Additive and Multiplicative Models

Additive and multiplicative models are two alternative approaches to modeling effect of risk factors on disease. In additive models, risk—or disease incidence—changes by some fixed amount when a risk factor is present. The statement ‘If you take up X, you will increase your risk of Y by 10%’ is an example of an additive model of risk. In contrast, multiplicative models represent the changes in risk as a proportion of the baseline risk. The statement ‘If you take up X, you will double your risk of Y’ is an example of a multiplicative model of disease risk.

The distinction between additive and multiplicative models becomes especially important when considering the effect of multiple risk factors. For instance, consider the situation described in Table 1.

The incidence of disease we expect to see in those with exposure to both A and B differs depending on whether we take a multiplicative or additive approach to disease risk. Under the additive model, we would expect to see 40 cases per 100,000 in those with both exposures, since exposure A increases incidence by 10 per 100,000, exposure B increases incidence by 20 per 100,000, and the baseline rate is 10 per 100,000 (10 + 10 + 20 = 40). Under the multiplicative model, we expect to see 60 cases per 100,000 in the group with both exposures, since exposure A doubles the incidence, exposure B triples the incidence, and the baseline incidence is 10 per 100,000 (10 × 2 × 3 = 60). When the observed incidence or risk of disease in people with multiple exposures differs from what is expected based on the model being used (whether additive or multiplicative), there is said to be interaction or effect modification between the exposures on that scale.

In the analysis of epidemiologic data, the choice of an additive or multiplicative model determines the type of regression analysis performed and the risk measures that will be reported. If an investigator is modeling risk as additive, he or she will generally use linear regression and report risk differences. An investigator who is modeling risk on a multiplicative scale will generally perform a logistic regression and report a relative risk or odds ratio. Epidemiologic investigations that are concerned with disease etiology usually use multiplicative models, while those focused on public health impact are more likely to use an additive risk model.