5.1 Attributable risk among the exposed
Definition 5.1.1. The attributable risk among the exposed is the risk difference \(p_1 - p_0\): the excess risk in the exposed over what they would have had without exposure. The attributable fraction among the exposed is the proportion of the exposed group’s cases that is due to exposure, \[ \text {AF}_E = \frac {p_1 - p_0}{p_1} = \frac {\RR - 1}{\RR } . \]
The second form follows by dividing numerator and denominator by \(p_0\); it can be estimated from a case-control study by putting the odds ratio in place of \(\RR \). With \(\RR = 2.5\) (Example 4.2.3) the attributable risk among the exposed is \(0.05 - 0.02 = 0.03\) over five years and \(\text {AF}_E = 1.5/2.5 = 0.6\): three in five cases among exposed workers would not have occurred without the exposure.
When an exposure lowers risk, as a vaccine does, the corresponding quantity is the preventable fraction among the exposed, \[ \text {PF}_E = \frac {p_0 - p_1}{p_0} = 1 - \RR , \] the proportion of the cases the exposed would have had that the exposure prevented. In vaccine trials it is called the vaccine efficacy.
The risk difference also has a clinical form. In a trial with a treatment that lowers the risk by \(\RD = p_0 - p_1\), the number needed to treat, \(\text {NNT} = 1/\RD \), is the number of patients who must be treated to prevent one event; for a harmful exposure \(1/(p_1 - p_0)\) is the number needed to harm. In Example 4.2.3, one extra case occurs for every \(1/0.03 = 33\) workers exposed for five years.
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