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Some of the relationships just presented can be readily demonstrated using simple algebra. Let ai, bi,
ci, di, n1i, and n0i in each stratum take on the values implied by the table below, and let their
respective totals across all strata by a, b, c, d, n1, and n0 (i.e., a = all exposed cases, b = all
unexposed cases, c = all exposed noncases, d = all unexposed noncases, n1 = all exposed persons,
n0 = all unexposed persons).
Exposure
Disease Yes No Total
Yes ai bi m1
(ai + bi)
No ci di m2 (ci + di)
Total n1i n0i ni
(ai + ci) (bi + di)
The incidence in exposed persons is ai/n1i within each stratum and a/n1 when the strata are ignored
(i.e., the total, or crude table). The (weighted) average incidence in the exposed across the strata is:
Σ (
n1i ai ai a
––– × ––– = ––– = –––
n
n
n n1i
) Σ (
where the summation goes over all strata. a/n is simply the crude incidence in the exposed.
Similarly, the weighted average of the stratum-specific risk ratios can be expressed as the sum across
all strata of:
wi ai /n1i wi ai n0i
––– × ––– = ––– × –––––
W bi /n0i W bi n1i
where wi are the weights for each stratum and W is the sum of the wi. If we let wi = bin1i /n0i, then
we have the sum across strata of:
Meanwhile, W is the sum across all strata of:
bi n1i /n0i ai n0i ai wi
––––––––– × ––––– = –––––
W bi n1i W
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www.epidemiolog.net, © Victor J. Schoenbach 2000 11. Multicausality: Confounding - 374
rev. 10/28/2000, 11/2/2000, 5/11/2001
)