MATH38052 Generalised Linear Models
9
1
Log linear models with Poisson response
We have seen examples with Poisson responses and a log link. The mean response
is linear after a log transformation. Such models are used sometimes as alternatives to
logistic

MATH38052 Generalised Linear Models
7
1
The General Linear model
7.1
Overview
The general linear model is of the form
y = X + ,
where Nn (0, 2 I ), I being the n n identity matrix, and 2 > 0 is usually unknown.
It is a GLM with normal response y N (, 2 )

MATH38052 Generalised Linear Models
1.1
1
Introduction
This course is concerned with statistical analysis of the dependence of one variable on
several others given data on a group of individuals or subjects. Examples of variables are
height and weight of

MATH38052 Generalised Linear Models
10
1
Contingency tables
Suppose we classify a random sample of n subjects according to two characteristics A
and B, e.g. gender and blood type. Let y i j be the number of subjects that fall into category
i of A and cate

MATH38052 Generalised Linear Models
8
1
Logistic regression with binomial response
For binomial response data, the logistic regression model is
=
yi
logit (i ) =
n i i + i B (n i , i ) independent,
x i , i = 1, . . . , n.
where logit () = log( 1 ) is the

MATH38052 Generalised Linear Models
6
1
Hypothesis testing for model reduction
After tting a GLM (and checking the residuals), the estimated parameters can be tested
for their signicance one by one using z-tests, i.e. comparing estimated values with their

MATH38052 Generalised Linear Models
3
1
Generalised linear models
3.1
Introduction
A generalised linear model (GLM) has two equations
y = + , and g () = x ,
(3.1)
where y is the response to some covariates x = (x 1 , . . . , x p ) , = E[y] is the mean res

MATH38052 Generalised Linear Models
4
1
The iteratively re-weighted least squares algorithm
The likelihood equation X W X = X W cannot always be solved by simply writing =
(X W X )1 X W , because generally speaking W and depend on .
An idea is to set some

MATH38052 Generalised Linear Models
2
1
The exponential family of distributions
We generalise the Normal distribution to a wider family of distributions. Recall that the
normal density is
f (y; , 2 )
=
=
=
cfw_
1
exp 2 (y )2 , < y <
2
2
cfw_
1
1
exp 2

MATH38052 Generalised Linear Models
5
1
Goodness-of-t
The goodness-of-t of a GLM (or any model) is how well it ts the data. With a linear
model, we look at the residual sum of squares (SSE, smaller is better), or equivalently
R 2 = 1 SSE /SST (larger is b