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race, and census tract. Regression is used to adjust for income, education,
marital status, and occupational exposure.
Lieberson [1985] warns that if the strata differ in the levels of some
third unmeasured factor that influences the outcome variable, the results
may be bogus.
Simpson's Paradox
A third omitted variable may also result in two variables appearing to be
independent when the opposite is true. Consider the following table, an
example of what is termed Simpson's paradox:
Treatment Group
Control
Treated
Alive
6
20
Dead
6
20
We don't need a computer program to tell us the treatment has no effect
on the death rate. Or does it? Consider the following two tables that
result when we examine the males and females separately:
Treatment Group
Control
Treated
Alive
4
8
Dead
3
5
Treatment Group
Control
Treated
Alive
2
12
Dead
3
15
In the first of these tables, treatment reduces the male death rate from 3 out
of 7 (0.43) to 5 out of 13 (0.38). In the second, the rate is reduced from 3
out of 5 (0.6) to 15 out of 27 (0.55). Both sexes show a reduction, yet the
combined population does not. Resolution of this paradox is accomplished
by avoiding a knee-jerk response to statistical significance when association
is involved. One needs to think deeply about underlying cause-and-effect
relationships before analyzing data. Thinking about cause and effect in the
preceding example might have led us to think about possible sexual differ-
ences and to stratify the data by sex before analyzing it.
