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

XII. Generalizing the
“Score-like”
Equations to Form
GEE Models

GEE models:

 For cluster-correlated data

model parameters:

b and a

correlation

parameters

regression

parameters

Matrix notation used to describe
GEE

Matrices needed specific to each
subject (cluster): Yi, mi, Di, Ci,
andWi

Yi¼

Yi 1

Yi 2

...

Yini

8

>>

>>

>>

<

>>

>>

>>

:

9

>>

>>

>>

=

>>

>>

>>

;

vector ofith subject’s

observed responses

mi¼

mi 1

mi 2

...

mini

8

>>

>>

>>

<

>>>

>>

>:

9

>>

>>

>>

=

>>>

>>

>;

vector ofith subject’s

mean responses

Ci¼working correlation matrix
(nini)

The estimating equations we have presented so

far have assumed one response per subject.

The estimating equations for GEE are “score-

like” equations that can be used when there are

several responses per subject or, more gener-

ally, when there are clustered data that con-

tains within-cluster correlation. Besides the

regression parameters (b) that are also present

in a GLM, GEE models contain correlation

parameters (a) to account for within-cluster

correlation.

The most convenient way to describe GEE

involves the use of matrices. Matrices are

needed because there are several responses

per subject and, correspondingly, a correlation

structure to be considered. Representing these

estimating equations in other ways becomes

very complicated.

Matrices and vectors are indicated by the use

of bold letters. The matrices that are needed

are specific for each subject (i.e.,ith subject),

where each subject has ni responses. The

matrices are denoted asYi,mi,Di,Ci, andWi

and defined as follows:

Yiis the vector (i.e., collection) of theith sub-

ject’s observed responses.

mi is a vector of the ith subject’s mean

responses. The mean responses are modeled

as functions of the predictor variables and the

regression coefficients (as in GLM).

Ciis theninicorrelation matrix containing

the correlation parameters.Ciis often referred

to as the working correlation matrix.

524 14. Logistic Regression for Correlated Data: GEE