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Different control groups for hypothetical case-control study
of an occupational exposure and smoking
Smokers Nonsmokers
––––––––––––––––– –––––––––––––––––
Row Exposed Unexposed Exposed Unexposed
––––––– ––––––– ––––––– –––––––
# (1) (2) (3) (4)
1. Number of cases 300 90 70 21
2. Proportional controls 150 150 350 350
(OR = 3.3) (OR = 3.3)
3. Biased controls 250 150 250 350
(OR = 2.0) (OR = 4.7)
Suppose, however, that controls are selected in a biased fashion, producing a biased control group
(row 3 in the second table) in which smoking and exposure are associated (verify this fact; try, for
example, computing the OR for smoking in relation to exposure). Reflecting the biased control
group, the stratum-specific IDR's are no longer 3.3. However, in this chapter our focus is the crude
association and whether it accurately represents the true situation (which in this instance we
constructed, rather than having to regard the stratified associations as the true situation). The crude
OR from the above table, using the cases on row 1 and controls from row 3, is (do try computing
this before reading the answer) (370 × 500) / (111 × 500) = 3.3.
Thus, even with this biased control group the crude OR remains unconfounded. Yet, the potential
confounder (smoking, a causal risk factor for the outcome) is indeed associated with the exposure in
the (biased) controls. [Several ways to see this association are:
The odds of exposure among smokers (cols. 1 and 2) are 250/150, quite different from the odds
of exposure among nonsmokers (cols. 3 and 4: 250/350), producing an odds ratio between
smoking and exposure of OR = 2.3).
Proportionately more smokers are exposed [250/(250 + 150) = 0.63] than are nonsmokers
[250/(250 + 350) = 0.42].
The odds of smoking among exposed (cols. 1 and 3) are 250/250, quite different from the odds
of smoking among the unexposed (cols. 2 and 4): 150/350), producing, of course, the same
odds ratio, 2.3).
Proportionately more exposed are smokers [250/(250+250) = 0.5] than are unexposed
[150/(150 + 350 ) = 0.3].
The potential confounder, smoking, is also associated with the outcome in the unexposed (e.g., IDR
= 30 per 1,000py / 3 per 1,000py in the study base, OR = (90 × 350) / (21 × 150) in the casecontrol
study with either control group. Thus, it is possible to have a risk factor that is associated
with exposure in the noncases yet not have confounding.
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www.epidemiolog.net, © Victor J. Schoenbach 2000 11. Multicausality: Confounding - 350
rev. 10/28/2000, 11/2/2000, 5/11/2001