data)

Specifying models in MLwiN• Three-level variance components for attainment

Specifying models in MLwiN• Are there classes and/or schools where the gender gap is large, small or

inverse?• Student gender in fixed part and Variance functions at each level

21

211010

20

20 2 ijkuijkijkuuijku xxxx

21

211010

20

20 2 ijkijkijkijk xxxx

ijkijkeeijke xxx 101020

20 2

Level 2 variance

Level 3 variance

Level 1 variance

Specifying models in MLwiN• Is the Gender gap differential according to teaching style?• Cross-level interactions between Gender and Teaching style in the

fixed part of the model• IE main effects for gender & style, and first order interaction between

Student Gender and Class Teaching Style

Fixed partCons: mean score for Male in Formally-taught class

Female: differential for female in formal class

Informal: differential for male in informal class

Female*Informal: differential for female in informal class

Residuals

• Key notion is that highest level residual is a random, allowed-to-vary departure from general relationship

• Each lower level residual is allowed-to-vary random departure from the higher-level departure

Level 3 residuals: school departures from grand mean line

ijkkijk xy 10

ijkijk xy 10

ijkkijk xy 10

kv0

Level 2 residuals: class departures from the associated school line

ijkjkijk xy 10

jku0

ijkjkijk xy 10

ijkjkijk xy 10

ijkjkijk xy 10

Level-1 residuals: student departures from the associated class line

ijkjkijk xy 10

ijke

Applying the model: the repeated cross-sectional model; changing school performance

Sc1 Sc2 Sc3....

1985 1986 1987 1985 1987 1986 1987

St1…St9 St1… St25 St1 …St32 St1… St22 St1… St12 St1… St29 St1… St13

School

Student

Cohort

• Modelling Exam scores for groups of students who entered school in 1985 and a further group who entered in 1986.• In a multilevel sense we do not have 2 cohort units but 2S cohort units where S is the number of schools. • The model can be extended to handle an arbitrary number of cohorts with imbalance

Applying the model: the repeated cross-sectional model; changing school performance

• Modelling Exam scores aged 16 for Level 3 139 state schools from the Inner London Education Authority, Level 2 304 cohorts with a maximum of 3 cohorts in any one school, and Level 1 115,347 pupils with a maximum of 135 pupils in any one school cohort

•pupil level variables: Sex, Ethnicity, Verbal Reasoning aged 11

• cohort-level variables: % of pupils in each school who were receiving Free-school meals in that year, % of pupils in the highest VRband in that year, the year that the cohort graduated

• school level variables: the ‘sex’ of the school (Mixed Boys and Girls); the schools’ religious denomination (Non-denominational, CofE, Catholic)

Further levels - as structures, etcSome examples of 4-level nested structures:

• student within class within school within LEA• people within households within postcode sectors within regions•Finally, Repeated measures within students within cohorts within schools

O1 O2 O1 O2 O1 O2 O1 O2 O1 O2 O1 O2 O1 O2 O1 O2

Cohorts are now repeated measures on schools and tell us about stability of school effects over time Measurement occasions are repeated measures on students and can tell us about students’ learning trajectories.

1990 1991 1990 1991

Sc1 Sc2...

Cohort

Msmnt occ

student

School

St1 St2... St1 St2.. St1 St2.. St1 St2..

Further levels - in MLwiN• Click on extra subscripts!

• Default is a maximum of 5 but can be increased