3.2 Risk
Definition 3.2.1. The risk or cumulative incidence of a disease over an interval \((0, t]\) is the probability that a person free of the disease at time \(0\) develops it during the interval.
In a closed cohort of \(N\) people at risk at time \(0\), followed completely for the whole period, the number \(X\) who develop the disease is \(\text {Bin}(N, R)\), where \(R\) is the risk. The estimate \(X/N\) and the intervals of the last subsection apply unchanged. An attack rate is the same quantity for an outbreak, where the period is the short duration of the outbreak (the proportion of those who ate at a wedding who became ill), and the case-fatality is the risk of death over a stated period among those with the disease.
Despite the word “rate” in these names, all are proportions. Two complications prevent them from being as simple in practice as in the definition:
- 1.
- Losses. People leave the study before the end of the period, by migration or refusal, without having had the event. Their outcome over the full period is unknown, so neither the numerator nor the denominator is known exactly. Section 11 handles this with life tables; meanwhile, the rate defined next uses the time each person was actually observed.
- 2.
- Competing risks. People die of other causes before they could develop the disease. The risk of the disease then depends on the death rate from other causes as well as on the disease itself (Problem 3.6.3).
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