Additive vs Multiplicative Effect Modification Section
Hammond et al (1979) sudied smoking and asbestos exposure as risks for lung cancer. The primary objective was to examine the association between smoking and lung cancer, but a secondary objective was to determine if a history of asbestos exposure modified that risk.
- In both strata of asbestos exposure, smokers had higher incident rate of lung cancer than did non-smokers - Primary result
- Among smokers, the incident rate was higher in those exposed to asbestos than those not exposed to asbestos (602 vs. 123).
|Hx. of Asbestos +||Hx of Asbestos -|
Another way to examine the data:
- The background cancer rate = 11/100,000 person-yrs =1, the reference group (persons not exposed to either factor).
- Increased rate due to smoking = 123 - 11 = 112 (smoker’s rate - background rate).
- Increased rate due to asbestos = 58 - 11 = 47 (asbestos’ rate - background rate).
- Total EXPECTED increased rate due to both factors = 112 + 47 = 159.
- Total OBSERVED increased rate = 602 - 11 = 591 (persons exposed to both - background).
Conclusion: The observed increase in the cancer rate due to both factors greatly exceeded the expected increase in cancer rate if the risks were just additive, that is, we consider 591 vs. 159. Interection is occurring, but not according to an additive model.
Another way to look at the data...
First quantify the associations using relative risk (or use an odds ratio):
- The background risk (rate of 11 / 100,000 person-yrs) = 1 (reference group - persons not exposed to neither factors).
- Relative risk (IDR) due to smoking = 123/11 = 11 (smoker’s rate / background rate).
- Relative risk (IDR) due to asbestos = 58/11 = 5 (asbestos’ rate / background rate).
- Total EXPECTED relative risk due to both factors = 11 × 5 = 55.
- Total OBSERVED relative risk due to both factors = 602 / 11 = 55.
Conclusion: The observed relative risk of cancer due to both factors is identical to the expected relative risk (from a multiplicative model of both factors), that is RR-observed = 55 vs. RR-expected = 55. Asbestos is effect modifier for the relationship between smoking and lung cancer incidence on a multiplicative scale!
Why are we doing this?
Previously, we talked about how effect modification is different from statistical interaction. We looked for effect modification because it may represent different biology. Seeing multiplicative interaction strengthens the expectation that there is biologic plausibilty for a relationship between a primary risk factor and disease. Demonstration of multiplicative effect modification is important in the search for etiologic pathways. We can hypothesize from the data that we looked at earlier that asbestos somehow causes an initial insult to the lung which makes it more susceptible to smoking and therefore the effect is multiplicative. It is not just additive. When there are multiplicative effects, biologically important interactions is occurring.
We also assess effect modification to determine the potential for removal of a risk factor. If the additive model holds, then we note that the removal of one agent can only be expected to eliminate the risk that arises from that particular agent but none of the risks that involves other agents. Additive interaction may be sufficient to suggest public health importance. If there is effect modification on multiplicative scale, removal of either of the agents involved will reduce some of the risk from the other factor as well.
We are faced with a situation where the decision about effect modification depends upon what model we employ to arrive at an expected joint effect to compare with the observed joint effect.
The additive model assumes that the excess-risk for two factors operating simultaneously is equal to the excess risk for the first factor plus the excess risk for the second factor. The additive model, expressed in terms of excess risk, is therefore:
The multiplicative model, in contrast, assumes that the relative risk (or risk ratio) for the two factors operating together is equal to the product of the relative risk for each of the two factors operating alone:
The existence of statistical interaction can be model dependent. Thus, consider the biological plausibility of an effect modifier. Choose an effect modification model that is important to the research objectives. For addressing public health measures to reduce disease frequency, an additive model is more relevant. For investigating disease etiology, the multiplicative model is often more relevant.