Understanding the Relationship Between Risks and Odds Ratios

BACKGROUND: Many articles provide only odds ratios (OR) and non relative risks (RR) as the effect estimate. For a variety of important reasons, multiple logistic regression used to adjust for confounders routinely provides only the adjusted OR (ORadj). However, from the clinician's perspective, the ORadj is only easily interpretable when it approximates the adjusted RR (RRadj). In general, the relationship between the OR and RR (adjusted or nonadjusted) is dependent on prevalence of disease in the control group (Po) and has always been presented as nonlinear. Therefore, it is difficult for the clinician to convert the OR to RR when reading the published data. A formula was proposed by Zhang and Yu, but the relationship remains nonlinear. OBJECTIVE: To develop a simple method to convert OR to RR without the use of computer. METHODS: Algebraic manipulation. RESULTS: Through algebraic manipulation, we show that although the OR and RR relationship is nonlinear over the range Po, the ratio OR/RR has a linear relationship with Po with a slope of "OR-1": OR/RR=(OR-1)xPo+1. This makes the prediction of RR on the basis of OR more transparent. It is clear that if Po is small, the RR approximates the OR, but only if the OR is also small. Previous problems with confidence intervals noted with the Zhang and Yu formula remain (ie, they are too narrow under some conditions) and the result should be interpreted with this limitation. Relationships between ORadj and risk difference or number needed to treat remain curvilinear, but some overall approximations can be made. CONCLUSION: A simple relationship exists that allows readers to easily convert ORadj to RRadj. Limitations of the approach remain but seem to be less restrictive than the limitations of not converting ORadj to RRadj.