Department of Health & Social Care
Odds ratio and relative risk
When we wish to compare proportions, for example to compare smoking prevalence in two different areas, there are two common ways to do this: the odds ratio and the relative risk.
As explained on the proportions page, proportions and odds are two ways of expressing the same thing. We can calculate relative risk by dividing one proportion by the other, or we can calculate the odds ratio by dividing one odds by the other.
The two-by-two contingency table below summarises two proportions:
- \(a\), \(b\), \(c\) and \(d\) are the numbers of individuals in the contingency table as follows:
| (Local) group of interest | (National) reference group | |
|---|---|---|
| Cases | \(a\) | \(b\) |
| Non-cases | \(c\) | \(d\) |
We have two populations, which we will refer to as the ‘local’ group and the ‘reference’ group. These groups could be of equal status if the comparison is simply between two populations, but usually there is a reference group with which the other groups are compared.
Calculation of odds ratios
The odds ratio \(OR\) is given by:
\[
OR = \frac{(\frac{a}{c})}{(\frac{b}{d})} = \frac{ad}{bc}
\]
Calculation of confidence intervals for odds ratios
\[ OR _{lower} = e^{ln(OR) - z\sqrt{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}}} \]
\[ OR _{lower} = e^{ln(OR) + z\sqrt{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}}} \]
where:
- \(a\), \(b\), \(c\) and \(d\) are the numbers of individuals in the contingency table as follows:
| (Local) group of interest | (National) reference group | |
|---|---|---|
| Cases | \(a\) | \(b\) |
| Non-cases | \(c\) | \(d\) |
- \(z\) is the \(100\left(1 - \frac{\alpha}{2}\right)\) th percentile value from the standard normal distribution
For example, for a 95% confidence interval, \(\alpha = 0.05\) and \(z \cong 1.96\) (the 97.5th percentile value from the standard normal distribution).
Calculation of relative risk
The relative risk \(RR\) is given by:
\[ RR = \frac{(\frac{a}{a+c})}{(\frac{b}{b+d})} \]
Approximate confidence intervals for the relative risk can be calculated using a normal approximation method as the log of the relative risk is approximately normally distributed. However the relative risk is not used as an indicator methodology currently by OHID.
Relationship between odds ratios and relative risks
Odds ratios and relative risks are closely related, and in many circumstances are very similar. Odds ratios have sometimes been referred to as ‘approximate relative risk’. This term infers that the odds ratio is in some way inferior, but this is not the case. In research studies, the statistic resulting from the study depends on the study design: cohort studies produce relative risks, whereas case-control studies produce odds ratios - when researchers have wished to publish relative risks from case-control studies they have used the term approximate relative risk because they only have odds ratios.
In fact, odds ratios have significant advantages over relative risks. When the proportions (prevalences) are small, the two are near identical, but in any other cases the odds ratio is preferable. Relative risks are perceived to be easier to understand, but can be misleading: the easiest way to demonstrate the problem with relative risk is by example:
If the prevalence of smoking is 20% in group 1, and 10% in group 2, the relative risk of a person being a smoker in group 1, compared with group 2, is \(\frac{0.2}{0.1} = 2\) (double the risk). This appears very straightforward. However, if we present the results the other way round and compare the risk of being a non-smoker in group 1, compared with group 2, we might expect that to be the inverse, ie 0.5, but it is in fact \(\frac{0.8}{0.9} = 0.\dot8\).
However, the same exercise with odds ratios gives more logical results: the odds of being a smoker in group 1 are \(\frac{0.2}{0.8} = 0.25\) and the odds in group 2 \(\frac{0.1}{0.9} = 0.\dot1\) so the odds ratio for being in a smoker in group 1, compared with group 2, is \(\frac{0.25}{0.\dot1} = 2.25\), which is quite close to the relative risk of 2. However, the odds ratio for being a non-smoker in group 1, compared with group 2, is the inverse of the odds ratio for being a smoker, so the statistic has the logical consistency lacking in the relative risk: the odds of being a non-smoker in group 1 are \(\frac{0.8}{0.2} = 4\) and the odds in group 2 \(\frac{0.9}{0.1} = 9\) so the odds ratio for being in a smoker in group 1, compared with group 2, is \(\frac{4}{9} = 0.\dot4\), which is \(\frac{1}{2.25}\).
Page last updated: December 2025