# Odds Ratio to Risk Ratio

## Calculates a risk ratio from an odds ratio

### Study Data

 Odds ratio (OR)
 Non-exposed prevalence %

RESULTS
Odds Ratio   Risk Ratio

### 0.94

#### Equation 1

$$\\ RR = \frac{OR}{(1-P_{ref})+(P_{ref}*OR)} = 0.94$$
RR = risk ratio; OR = odds ratio
Pref = Prevalence of the outcome in the reference group

##### Risk Ratio (RR)
$$RR = \frac{Probability_{exposed}}{Probability_{non-exposed}}$$
##### Odds Ratio (OR)
$$OR = \frac{Odds_{exposed}}{Odds_{non-exposed}}$$

In medical literature, the relative risk of an outcome is often described as a risk ratio (the probability of an event occurring in an exposed group divided by the probability in a non-exposed group). Certain types of trial designs, however, report risk as an odds ratio. This format is commonly expressed in cohort studies using logistic regression.

When the incidence of an outcome is low (<10%), the odds ratio is very similar to the risk ratio.1 However, the odds ratio becomes exponentially more different from the risk ratio as the incidence increases, which exaggerates either a risk or treatment effect.

#### Exaggerated Odds Ratio Effect

Using the following simulated data set, it would appear that having ICU delirium results in a 2-fold increase in the risk of mortality:

 Mortality in patients with ICU Delirium Mortality in patients without ICU Delirium Adjusted OR 70% 45% 2

After converting the odds ratio to a risk ratio, the actual risk is 1.4 (mortality is 1.4 times more likely in patients with ICU delirium compared to those without ICU delirium). Because the incidence rate in the non-delirium group is high, the odds ratio exaggerates the true risk demonstrated in the study.

1. Zhang J, Yu KF. What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes. JAMA. 1998 Nov 18;280(19):1690-1. PMID 9832001
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