The next set of exercises is designed to illustrate how an
individual level odds ratio can differ from a population
averaged (marginal) odds ratio. Consider a fictitious data
set in which there are only 2 subjects, contributing 200
observations apiece. For each subject, 100 of the observa-
tions are exposed (E¼1) and 100 are unexposed (E¼0),
yielding 400 total observations. The outcome is dichoto-
mous (D¼1 andD¼0). The data are summarized using
three 22 tables. The tables for Subject 1 and Subject
2 summarize the data for each subject; the third table pools
the data from both subjects.
Subject 1
E¼ 1 E¼ 0D¼ 15025
D¼ 05075
Total 100 100
Subject 2
E¼ 1 E¼ 0
D¼ 12510
D¼ 07590
Total 100 100Pooled subjects
E¼ 1 E¼ 0
D¼ 17535
D¼0 125 165
Total 200 200- Calculate the odds ratio for Subject 1 and Subject 2 sep-
arately. Calculate the odds ratio after pooling the data
for both subjects. How do the odds ratios compare?
Note: The subject-specific odds ratio as calculated
here is a conceptualization of a subject-specific effect,
while the pooled odds ratio is a conceptualization of a
population-averaged effect. - Compare the baseline risk (whereE¼0) of Subject 1
and Subject 2. Is there a difference (i.e., heterogene-
ity) in the baseline risk between subjects? Note that
for a model containing subject-specific random
effects, the variance of the random intercept is a mea-
sure of baseline risk heterogeneity. - Do Subject 1 and Subject 2 have a different distribu-
tion of exposure? This is a criterion for evaluating
whether there is confounding by subject. - Suppose an odds ratio is estimated using data in
which there are many subjects, each with one obser-
vation per subject. Is the odds ratio estimating an
individual level odds ratio or a population averaged
(marginal) odds ratio?
For Exercise 14 and Exercise 15, consider a similar scenario
as was presented above for Subject 1 and Subject 2. How-
ever, this time the risk ratio rather than the odds ratio is the
measure of interest. The data for Subject 2 have been altered
slightly in order to make the risk ratio the same for each
subject allowing comparability to the previous example.
Practice Exercises 593