Risk and relative risk

Risk is a conditional probability

• Risk is the probability of an unwanted outcome, and in practice it is nearly always conditional — the risk for a particular group.

$$\text{risk of the outcome for group } G = P(\text{outcome} \mid G)$$

• Because it is conditional, everything from the previous two pages applies: the group is the denominator, and reversing it changes the meaning entirely.

Absolute risk and relative risk

• Absolute risk is the probability itself: 3 in 1 000, or 0.003.
• Relative risk compares two groups by dividing:

$$\text{relative risk} = \frac{P(\text{outcome} \mid \text{exposed})}{P(\text{outcome} \mid \text{not exposed})}$$

• Reading a relative risk:
  • $\text{RR} = 1$ → no difference between the groups
  • $\text{RR} = 2$ → the exposed group's risk is twice that of the unexposed group
  • $\text{RR} = 0.5$ → the exposed group's risk is half — the exposure is protective.

Absolute risk reduction and increase

• The absolute difference is the subtraction rather than the division:

$$\text{absolute risk difference} = P(\text{outcome} \mid \text{exposed}) - P(\text{outcome} \mid \text{not exposed})$$

• The two measures answer different questions, and both are needed:
  • Relative risk tells you how much the exposure matters as a multiplier — useful for judging whether an effect is real and for comparing across contexts.
  • Absolute risk difference tells you how many people are affected — the only one that supports a decision.

Why a relative risk alone is misleading

• The same relative risk of 2 describes both of these:

• In Situation A "the risk doubles" is technically true and practically meaningless; in Situation B it is a public health emergency. A relative risk without its baseline cannot be interpreted.
• This is why headlines quote relative risks: it is the larger, more alarming number, and it is true.

Number needed to treat

• A useful way to make an absolute difference concrete:

$$\text{number needed to treat} = \frac{1}{\text{absolute risk difference}}$$

• If a programme reduces risk from 4% to 3%, the absolute reduction is 0.01, so $1/0.01 = 100$ — 100 people must take part to prevent one case.
• The same idea in reverse gives the number needed to harm for a side effect.

Reading risk statements critically

• Ask these every time:
  • Risk of what, exactly? A "50% higher risk of hospital admission" is not the same as of death.
  • Compared with whom? The comparison group is often unstated.
  • Over what period? A lifetime risk and an annual risk differ enormously.
  • Absolute or relative? If only one is given, it is nearly always relative.
  • Is it causal? A relative risk from an observational study is an association — everything about confounding applies.

Risk in context

• Risks are compared badly by human intuition, which over-weights dramatic, unfamiliar and involuntary risks and under-weights familiar, voluntary ones.
• A statistically literate answer notes the absolute size of a risk alongside its relative change, and compares it with a familiar baseline risk where one is available.