ThreeLevel Generalised Linear Models

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Three-Level Generalised Linear Models
Session 8
Damon Berridge
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Three-Level Random Intercept Models

• E.g. where students were nested within schools.
• The response variable now needs to acknowledge
  the extra level
• Student i (level 1)
• Class j (level 2)
• School k (level 3)

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Simple example linear predictor one level-1
covariate

• Hence

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The Three-level GLM
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The Three-level GLM
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Linear model

• The total variance between all level-one units
• Total variance of the level-two units

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Intraclass correlations (1)

• Proportion of the total variance from level one
• Proportion of the total variance from level two
• Proportion of the total variance from level three
  

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Intraclass correlations (2)

• Proportion of the total variance from levels one
  and two
• The correlation between different level-1 units
  (e.g. pupils) of a given level-2 unit (e.g.
  class) and level-3 unit (e.g. school)

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Intraclass correlations (3)

• Correlation between different level-1 units of
  different level-2 units for a given level-3 unit
  is

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Binary Response Model

• With binary response model easier to focus on
  correlations between the different latent
  responses
• For the probit model these correlations are
• As , but for the logit

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Three-Level Generalised Linear Model Likelihood

• where
• and

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Example 3LC2. Binary response model Guatemalan
mothers using prenatal care for their children
(1558 mothers in 161 communities) 

• Rodríguez and Goldman (2001), use of modern
  prenatal care.
• Binary indicator for whether the mother had
  prenatal care, there are 25 covariates.  
• The variables include the level-2 mother
  identifier (mom).
• The community or cluster (level-3) identifier.
• Binary indicator of the use of prenatal care for
  each child and other child-family.
• Community-level explanatory variables.
• The explanatory variables are either continuous
  variables (pcind81 proportion indigenous in 1981
  and ssdist distance to nearest clinic)
• or 0-1 dummy variables (all others) representing
  discrete factors coded using the reference
  categories.
• Reference categories are child aged 0-2 years,
  mother aged lt25 years, birth order 1 (eldest
  child), ladino (a Spanish term used to describe
  various socio-ethnic categories in Central
  America), mother with no education, husband with
  no education, husband not working or in unskilled
  occupation, no modern toilet in household, and no
  television in the household.

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Discussion

• The level 3 (community ) scale parameter estimate
  is 3.6790 (0.61058).
• The level 2 (mom) scale parameter estimate is
  7.0869 (0.94235).
• More dependence at the lower level.