Risk and relative risk

Absolute risk

• Risk is simply the probability of an outcome for a group — the proportion of that group who have the outcome.
• Absolute risk of an outcome in a group:

$$\text{risk} = \frac{\text{number with the outcome}}{\text{total in the group}}$$

• Like any probability it is between 0 and 1, and can be written as a fraction, decimal or percentage.

Relative risk

• Relative risk compares the risk in one group with the risk in another, as a ratio (a division):

$$\text{relative risk} = \frac{\text{risk in group A}}{\text{risk in group B}}$$

• The group on the bottom is the baseline (comparison) group.
• Method: find each group's absolute risk first, then divide one by the other.

Interpreting relative risk

• Relative risk $= 1$ → the two groups have the same risk.
• Greater than 1 → group A is more at risk. A value of 2 means "twice as likely".
• Less than 1 → group A is less at risk. A value of 0.5 means "half as likely".
• Always say it in words, and name the comparison group. "The relative risk is 2.4" is incomplete; "people who did not wear a helmet were 2.4 times as likely to be injured as those who did" is the answer.
• Turning the ratio into plain English:
  • subtract 1 and read the rest as a percentage — a relative risk of $1.35$ is a 35% increase in risk
  • a relative risk of $0.80$ is a 20% decrease

Why relative risk can mislead

• A large relative risk can come from two tiny risks. Going from a 1-in-a-million risk to 3-in-a-million triples the relative risk, while changing almost nothing in practice.
• So report the absolute risks alongside it. "Three times as likely" sounds alarming; "0.003% instead of 0.001%" puts it in proportion, and quoting both is what a full-mark answer does.
• The baseline group must be stated, because swapping it inverts the ratio: 2.5 one way is 0.4 the other, and both describe the same data.
• Relative risk does not prove cause. The two groups may differ in other ways — age, income, how much they do the activity — and a difference in risk on its own cannot separate those from the factor being studied.