## Estimating Relative Risk Reduction from Odds Ratios

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Estimating Relative Risk Reduction from Odds Ratios

Odds are hard to work with because they are the likelihood of an event occurring compared to not occurring—e.g., odds of two to one mean that likelihood of an event occurring is twice that of not occurring. Contrast this with probability which is simply the likelihood of an event occurring.

An odds ratio (OR) is a point estimate used for case-control studies which attempts to quantify a mathematical relationship between an exposure and a health outcome. Odds must be used in case-control studies because the investigator arbitrarily controls the population; therefore, probability cannot be determined because the disease rates in the study population cannot be known. The odds that a case is exposed to a certain variable are divided by the odds that a control is exposed to that same variable.

Odds are often used in other types of studies as well, such as meta-analysis, because of various properties of odds which make them easy to use mathematically. However, increasingly authors are discouraged from computing odds ratios in secondary studies because of the difficulty translating what this actually means in terms of size of benefits or harms to patients.

Readers frequently attempt to deal with this by converting the odds ratio into relative risk reduction by thinking of the odds ratio as similar to relative risk. Relative risk reduction (RRR) is computed from relative risk (RR) by simply subtracting the relative risk from one and expressing that outcome as a percentage (1-RR).

Some experts advise readers that this is safe to do if the prevalence of the event is low. While it is true that odds and probabilities of outcomes are usually similar if the event rate is low, when possible, we recommend calculating both the odds ratio reduction and the relative risk reduction in order to compare and determine if the difference is clinically meaningful. And determining if something is clinically meaningful is a judgment, and therefore whether a conversion of OR to RRR is distorted depends in part upon that judgment.

a = group 1 outcome occurred
b = group 1 outcome did not occur
c = group 2 outcome occurred
d = group 2 outcome did not occur

OR = (a/b)/(c/d)
Estimated RRR from OR (odds ratio reduction) = 1-OR

RR = (a/ group 1 n)/(c/ group 2 n)
RRR – 1-RR