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Slide 1
School of Veterinary Medicine and Science Multilevel modelling
Chris Hudson
Slide 2
School of Veterinary Medicine and Science Regression
models..with 1 predictor outcome predictor
Slide 3
School of Veterinary Medicine and Science Regression models
y=0.06 + 0.31x..with 1 predictor
Slide 4
School of Veterinary Medicine and Science Regression
models..with >1 predictor
Slide 5
School of Veterinary Medicine and Science More complex data
structures In real life, things are often less simple! Are your
units of data really independent? Repeated measures within
individuals Pupils within schools Individuals within households
within neighbourhoods These are multilevel structures e.g. pupils
from the same school are more likely to be similar than pupils from
different schools
Slide 6
School of Veterinary Medicine and Science A real example Using
a multilevel/hierarchical structure 2d
Slide 7
School of Veterinary Medicine and Science A real example Using
a multilevel/hierarchical structure
Slide 8
Why should we use multilevel models? Slides from
www.bris.ac.uk/cmmwww.bris.ac.uk/cmm
Slide 9
How do we deal with this? Contextual analysis. Analysis
individual-level data but include group-level predictors Problem:
Assumes all group-level variance can be explained by group-level
predictors; incorrect SEs for group-level predictors Do pupils in
single-sex school experience higher exam attainment? Structure:
4059 pupils in 65 schools Response: Normal score across all London
pupils aged 16 Predictor: Girls and Boys School compared to Mixed
school Parameter Single level Multilevel Intercept (Mixed
school)-0.098 (0.021) -0.101 (0.070) Boy school 0.122 (0.049) 0.064
(0.149) Girl school 0.245 (0.034) 0.258 (0.117) Between school
variance( u 2 ) 0.155 (0.030) Between student variance ( e 2 )
0.985 (0.022) 0.848 (0.019) Parameter Single level Intercept (Mixed
school)-0.098 (0.021) Boy school 0.122 (0.049) Girl school 0.245
(0.034) Between school variance( u 2 ) Between student variance ( e
2 ) 0.985 (0.022)
Slide 10
How do we deal with this? Analysis of covariance (fixed effects
model). Include dummy variables for each and every group Problems
What if number of groups very large, eg households? No single
parameter assesses between group differences Cannot make inferences
beyond groups in sample Cannot include group-level predictors as
all degrees of freedom at the group-level have been consumed Target
of inference: individual School versus schools
Slide 11
How do we deal with this? Fit single-level model but adjust
standard errors for clustering (GEE approach) Problems: Treats
groups as a nuisance rather than of substantive interest; no
estimate of between-group variance; not extendible to more levels
and complex heterogeneity Multilevel (random effects) model
Partition residual variance into between- and within-group (level 2
and level 1) components Allows for un-observables at each level,
corrects standard errors Micro AND macro models analysed
simultaneously Avoids ecological fallacy and atomistic fallacy
Slide 12
School of Veterinary Medicine and Science ML models 1.Account
appropriately for clustering (even in complex structured data)
2.Allow inferences to be made about differences between levels
(including generalisation to wider popluations) cf treating this as
a nuisance
Slide 13
School of Veterinary Medicine and Science Random effects Random
intercepts Each higher-level unit has a different intercept These
are assumed to come from a Normal distribution
Slide 14
School of Veterinary Medicine and Science Random effects Random
slopes Its also possible to let the slope of each line vary between
higher-level units
Slide 15
School of Veterinary Medicine and Science An example
Slide 16
School of Veterinary Medicine and Science Resources
www.bris.ac.uk/cmm www.statmenthods.net