Generalised Linear Models

Question 1

What two equations define a GLM? Define all terms used

Answer 1

The first is y = µ + ε

• Where y is the vector of responses
• µ is the expected response
• ε is the vector of iid random errors ~ N(0,1)

The second is g(µ) = x’β where x is the transpose of the predictor variables and β is the vector of parameters.
g is our link function, linking the mean response to the linear predictor η = x’β

The idea is that µ need not be linear as long as g(µ) = η is

Question 2

State the three components of a GLM

Answer 2

• An ‘error’ distribution - not to be confused with ε. It is our distribution from the exponential family for the responses y
• Link function g(µ) connecting the mean response to the linear predictor η
• Predictor η = x’β that is a linear combination of the expected variables x and the parameters β

Question 3

What assumptions do we make for a GLM?

Answer 3

We assume the distribution of y is a member of the exponential family and that our vector y for different covariate values are independent.

In addition, only θ can change with i, all the rest are the same for all i

Question 4

Describe the concept of a Canonical Link

Answer 4

We write the natural parameter from the yθ in the exponential family as a function of µ. That is, we write b(θ) in θ but in terms of µ

Question 5

State the IRLS Algorithm

Answer 5

β^(k+1) = β(k) + (X’ W^(k) X)^-1 X’ W^(k) e^(k)

Question 6

What happens to the Newton Raphson Algorithm when we use the Fisher Info Matrix instead of Observed Fisher info?

Answer 6

We go from the N-R algorithm to the Fisher Scoring Algorithm

Question 7

Which 3 Algorithms are identical?

Answer 7

The N-R, the Fisher Scoring and IRLS

Question 8

Describe what’s meant by a Saturated Model

Answer 8

This means that all our natural parameters θ_i are free to vary

Question 9

Outline Scaled Deviance

Answer 9

Of a GLM is twice the difference in maximum log-likelihood value when comparing it with the saturated model.

Question 10

Give the formula for the Generalised Pearson Statistic

Answer 10

x^2 = ∑ (y_i - µ)^2 / V(µ)a(ϕ) which should be ~ X^2_(n-p) if the model is true

Question 11

What do we do after fitting a GLM to test the significance of parameters?

Answer 11

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