Multilevel Generalised Linear Models

1
Multilevel Generalised Linear Models
Session 7
Damon Berridge
2
GLMs

• The models we have considered so far, namely
  linear, binary response and Poisson models, are
  special cases of the generalised linear model
  (GLM) or exponential family.
• It will help us in considering extensions of
  these models to 3 levels, and to multivariate
  responses, if we can start to write each of the
  models using GLM notation.

3
GLMs

• In generalised linear models, the explanatory
  variables and the random effects for a 2-level
  model affect the response via the linear
  predictor
• The GLM is obtained by specifying some function
  of the response conditional on the linear
  predictor and other parameters, i.e.
• scale parameter
• function that gives the conditional
  mean and variance

4
GLMs

• The mean and variance are related so that
• the variance function
• the link function
• the inverse link function
• differ for different members of the exponential
  family

5
The Linear Model
6
The Linear Model

• Mean and variance functions
• Mean and variance not related
• Link function
• We define this model by Gaussian error, identity
  link

7
Binary Response Models

• Rewrite the binary response model from earlier as
• Then
• So that

8
Binary Response Models

• Mean and variance functions
• The mean and variance are related as
• Also
• Logit link
• Probit link
• Cloglog link

9
Poisson Model

• If we rewrite the Poisson model from an earlier
  section as

• Then we can write

• So that

10
Poisson Model

• The mean, (mij), and variance functions are

• Note that in the Poisson model, the mean and
  variance are related as

• The link function is the log link as qij ln
  mij . We define the Poisson model with Poisson
  error p, and logit g, probit p or cloglog c link.

11
Two-Level Generalised Linear Model Likelihood

• We can now write the 2-level Generalised Linear
  Model likelihood for the linear model, binary
  response and Poisson models in a general form,
  i.e.

• where

• and