5.2 The population attributable fraction
Definition 5.2.1. Let \(p\) be the risk in the whole population and \(p_0\) the risk among the unexposed. The population attributable risk is \(p - p_0\), and the population attributable fraction is \[ \text {PAF} = \frac {p - p_0}{p} , \] the proportion of all cases in the population that would not have occurred had nobody been exposed.
The definition uses \(p_0\) as the risk the whole population would have without exposure, which is the assumption that the unexposed are exchangeable with the exposed (Section 1.5). Two formulas compute the PAF from quantities that the usual designs estimate.
Theorem 5.2.2 (Levin’s formula). If a proportion \(P_e\) of the population is exposed, then \[ \text {PAF} = \frac {P_e(\RR - 1)}{1 + P_e(\RR - 1)} . \]
Proof. The population risk is the average of the group risks, \(p = P_e p_1 + (1 - P_e)p_0 = p_0\{1 + P_e(\RR - 1)\}\). Hence \(p - p_0 = p_0 P_e(\RR - 1)\), and dividing by \(p\) gives the result. □
Theorem 5.2.3 (Miettinen’s formula). If a proportion \(P_c\) of the cases is exposed, then \[ \text {PAF} = P_c\,\frac {\RR - 1}{\RR } = P_c\,\text {AF}_E . \]
Proof. All cases due to exposure are among the exposed cases, and among those a fraction \(\text {AF}_E\) is due to exposure. Formally, the number of excess cases per head of population is \(P_e(p_1 - p_0)\), and the total number of cases per head is \(p\), so \[ \text {PAF} = \frac {P_e(p_1 - p_0)}{p} = \frac {P_e p_1}{p} \cdot \frac {p_1 - p_0}{p_1} . \] The first factor is the proportion of cases that are exposed, \(P_c\), and the second is \(\text {AF}_E\). □
The PAF depends on both the strength of the association and how common the exposure is. Levin’s formula needs the prevalence of exposure in the population and suits cohort or cross-sectional data. Miettinen’s formula needs the prevalence of exposure among cases, which a case-control study measures directly, with the odds ratio in place of \(\RR \). Miettinen’s form has a further advantage that matters in later sections: when the risk ratio has to be adjusted for confounding, substituting the adjusted ratio into Miettinen’s formula gives a valid PAF, while substituting it into Levin’s does not.
Example 5.2.4. In Example 4.2.3 a third of the cohort is exposed and the overall risk is \(108/3600 = 0.03\). Directly, \(\text {PAF} = (0.03 - 0.02)/0.03 = 1/3\). By Levin, \(P_e = 1/3\) and \[ \text {PAF} = \frac {(1/3)(1.5)}{1 + (1/3)(1.5)} = \frac {0.5}{1.5} = \frac 13 . \] By Miettinen, \(P_c = 60/108 = 0.556\) and \(\text {PAF} = 0.556 \times 0.6 = 1/3\). If the cohort is representative, a third of the cases in the population would be avoided by removing the exposure.
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