THREELEVEL MODEL

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THREE-LEVEL MODEL

• Two 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!

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Session outline

• Unit and classification diagrams, dataframes
• Some examples of applied research
• Algebraic specification of 3 level
  random-intercepts model
• Residuals
• Various forms of the VPC
• Correlation (dependency) structure for 3 level
  model
• Specifying models in MLwiN
• Applying the model
• - the repeated cross-sectional model changing
  school performance
• Further levels
• - as structures etc
• - in MLwin

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Three-level modelsUnit and classification
diagrams

• Student achievement affected by student
  characteristics, class characteristics and
  school characteristics
• Need more than 1 class per school imbalance
  allowed
• Need lots of pupils,classes and schools!

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Data Frame for 3 level model
NB must be sorted correctly for MLwiN, recognises
units by change in higher-level indices
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Some examples (with references)

• West, B T et al (2007) Linear mixed models,
  Chapman and Hall, Boca Raton
• Dependent variable students gain in Maths
  score, kindergarten to first grade
• Explanatory variables
• 1 Students (1190) Maths score in kindergarten,
  Sex, Minority, SES
• 2 Classrooms (312) Teachers years of teaching
  experience, Teachers maths experience, teachers
  maths knowledge
• 3 Schools (107) households in nhood of school
  in poverty
• NB lacks power to infer to specific
  classes/schools?

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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

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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 womens
  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

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Algebraic specification of random intercepts model
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Level 3 residuals school departures from grand
mean line
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Level 2 residuals class departures from the
associated school line
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Level-1 residuals student departures from the
associated class line
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Shrinkage

• Still applies!
• Level 3 school means shrunk towards the grand
  mean
• Level 2 class means are shrunk towards the
  associated shrunken school mean
• Greatest shrinkage when raw means most extreme
  and when fewest pupils
• Formulae Raudenbush, S. W., and A. S. Bryk
  (2002) Hierarchical Linear Models Applications
  and Data Analysis Methods. SAGE,230, 250-251
• Appreciation

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Shrinkage at level 3Balanced design

• 4 pupils in each class, 3 classes in each of 20
  schools
• shrinkage of school means to grand mean
• greatest shrinkage for schools with most extreme
  raw means
• no crossing in balanced case

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Shrinkage at level 2Balanced design

• 4 pupils in each class, 3 classes in each of 20
  schools
• shrinkage of class means to school shrunken mean
  
• crossing even in balanced design as shrink
  towards shrunken
• school mean

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Various forms of the VPC for random intercepts
model
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Correlation structure of 3 level model
Intra-class correlation (within same school
same class) r1 Intra-school correlation (within
same school, different class) r2
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Example pupils within classes within schools
(Snijder Bosker data)
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Variance Partition Coefficients pupils within
classes within schools (Snijder Bosker data)
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Specifying models in MLwiN

• Three-level variance components for attainment

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Specifying models in MLwiN

• Are there classes and/or schools where the gender
  gap is large, small or inverse to the sample as
  whole?
• Student gender in fixed part and Variance
  functions at each level

Level 3 variance
Level 2 variance
Level 1 variance
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Specifying models in MLwiN

• Is the Gender gap for pupils affected by class
  teaching style?
• Cross-level interactions between Gender (student)
  and Teaching style (of the class) 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 part B0 mean score for Male in
Formally-taught class B0 B1 mean score for
Females in Formally-taught class B0 B2 mean
score for Males in Informally-taught class B0
B1 B2 mean score for Females in
Informally-taught class
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Applying the model the repeated cross-sectional
model changing school performance

• 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

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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)
  

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Applying the modelcontinued

• script for West et al s (2007) exampleoriginal
  using SPSS
• script for Bickels (2007) exampleoriginal using
  HLM, SPSS, SAS, Stata, R

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Further levels - as structures, etc

• Some 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

St1 St2... St1
St2.. St1 St2..
St1 St2..
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.
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Further levels - in MLwiN

• Click on extra subscripts!
• Default is a maximum of 5 but can be increased