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and
R111 = R100 × R010 × R001 / (R000) 2 (M6)
Again, there is a baseline risk or rate in the denominator of each relative risk, so when the relative
risks are converted to risks, the R000 in the numerator eliminates one of the resulting three R000's,
leaving two remaining in the denominator. As before, "by itself" means without other specified
factors, but including baseline risk.
Note, however, that the multiplicative model can also be written as an additive model on the
logarithmic scale (because addition of logarithms is equivalent to multiplication of their arguments):
ln(R111) = ln(R100) + ln(R010) + ln(R001) – 2 × ln(R000) (M7)
For this reason, the difference between the additive and multiplicative models can be characterized
as a transformation of scale. So "effect modification" is scale-dependent.
Optional aside – It can also be shown that a multiplicative model can be expressed as an
additive model on the natural scale plus an interaction term. For two factors: (R10 –
R00)(R01 – R00)/R00, or equivalently, (R00)(RR10–1)(RR01–1) – essentially, we add a "fudge
factor".
Additive model:
R11 = R10 + R01 – R00
Additive model with interaction term:
R11 = R10 + R01 – R00 + R00 × (RR10–1) × (RR01–1)
Multiplying out the interaction term:
R11 = R10 + R01 – R00 + R00 × RR10 × RR01 – R00 × RR10 – R00 × RR01 + R00
Dividing both sides by R00:
RR11 = RR10 + RR01 – 1 + RR10 × RR01 – RR01 – RR10 + 1
Simplifying:
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www.epidemiolog.net, © Victor J. Schoenbach 12. Multicausality: Effect modification - 403
rev. 11/5/2000, 11/9/2000, 5/11/2001