# Risk and Odds

Understand the concepts of risk and Odds Ratio

## Risk

**Absolute Risk**is the risk of an event occurring in the exposed group**Relative Risk**(or risk ratio) is the risk of an event occurring in the exposed group relative to the unexposed group.

**Absolute Risk Reduction**is the decrease in risk provided by an exposure:

Is a clinically useful measure of the value of an intervention because contextualises the efficacy of the intervention with the prevalence of the outcome. ARR is better (more intuitively) expressed as:**Number Needed to Treat (NNT)**is the number of individuals who must receive a treatment to prevent one event:

**Relative Risk Reduction**is the decrease in incidence provided by treatment. It is not as useful a measure of the value of an intervention, but drug companies like it because the numbers are bigger than absolute risk reduction.

## Odds

**Odds**are the probability of an event happening compared to the probability of it not happening, usually expressed as a fraction- The
**Odds Ratio**is the ratio of the odds of the outcome occurring in the exposed compared to the odds of it occurring in the unexposed- An OR < 1 suggests the risk is lower in the exposed group
- An OR > 1 suggests the risk is higher in the exposed group
- An OR = 1 suggests that the groups are equivalent

- In general, the OR overstates risk compared to the RR
- It is approximately equal to the RR when the outcome is rare (< 10%)

This is because**when the event rate is small, the number of non-events in a group is very similar to the overall number of individuals in the group**. - It is used when:
- The denominator is uncertain, i.e.:
- In retrospective designs, such as case-control studies when patients with the disease were identified, and then exposures ascertained

- When it statistically appropriate (ORs are much easier to use in statistical tests), i.e.:
- Multivariate regression
- Systematic Reviews

- The denominator is uncertain, i.e.:

## Risk versus Odds

**Relative Risk** and **Odds Ratios** are both methods of comparing the likelihood of an outcome occurring between two groups. The difference, and particularly the concept of odds ratios, are **commonly confused**. Relative risk tends be much more intuitive than odds ratios. Imagine a trial has been performed, where group A was exposed group:

- In group A, the mortality was 50%
- In group B, the mortality was 25%

The **relative risk** is intuitive:

The **odds ratio** is not:

A RR of 2 is intuitive, but the OR of 3 is not. Now, imagine another trial where:

- In group A, the mortality was 90%
- In group B, the mortality was 10%

The relative risk is 9, but the OR is 81!

So why use odds ratios at all? Odds ratios are:

- Required when research subjects are selected on the basis of outcome rather than the basis of exposure
- Used by many statistical tests because the log odds ratio is normally distributed, which is a mathematically useful property

Relative Risk has a weakness as well - it is dependent on how the question is framed. Using the first trial above, we calculated that RR for death was 2 and the OR was 3. Rather than calculating mortality, an alternative method could be to look at survival:

- In group A, the survival was 50%
- In group B, the survival was 75%

Note that the relative risk is not 0.5 (as you may initially assume), however the odds ratio is just the inverse of the previous value.

## References

- Myles PS, Gin T. Statistical methods for anaesthesia and intensive care. 1st ed. Oxford: Butterworth-Heinemann, 2001.
- Course notes from "Introduction to Biostats", University of Sydney, School of Public Health, circa 2013.
- Simon S. Odds ratio vs. relative risk. "Steve's Attempt to Teach Statistics (StATS)". Children's Mercy Hospital, 2006.
- Bland JM, Altman D. Bland J Martin, Altman Douglas G. The odds ratio. BMJ 2000; 320 :1468.