Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Exponential family distributions
include:
Binomial
Normal
Poisson
Exponential
Gamma

Generalized linear model

gðmÞ¼b 0 þ~

p

h¼ 1

bhXh;

where:mis the mean responseE(Y)
g(m) is a function of the
mean

Three components for GLM:

1. Random component


2. Systematic component


3. Link function


4. Random component

Yfollows a distribution from

the exponential family

2. Systematic component

TheXs are combined in the

model linearly, (i.e.,b 0 þSbhXh)

Logistic model:

P(X) =

1 + exp[–(b 0 + Σ bhXh]

1

linear component

The binomial distribution belongs to a larger

class of distributions called the exponential

family. Other distributions belonging to the

exponential family are the normal, Poisson,

exponential, and gamma distributions. These

distributions can be written in a similar form

and share important properties.

Letmrepresent the mean responseE(Y), and

g(m) represent a function of the mean response.

A generalized linear model withpindependent

variables can be expressed as g(m) equals

b 0 plus the summation of thepindependent

variables times their beta coefficients.

There are three components that comprise

GLM: (1) a random component, (2) a system-

atic component, and (3) the link function.

These components are described as follows:

1. The random componentrequires the out-
   come (Y) to follow a distribution from the
   exponential family. This criterion is met for
   a logistic regression (unconditional) since the
   response variable follows a binomial distri-
   bution, which is a member of the exponential
   family.


2. The systematic component requires that
   theXs be combined in the model as a linear
   functionðb 0 þSbhXhÞof the parameters. This
   portion of the model is not random. This crite-
   rion is met for a logistic model, since the model
   form contains a linear component in its
   denominator.

504 14. Logistic Regression for Correlated Data: GEE