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

IfYinot independent but MV normal

+

Lspecified

IfYinot independent andnot

MV normal

+

Quasi-likelihood theory

Quasi-likelihood:

 No likelihood

 Specify mean variance
relationship

 Foundation of GEE

VI. GEE Models

GEE: class of models for correlated
data Link functiongmodeled as

gðmÞ¼b 0 þ~

p

h¼ 1

bhXh

ForYð 0 ; 1 Þ)logit link

gðmÞ¼logit PðY¼ 1 jXÞ

¼b 0 þ~

p

h¼ 1

bhXh

For nonindependent outcomes whose joint dis-

tribution is multivariate (MV) normal, the like-

lihood is relatively straightforward, since the

multivariate normal distribution is completely

specified by the means, variances, and all of the

pairwise covariances of the random outcomes.

This is typicallynotthe case for other multi-

variate distributions in which the outcomes

arenotindependent. For these circumstances,

quasi-likelihood theory offers an alternative

approach for model development.

Quasi-likelihood methods have many of the

same desirable statistical properties that maxi-

mum likelihood methods have, but the full like-

lihood does not need to be specified. Rather,

the relationship between the mean and vari-

ance of each response is specified. Just as the

maximum likelihood theory lays the founda-

tion for GLM, the quasi-likelihood theory lays

the foundation for GEE models.

GEE represent a class of models that are often

utilized for data in which the responses are

correlated (Liang and Zeger, 1986). GEE mod-

els can be used to account for the correlation of

continuous or categorical outcomes. As in

GLM, a function of the meang(m), called the

link function, is modeled as linear in the regres-

sion parameters.

For a dichotomous outcome, the logit link is

commonly used. For this case,g(m) equals logit

(P), where P is the probability that Y¼1.

If there arepindependent variables, this can

be expressed as: logit P(Y¼1|X) equals

b 0 plus the summation of thepindependent

variables times theirbcoefficients.

506 14. Logistic Regression for Correlated Data: GEE