Estimation techniques for clustered hierarchical data

1
Estimation techniques for clustered
(hierarchical) data

• Cluster-robust linear regression
• and
• Multilevel modelling

2
Estimators a trade off

• Trade-off
• Simplicity
• OLS the simplest and best understood estimator
• Restrictive assumptions
• OLS makes assumptions about the data that often
  do not apply
• e.g., independence
• Other estimators
• More realistic assumptions
• e.g., that observations are inter-related in
  various ways
• e.g., clustering
• pupils in schools (or classes)
• Statistical impact ? serial correlation
• intra-group
• More complex
• Computation done by software
• Need an intuitive understanding of what they do
  and what they dont do

3
Problems arising from clustering (hierarchical
data)

• OLS
• Assumes observations independent
• ? Maximum information
• Survey data
• Observations often clustered
• Individuals in families
• Firms in industries or locations
• Students in classes
• ? observations not fully independent
• reflected in the residuals
• ? OLS underestimates SEs of regression
  coefficients
• ? spurious precision

4
The problem of dependent observations

• Units are clustered ( grouped)
• e.g., students within a particular school
• tend to be more like each other than students at
  other schools
• ? a sample of students from a single school
• less varied data than a random sample of the
  same sizefrom all students
• ? loss of information
• ? cannot use OLS
• dependence between observations has to be modelled

5
Implications for estimation

• OLS not appropriate
• ? use different estimators
• Cluster robust linear regression
• Adjusts SEs to account for loss of independence
• ? clustering a nuisance to control for
• Necessary for honest estimates of standard
  errors
• Multilevel modelling ( Hierarchical linear
  modelling)
• Benefit
• Explicitly model effects at each level
• e.g., school/classroom/pupil
• Identifies where and how effects occur
• Cost
• More powerful assumptions
• ? results more dependent on assumption of random
  and normally distributed effects
• i.e., sensitive to outliers and skewed error
  distribution
• If assumptions do not hold, MLM
• underestimates SEs of higher-level parameters

6
Estimation and data requirements

• CRLR
• STATA 8
• LIMDEP 8
• Multilevel modelling
• MLwiN
• Data requirements
• Most common
• individual data (e.g., pupils)
• Variables to indicate belonging to higher level
  units
• Class (and/or teacher)
• School
• Number of levels?
• In practice, no more than 3 or 4

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Multilevel modelling continuous response
2-level, 2 variable example

• Single-level
• Pupils only
• Multilevel (1)
• Pupils
• Schools
• different intercepts for each school
• Multilevel (2)
• Pupils
• Schools
• different intercepts
• different slopes

8
Single level model

• Individual pupils
• yi individual test scores
• xi individual ability
• ei individual error terms
• difference between actual predicted scores
• i indexes pupils 1n
• ?0 overall intercept (fixed all the same)
• ?1 overall slope (fixed all the same)
• Shows how individual test scores related to
  individual ability
• ?1 measures the average relationship
• Shortcoming
• No measurement of how this average relationship
  varies between schools
• ? model both pupil and school effects together

9
Two-levels pupils in schools

• Random intercepts and slopes
• Most general model for 2 variables and 2 levels
• Fixed part of the model ( regression
  coefficients)
• ?0 overall intercept (fixed)
• Subscript 0 ? associated with the intercept
• ?1 overall slope (fixed)
• Subscript 1 ? associated with the slope
• Random part of the model
• Random effects
• j indexes schools
• uoj a school-level error term (between-school
  effect)
• gives each school an individual intercept (? ?0)
• u1j gives each school an individual slope
• Random slope coefficient (?1u1j)xij
• j ? between-school effect
• eij within-school error term for ith pupil in
  the jth school
• pupil-level error term

10
Variance of the random effects

• Additional information

11
Example from MLwiN

• Fixed Component
• Overall constant -0.012 (not significant) (not
  interesting!)
• Positive mean association between ability score
  (0.557)
• Random component
• School-level variance component 0.090
• Variation of individual schools slopes around
  mean 0.015
• Positive cov. between schools intercepts and
  slopes 0.018
• Pupil-level variance (within-school variation)
  0.554

12
Further implications

• Between-school variation around the regression
  line as a proportion of total variation
• 14 of variation accounted for at school level
• Positive covariance between schools intercepts
  and slopes
• ? schools with steep slopes have high intercepts
  and vice versa
• i.e., fanning out of school regression lines

13
Use in school effectiveness research

• Specify additional variables at all levels
• Control for variations in individual pupil
  characteristics (subscript ij)
• Specify variables at individual level
• Account for differences in school performance
  (subscript j)
• Specify variables at school level
• Contextual (function of the pupil-level data)
• Proportion with FSM
• Mean and/or SD of attainment
• True school-level variable
• Mixed or single-sex
• Homework policy
• Can be specified with or without random elements

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Want to know more?

• In order of difficulty
• Tranmer, M. (2002) Multilevel Modelling
  Coursebook, Manchester University CCSR.
• Rasbash, J. et al. (2002) A Users Guide to MLwiN
  (Version 2.1d), Centre for Multilevel Modelling,
  Institute of Education, University of London.
• Goldstein, H. (1995 2nd Ed.) Multilevel
  Statistical Models, London Arnold
• Available online http//www.ioe.ac.uk/hgpersonal/