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

Heartburn Relief Study:

GEE: marginal model

expðb^ 1 ÞispopulationdOR

MLM:

expðb^ 1 ÞisdOR for anindividual

What is an individual OR?

Each subject has separate prob-

abilities

PðX¼ 1 jRX¼ 1 Þ

PðX¼ 1 jRX¼ 0 Þ

+

OR compares RX¼1 vs. RX¼ 0

for an individual

Goal OR

Population

inferences

) marginal

Individual

inferences

) individual

The odds ratio for a marginal model is the ratio

of the odds of heartburn for RX¼1 vs. RX¼ 0

among the underlying population. In other

words, the OR is a population average. The

odds ratio for a model with a subject-specific

effect, as in the mixed logistic model, is the

ratio of the odds of heartburn for RX¼1 vs.

RX¼0foranindividual.

What is meant by an odds ratio for an individ-

ual? We can conceptualize each subject as hav-

ing a probability of heartburn relief given the

active treatment and having a separate proba-

bility of heartburn relief given the standard

treatment. These probabilities depend on the

fixed treatment effect as well as the subject-

specific random effect. With this conceptualiza-

tion, the odds ratio that compares the active vs.

standard treatment represents a parameter

that characterizes an individual rather than a

population (see Practice Exercises 10–15). The

mixed logistic model supplies a structure that

gives the investigator the ability to estimate

an odds ratio for an individual, while simulta-

neously accounting for within-subject and

between-subject variation.

The choice of whether a population averaged

or individual level odds ratio is preferable

depends, in part, on the goal of the study. If

the goal is to make inferences about a popula-

tion, then a marginal effect is preferred. If the

goal is to make inferences on the individual,

then an individual level effect is preferred.

586 16. Other Approaches for Analysis of Correlated Data