6 Handling risk
It is often of interest to know about the risks associated with particular events or exposures. For example the risk of an adverse event for participants in a clinical trial comparing two treatments for wound infection.
Table 5: 2 x 2 table illustrating the calculation of risk
Exposure:
Total
Event:
Yes
No
Yes
a
b
a+b
No
c
d
c+d
Total
a+c
b+d
n
Risk of event for exposed $$= a/(a+c)$$
Risk of event for unexposed $$= b/(b+d)$$
Table 6: Adverse event rates by treatment group
Exposure:
Total
Event:
Treatment 1
Treatment 2
Adverse event
45
25
70
No adverse event
905
1098
2003
Total
950
1123
2073
Risk of adverse event on treatment 1 $$ = a/(a+c) = 45/950 =0.05 = 5 \mbox{ per }100$$ or 5%.
Risk of death for non-smokers $$ = b/(b+d) = 25/1123 = 0.02 = 2 \mbox{ per }100$$ or 2%.Contents
Absolute risk of an event is the probability of the event occurring (usually within a stated time period for a defined population):
Relative risk of a particular event for a given exposure is the ratio of the risk of the event occurring in the exposed group divided by the risk of the event occurring in the unexposed group:
Relative risk$$ = \frac{a/(a+c)}{b/(b+d)} = \frac{a(b+d)}{b(a+c)}= \frac{45*1123}{25*950} = 2.13$$
Odds of an event occurring is the ratio of the probability of the event occurring to the probability of the event not occurring:
odds of an event given exposure $$ = \frac{a}{c} = \frac{45}{905} = 0.05$$
odds of an event given not exposed $$ = \frac{b}{d}= \frac{25}{1098} = 0.02$$
Odds ratio is the ratio of the odds of an event in the exposed group compared to the unexposed group $$= (a/c)/(b/d) = ad/bc = \frac{45*1098}{25*905} = 2.18.$$ Note that when the event of interest is rare the odds ratio approximates to the relative risk and is often interpreted as a relative risk, as is the case with this example.
Absolute risk difference is the absolute additional risk of an event due to a particular exposure. It is calculated as the risk in the exposed group minus the risk in the unexposed group. If the risk is harmful, so that the risk is increased by the exposure this difference is called the absolute risk excess (ARE) (for example the absolute risk excess of an adverse event for treatment 1 $$0.05-0.02=0.03$$). If the risk is decreased by the exposure (for example using sunscreen to reduce the risk of melanoma) then this difference is called the absolute risk reduction (ARR)
The number needed to treat is a measure of the impact of a particular risk on patients often used in clinical practice. It is the additional number of people that would need to be given a new treatment in order to cure one extra person compared to the old treatment $$= 1/\mbox{ARR}$$. Alternatively, for a harmful exposure the number needed to treat is referred to as the number needed to harm and is calculated in the same way as the number needed to treat, but ignoring the sign. For the adverse event example above, the number needed to harm is $$1/0.03=33.3.$$ Thus approximately 33 people would need to be on treatment 1 compared to treatment 2 for one additional adverse event.
Note that as with any estimated quantity it is possible to construct confidence intervals for these measures.
Points to consider when communicating risk
Individuals who do not deal with numbers and data regularly can often struggle to understand measures of risk, and this case it can be useful to express risks in terms of natural frequencies rather than percentages. Thus if we assume that the success rate following a single cycle of IVF is about 33% then it is more easily understood by stating that of 100 women undergoing treatment 33 will become pregnant.