Adjusted Risk Ratios versus Adjusted Odds Ratios

Risk ratios (RR) and odds ratios (OR) are both used to analyze factors associated with increased risk of disease. They are different in that a risk ratio is the ratio of two risks and an odds ratio is the ratio of two odds. Risks and odds are calculated differently. For example, with one toss of a fair die, the risk of rolling a 1 is 1/6 = 0.167. The odds of rolling a 1 are 1/5 = 0.200. In textbook terms, risk is successes divided by total trials and odds are successes divided by failures. In this example rolling a 1 is a success and rolling 2 through 5 is a failure.

In R, glm (generalized linear model) is used to compute adjusted risk ratios (ARR’s) and adjusted odds ratios (AOR’s). Adjusting eliminates the influence of correlations between factors on the ratios. In statistical terms, adjusting controls for confounding. In addition to confounding, the RR’s and OR’s can be influenced by interaction which is also called effect modification. This occurs when the ratio for one group is different from the ratio for another group. For example, non-smokers who work with asbestos might have 4.0 times the risk of getting lung cancer compared to non-smokers who do not work with asbestos. This is for non-smokers. For smokers, this risk ratio might be much larger at 8.0, meaning smokers who work with asbestos have 8.0 times the risk of getting lung cancer compared to smokers who do not work with asbestos. In this case there is interaction between smoking and asbestos exposure because the RR for non-smokers is 4.0 and the RR for smokers is 8.0. Alternatively, it could be said that smoking modifies the effect of the asbestos exposure on lung cancer.

For a given data set, the AOR’s and ARR’s will almost always agree in terms of statistical significance. The AOR’s and ARR’s will be different but the p values will be close. This is not true for interaction terms. Often, using the same data, an analysis of interaction using OR’s will indicate significant interaction while the RR’s will indicate no interaction. This phenomenon is explained very well in the article “Distributional interaction: Interpretational problems when using incidence odds ratios to assess interaction” by Ulka B Campbell, Nicolle M Gatto and Sharon Schwartz. (Epidemiologic Perspectives & Innovations 2005, 2:1 doi:10.1186/1742-5573-2-1) https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1079911/.

The authors described the problem for RR’s and OR’s and included mathematical explanations. The purpose of this article is to show how this problem occurs when using adjusted RR’s and adjusted OR’s with the R glm function to assess interaction. > From <https://www.r-bloggers.com/simulations-comparing-interaction-for-adjusted-risk-ratios-versus-adjusted-odds-ratios/>