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

V. Generalized Linear
Models

General form of many statistical
models:

Y¼fðX 1 ;X 2 ;...;XpÞþE;

where:Yis random
X 1 ,X 2 ,...,Xpare fixed
Eis random

Specify:

1. A function (f) for the fixed
   predictors, e.g., linear


2. A distribution for the random
   error (E), e.g., N(0,1)

GLM models include:

Logistic regression

Linear regression

Poisson regression

GEE models are extensions of GLM

GLM: a generalization of the clas-
sical linear model

Linear regression
Outcome:
 Continuous
 Normal distribution

Logistic regression
Outcome:
 Dichotomous
 Binomial distribution:
EðYÞ¼m¼PðY¼ 1 Þ

Logistic regression used to model

PðY¼ 1 jX 1 ;X 2 ;...;XpÞ

For many statistical models, including logistic

regression, the predictor variables (i.e., inde-

pendent variables) are considered fixed and

the outcome, or response (i.e., dependent vari-

able), is considered random. A general formu-

lation of this idea can be expressed asY¼f(X 1 ,

X 2 ,...,Xp)þEwhereYis the response vari-

able,X 1 ,X 2 ,...,Xpare the predictor variables,

andErepresents random error. In this frame-

work, the model forYconsists of a fixed com-

ponent [f(X 1 ,X 2 ,...,Xp)] and a random com-

ponent (E).

A function (f) for the fixed predictors and

a distribution for the random error (E) are

specified.

Logistic regression belongs to a class of models

called generalized linear models (GLM). Other

models that belong to the class of GLM include

linear and Poisson regression. For correlated

analyses, the GLM framework can be extended

to a class of models called generalized esti-

mating equations (GEE) models. Before dis-

cussing correlated analyses using GEE, we

shall describe GLM.

GLM are a natural generalization of the classi-

cal linear model (McCullagh and Nelder, 1989).

In classical linear regression, the outcome is

a continuous variable, which is often assumed

to follow a normal distribution. The mean

response is modeled as linear with respect to

the regression parameters.

In standard logistic regression, the outcome is a

dichotomous variable. Dichotomous outcomes

are often assumed to follow a binomial distri-

bution, with an expected value (or mean,m)

equal to a probability [e.g., P(Y¼1)].

It is this probability that is modeled in logistic

regression.

Presentation: V. Generalized Linear Models 503