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

3. Link function:

gðmÞ¼b 0 þ~bhXh

g‘‘links’’EðYÞwithb 0 þ~bhXh

Logistic regression (logit link)

gðmÞ¼log

m

1 m



¼logitðmÞ

Alternate formulation

Inverse of link function¼g^1
satisfies

g^1 ðgðmÞÞ¼m

Inverse of logit function in terms
of (X,b)

g^1 ðX;bÞ¼m

¼

1

1 þexp  aþ~

p

h¼ 1

bhXh

"# !;

where

gðmÞ¼logit PðD¼ 1 jXÞ

¼b 0 þ~

p

h¼ 1

bhXh

GLM:

 Uses ML estimation

 Requires likelihood functionL
where

L¼

YK

i¼ 1

Li

ðassumesYiare independentÞ

IfYinot independent and not normal

+

Lcomplicated or intractable

3. Thelink functionrefers to that function of
   the mean response,g(m), that is modeled line-
   arly with respect to the regression parameters.
   This function serves to “link” the mean of the
   random response and the fixed linear set of
   parameters.

For logistic regression, thelog odds(orlogit)of

the outcome is modeled as linear in the regres-

sion parameters. Thus, the link function for

logistic regression is the logit function [i.e.,

g(m) equals the log of the quantitymdivided by

1 minusm].

Alternately, one can express GLM in terms of

theinverseof the link function (g^1 ), which is

the meanm. In other words,g^1 (g(m))¼m. This

inverse function is modeled in terms of the

predictors (X) and their coefficients (b) (i.e.,

g^1 (X,b)). For logistic regression, the inverse

of the logit link function is the familiar logistic

model of the probability of an event, as shown

on the left. Notice that this modeling of the

mean (i.e., the inverse of the link function) is

not a linear model. It is thefunctionof the

mean (i.e., the link function) that is modeled

as linear in GLM.

GLM uses maximum likelihood methods to

estimate model parameters. This requires

knowledge of the likelihood function (L),

which, in turn, requires that the distribution

of the response variable be specified.

If the responses are independent, the likeli-

hood can be expressed as the product of each

observation’s contribution (Li) to the likeli-

hood.

However, if the responses are not independent,

then the likelihood can become complicated,

or intractable.

Presentation: V. Generalized Linear Models 505