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

Correlated vs. independent

 Identical model

but
 Different assumptions

GEE:

 Generalization of quasi-
likelihood

 Specify a “working” correlation
structure for within-cluster
correlations

 Assume independence between
clusters

VII. Correlation Structure

Correlation and covariance sum-
marized as square matrices

Covariance matrix forY 1 andY 2

V¼

varðY 1 Þ covðY 1 ;Y 2 Þ

covðY 1 ;Y 2 Þ varðY 2 Þ

"

The logistic model for correlated data looks

identical to the standard logistic model. The

difference is in the underlying assumptions of

the model, including the presence of correla-

tion, and the way in which the parameters are

estimated.

GEE is a generalization of quasi-likelihood

estimation, so the joint distribution of the

data need not be specified. For clustered

data, the user specifies a “working” correlation

structure for describing how the responses

within clusters are related to each other.

Between clusters, there is an assumption of

independence.

For example, suppose 20 asthma patients are

followed for a week and keep a daily diary of

inhaler use. The response (Y) is given a value of

1 if a patient uses an inhaler on a given day and

0 if there is no use of an inhaler on that day.

The exposure of interest is daily pollen level.

In this analysis, each subject is a cluster. It is

reasonable to expect that outcomes (i.e., daily

inhaler use) are positively correlated within

observations from the same subject but inde-

pendent between different subjects.

The correlation and the covariance between

measures are often summarized in the form

of a square matrix (i.e., a matrix with equal

numbers of rows and columns). We use simple

matrices in the following discussion; however,

a background in matrix operations is not

required for an understanding of the material.

For simplicity consider two observations,Y 1

andY 2. The covariance matrix for just these

two observations is a 22 matrix (V) of the

form shown at left. We use the conventional

matrix notation of bold capital letters to iden-

tify individual matrices.

EXAMPLE

Asthma patients followed 7 days

Y: daily inhaler use (0,1)

E: pollen level

Cluster: asthma patient

Yiwithinsubjects correlated

but

Yibetweensubjects independent

Presentation: VII. Correlation Structure 507